Equations Of Motion Research Articles

Bidirectional pendulum-type tuned mass damper inerter for synchronously controlling the alongwind and crosswind responses of super-tall buildings

In the context of slender super-tall buildings, which have a high height-to-width ratio and low damping, significant bidirectional vibrations can occur under strong winds. Traditional TMDI device is mainly focused on controlling unidirectional wind-induced vibration responses. To address this issue, the present study introduces a bidirectional pendulum-type tuned mass damper inerter (BPTMDI) designed to synchronously mitigate alongwind and crosswind responses. The study derives nonlinear equations of motion for a multiple degrees of freedom (MDOF) structure integrated with the BPTMDI, using the Euler-Lagrange method. These equations capture the three-dimensional motion characteristics of the BPTMDI. Additionally, linear equations of motion are derived for a simplified planar model. The time-domain responses of the MDOF structure coupled with the BPTMDI are calculated by dividing the system into linear and nonlinear subsystems and applying the state space method and Runge-Kutta method, respectively. The effectiveness of the BPTMDI is evaluated through time history response analysis of a 400 m tall case-study building subjected to bidirectional wind loadings. Results show that the BPTMDI can significantly reduce both alongwind and crosswind responses, with reductions of at least 29.75 % in displacement and 48.64 % in acceleration, regardless of wind direction. Furthermore, the BPTMDI demonstrates improved control efficacy compared to BPTMD systems. Lastly, the simplified planar model of the BPTMDI shows basic similarity to the nonlinear planar-spherical model.

Molecular Dynamic Simulations for Biopolymers with Biomedical Applications.

Computational modeling (CM) is a versatile scientific methodology used to examine the properties and behavior of complex systems, such as polymeric materials for biomedical bioengineering. CM has emerged as a primary tool for predicting, setting up, and interpreting experimental results. Integrating in silico and in vitro experiments accelerates scientific advancements, yielding quicker results at a reduced cost. While CM is a mature discipline, its use in biomedical engineering for biopolymer materials has only recently gained prominence. In biopolymer biomedical engineering, CM focuses on three key research areas: (A) Computer-aided design (CAD/CAM) utilizes specialized software to design and model biopolymers for various biomedical applications. This technology allows researchers to create precise three-dimensional models of biopolymers, taking into account their chemical, structural, and functional properties. These models can be used to enhance the structure of biopolymers and improve their effectiveness in specific medical applications. (B) Finite element analysis, a computational technique used to analyze and solve problems in engineering and physics. This approach divides the physical domain into small finite elements with simple geometric shapes. This computational technique enables the study and understanding of the mechanical and structural behavior of biopolymers in biomedical environments. (C) Molecular dynamics (MD) simulations involve using advanced computational techniques to study the behavior of biopolymers at the molecular and atomic levels. These simulations are fundamental for better understanding biological processes at the molecular level. Studying the wide-ranging uses of MD simulations in biopolymers involves examining the structural, functional, and evolutionary aspects of biomolecular systems over time. MD simulations solve Newton's equations of motion for all-atom systems, producing spatial trajectories for each atom. This provides valuable insights into properties such as water absorption on biopolymer surfaces and interactions with solid surfaces, which are crucial for assessing biomaterials. This review provides a comprehensive overview of the various applications of MD simulations in biopolymers. Additionally, it highlights the flexibility, robustness, and synergistic relationship between in silico and experimental techniques.

Free vibration analysis of nanocomposite plate in contact with fluid using a novel quasi-3D shear deformation theory and artificial neural network

This article presents an analytical approach leveraging a novel quasi-three-dimensional shear deformation theory to analyze the free vibration response of functionally graded graphene platelet-reinforced composite (FG-GPLRC) plates in a liquid medium. The study meticulously determines material properties using the Halpin-Tsai micromechanical model, offering a thorough characterization of these advanced nanocomposite structures. Differential motion equations, incorporating transverse shear and normal stresses, are derived via Hamilton’s principle. The natural frequencies of the nanocomposite plates are computed using Galerkin’s method and validated against published results, affirming their accuracy and reliability. The key innovation of this research is introducing a new trigonometric shape function, which exhibits superior precision compared to previously proposed shape functions and existing theories. Furthermore, an artificial neural network (ANN) model is developed to train the computed results, enabling precise prediction of the natural frequencies without further computational runs. The Bayesian Regularization algorithm, implemented in Matlab, is utilized for this purpose. Across different GPLs patterns and varying input parameters, the error percentage between the ANN model predictions and the results from the Galerkin’s method is consistently low, often below 0.1%. The ANN model demonstrates robust generalization by effectively predicting outcomes with input data beyond its training range. Additionally, it provides immediate computational results, eliminating the delays of traditional methods. The article also examines the effects of various factors, including material characteristics, geometric parameters, and the fluid medium, on the free vibration behavior of FG-GPLRC plates.

Nonstationary response statistics of structures with hysteretic damping to evolutionary stochastic excitation

The damping of a structure has often been modeled as linear hysteretic damping (LHD), so its corresponding equation of motion (EOM) is an integro-differential equation that involves the Hilbert transform of response displacement. As a result, the system is non-causal in nature, and it is challenging to compute its nonstationary response statistics under evolutionary stochastic excitation. This article develops an efficient solution method to obtain closed-form solutions for various nonstationary response statistics, including the evolutionary power spectrum (EPS), correlation function and mean square values. The novel solution method utilizes the concept of causalization time to introduce a “causalized” impulse response function (IRF), by which causal response statistics are computed based on a pole-residue approach. This approach requires obtaining a pole-residue form of the transfer function (TF) from the frequency response function (FRF) of the system, which is readily obtained from the EOM. Subsequently, the desired response statistics are obtained by shifting the causal response statistics back to the original time. To obtain the pole-residue form of the TF, two steps are necessary: (1) taking the inverse Fourier transform of the FRF of the oscillator to obtain a discrete IRF and (2) using the Prony-SS method to decompose this discrete IRF to obtain the pole residues associated with the TF. The correctness of the proposed method is numerically verified by Monte Carlo simulations through examples of hysteretic damping and mixed viscous-hysteretic damping oscillators that are subjected to white noise, modulated white noise and modulated Kanai–Tajimi model random excitations.

Методика уточненного расчета давления в тормозных цилиндрах поезда с учетом влияния утечек сжатого воздуха, распределенных по длине тормозной магистрали

Objective: to develop a methodology for estimating the filling value of the brake cylinders of a freight train, taking into account the amount of air leakage distributed along the length of the brake line. As initial data for determining the pressure of the brake cylinders, use the pressure drop in the brake line along the length of the train, the amount of charging pressure in the head of the train and the discharge stage of the equalizing tank. Methods: experimental study of gas dynamic processes in key volumes of the braking system in the head and tail parts of the train, such as: brake line and brake cylinder. Approximation of experimental data obtained from the braking system of a real train with the derivation of generalized equations. Results: as a result of the research, refined mathematical models were obtained that allow determining the pressure in the brake cylinder of any car, as well as the average pressure in the brake cylinders along the length of the train, taking into account the amount of leakage of the charging pressure in the head of the train and the length of the train in the cars. Practical significance: The application of a new methodology for calculating the filling value of brake cylinders will allow taking into account the influence of the length of the brake line on the braking force of each train car. This will help to correct the equation of train movement and improve the accuracy of calculating the braking distance of freight trains, taking into account the technical condition of the braking network.

Resonance theory of vibrational strong coupling enhanced polariton chemistry and the role of photonic mode lifetime

Recent experiments demonstrate polaritons under the vibrational strong coupling (VSC) regime can modify chemical reactivity. Here, we present a complete theory of VSC-modified rate constants when coupling a single molecule to an optical cavity, where the role of photonic mode lifetime is understood. The analytic expression exhibits a sharp resonance behavior, where the maximum rate constant is reached when the cavity frequency matches the vibration frequency. The theory explains why VSC rate constant modification closely resembles the optical spectra of the vibration outside the cavity. Further, we discussed the temperature dependence of the VSC-modified rate constants. The analytic theory agrees well with the numerically exact hierarchical equations of motion (HEOM) simulations for all explored regimes. Finally, we discussed the resonance condition at the normal incidence when considering in-plane momentum inside a Fabry-Pérot cavity.

Influence of tangential hydrodynamic force and inclination angle calculation on the dynamic responses analysis of mooring line based on lumped mass method

Research on mooring line response offers a scientific foundation and technical backing for ocean engineering management. The lumped mass method is commonly used to calculate the dynamic responses of mooring line, pioneered by Walton and Polachek, although it ignores the tangential hydrodynamic force. Nakajima addressed this limitation, but the inclination angle calculation for each element is approximated by the average of the adjacent two nodes. To accurately assess the influences of these factors, this work introduces the tangential hydrodynamic force component into the equation of motion for mooring line dynamic analysis based on the lumped mass method. Meanwhile, we improve the discretization process of lumped mass nodes, analyzing half of the connected spring at both ends of each lumped mass node respectively. The detailed formulation and motion equation for static and dynamic mooring line analysis are presented. The time-domain solution procedure based on the finite difference method is adopted. Finally, different external excitations are applied to calculate the dynamic responses of mooring line. The results show that the top tension increases by about 2%–5% considering the contribution of tangential hydrodynamic force, and the inclination angle calculation significantly influences the dynamic configuration of the mooring line when it is intense change.

Nonlinear adaptive pose motion control of a servicer spacecraft in approximation with an accelerated tumbling target

Removing a limited number of large debris can significantly reduce space debris risks. These bodies are generally exposed to extreme environmental disturbance torques or consecutive accidents due to their large wet area, which causes them to experience accelerated high-rate tumbling motion. The existing literature has adequately explored the approximation operations with non-cooperative targets exhibiting 3-axis tumbling motion. However, the research gap lies in the lack of attention given to addressing this approximation for targets undergoing accelerated motion. Agile, accurate, and large-angle maneuvers are three common necessities for safely capturing such targets. Changes in the moment of inertia brought on by fuel slushing cannot be disregarded during such a maneuver. To deal with nonlinearities, adverse coupling effects, actuator saturation constraints, time-varying moment of inertia, and external disturbances that worsen during accelerated agile large-angle maneuvers, a novel adaptive control approach is developed in this paper. The controller's main advantage is its adjustable desired acceleration, which maintains its performance even when dealing with accelerated motion. The control law is directly synthesized from the nonlinear relative equations of motion, without any linearization or simplification of the system dynamics, making it robust to a variety of orbital elements and target behaviors. Adaptation laws are extracted from the Lyapunov stability theorem in a way that guarantees asymptotic stability. Moreover, control actuator roles (delay, saturation, and allocation) are accounted for in modeling and simulation. Finally, a comprehensive numerical simulation based on three different realistic and strict scenarios is carried out to demonstrate the effectiveness and performance of the proposed control approach. The controller's robustness against time-varying dynamic parameters (sharp and sudden change, smooth and slow change, and periodic change) is extensively demonstrated through simulation.

Vibration and stability of axially moving beam on multiple intermediate elastic supports with transfer matrix method

This study delves into the vibration characteristics and stability of an axially moving beam with multiple intermediate elastic supports. Initially, the motion equation of the beam is developed, setting the stage for an in-depth investigation of its free vibration characteristics through the transfer matrix method. This analysis covers the beam’s natural frequencies and critical velocity when moving at a constant velocity. Additionally, the dynamic behavior of the beam is investigated when it is subjected to constant axial movement and a concurrent traveling load. The multi-scale method is used to analyze the instability regions of the axially moving beam on multiple intermediate elastic supports as disturbance of the axial movement approaches subharmonic and sum-type combined resonances. The investigation further considers how alterations in the stiffness and configuration of the elastic supports influence the dynamic behavior and stability of the beam. Unique dynamic characteristics arise when the axially moving beam is supported by multiple elastic elements, contrasting with those observed in beams without such supports. An increase in support stiffness enhances the critical velocity at which the beam axially moves, potentially causing modal transformations of the structure prior to reaching this critical velocity, and diminishing the instability regions associated with sum-type combined resonances under disturbance velocities.

Математическое моделирование мягкого экзоскелета из магнитно-реологических модулей с интеллектуальным управлением

The article considers a fundamentally new generation of soft exoskeletons based on magnetic rheological modules. A model of an exoskeleton link with adjustable rigidity is proposed, for which a system of differential equations of motion is compiled. A model of an exoskeleton for human lower extremities, composed of modules with magnetic rheological fluid, has been developed. The problems of kinematics and dynamics that arise when controlling the movement of an exoskeleton are presented. A block diagram of an algorithm for controlling the movement of an exoskeleton has been developed, solving control problems taking into account neural network methods. A qualitative study by analytical methods of the proposed mathematical model of an exoskeleton with links with adjustable stiffness is carried out. The use of regulation of the rigidity of the links of the exoskeleton modules in the necessary way will make it possible to create devices that are comfortable for the user, both from the physiological and psychological points of view. Soft exoskeletons will be in demand in many industries and medicine. The use of electromagnetic actuators, which, as a rule, have smaller dimensions and energy consumption, an almost instantaneous response speed in comparison with pneumatic drives, hydraulic drives and electric drives, is promising and technically, energetically, environmentally, economically more acceptable. Under the action of electric or magnetic fields, the stiffness of electroelastic and magneto-rheological materials can change by more than an order of magnitude. This makes it possible to create exoskeleton links with adjustable stiffness. Regulation of the rigidity of the elements of the exoskeleton modules in real time is a complex control task. Therefore, exoskeleton motion control requires the development of adaptive type algorithms based on a neuro-fuzzy approach, using information received from feedback. Neural network algorithms for self-learning in the process of functioning are a logically expedient part of a soft bionic exoskeleton. Also, as a necessary component of the exoskeleton, along with the actuator part, a sensor system is needed with sensitive sensors for the user’s neuromuscular impulses and various sensors for position, speed, acceleration, force, moment of force of the exoskeleton. A comprehensive study of scientific and technical problems of the functioning of the model of the magnetic-rheological medium, exoskeleton modules and their compilation into a single multi-link structure, its movement using the technology of modern mathematical modeling was carried out. The construction of neuro-fuzzy control systems for soft exoskeletons is described as part of the development of new mathematical methods for modeling complex technical objects using modern mechatronics and cyborgics.

Comprehensive design strategy of the NSD–TMD controlled system with closed-form frequency-amplitude solutions

A negative stiffness device (NSD) utilizing pre-compressed helical springs has been integrated with diverse vibration control techniques. Recently, the incorporation of NSD has been proposed to enhance the efficacy of a tuned mass damper (TMD) controlled system. This study introduces two design strategies for the NSD–TMD system, which functions either as a TMD with an adjustable operating bandwidth or as a pure non-linear energy sink. First, the non-linear force of NSD was expressed in a Taylor series approximation form incorporating a negative linear stiffness and a positive cubic stiffness term. Second, the equations of motion were formulated, and closed-form solutions were derived using the complexification–averaging method. The accuracy of the Taylor series approximation was substantiated within a specified range through a comparison of force–displacement curves. Third, the accuracy of the dynamic closed-form solutions was affirmed by comparing the amplitude–frequency curves with the numerical simulations. For a case study, an actual timber pedestrian bridge with decaying natural frequency was selected, and the two design strategies were examined individually. The vibration control performance of the TMD and NSD–TMD was also comparatively analyzed. The assessment focused on the peak force transmissibility value and the effective operating frequency bandwidth. Results highlighted a notable enhancement in operational performance, particularly for primary structures exhibiting diminishing modal frequencies. In a TMD-controlled system, the efficacy of vibration control deteriorates and may even amplify the dynamic response when a modal frequency shift occurs in the primary structure. Conversely, both design strategies of NSD–TMD can achieve lower transmissibility and broaden the operating bandwidth.

Motion Equations and Attitude Control in the Vertical Flight of a VTOL Bi-Rotor UAV: Part 2

This paper gathers the dynamical modeling of an unmanned aircraft and the design and simulation of the control system, allowing it to perform a Vertical Take-Off (VTOL) maneuver, a fixed-wing (FW) flight and a transition between the two configurations using two tilting rotors (Bi-Tilt). These Unmanned Aerial Vehicles (UAVs) operating in this configuration are categorized as Hybrid UAVs, for their capability of having a dual flight envelope: flying like a multi-rotor and navigating like a traditional fixed-wing aircraft, allowing the drone to perform complex missions where these two flight configurations are essential. This work exhibits the Bi-Rotor non-linear dynamics, valid for both flight configurations, the design of the control algorithm for stability and navigation, and a simulation of a complete flight mission.

Kalman Filter for a Particular Class of Dynamic Object Images

We discuss the problem of estimating the state of a dynamic object by using observed images generated by an optical system. The work aims to implement a novel approach that would ensure improved accuracy of dynamic object tracking using a sequence of images. We utilize a vector model that describes the object image as a limited number of vertexes (reference points). Upon imaging, the object of interest is assumed to be retained at the center of each frame, so that the motion parameters can be considered as projections onto the axes of a coordinate system matched with the camera's optical axis. The novelty of the approach is that the observed parameters (the distance along the optical axis and angular attitude) of the object are calculated using the coordinates of specified points in the object images. For estimating the object condition, a Kalman-Bucy filter is constructed on the assumption that the dynamic object motion is described by a set of equations for the translational motion of the center of mass along the optical axis and variations in the angular attitude relative to the image plane. The efficiency of the proposed method is illustrated by an example of estimating the object's angular attitude.

Discovering dynamic laws from observations: The case of self-propelled, interacting colloids.

Active matter spans a wide range of time and length scales, from groups of cells and synthetic self-propelled colloids to schools of fish and flocks of birds. The theoretical framework describing these systems has shown tremendous success in finding universal phenomenology. However, further progress is often burdened by the difficulty of determining forces controlling the dynamics of individual elements within each system. Accessing this local information is pivotal for the understanding of the physics governing an ensemble of active particles and for the creation of numerical models capable of explaining the observed collective phenomena. In this work, we present ActiveNet, a machine-learning tool consisting of a graph neural network that uses the collective motion of particles to learn active and two-body forces controlling their individual dynamics. We verify our approach using numerical simulations of active Brownian particles, active particles undergoing underdamped Langevin dynamics, and chiral active Brownian particles considering different interaction potentials and values of activity. Interestingly, ActiveNet can equally learn conservative or nonconservative forces as well as torques. Moreover, ActiveNet has proven to be a useful tool to learn the stochastic contribution to the forces, enabling the estimation of the diffusion coefficients. Therefore, all coefficients of the equationof motion of Active Brownian Particles are captured. Finally, we apply ActiveNet to experiments of electrophoretic Janus particles, extracting the active and two-body forces controlling colloids' dynamics. On the one side, we have learned that the active force depends on the electric field and area fraction. On the other side, we have also discovered a dependence of the two-body interaction with the electric field that leads us to propose that the dominant force between active colloids is a screened electrostatic interaction with a constant length scale. We believe that the proposed methodological tool, ActiveNet, might open a new avenue for the study and modeling of experimental suspensions of active particles.

Vibration and aeroelastic instability analysis in GPL-porous multi-layered beam with the rotation effect

This research studies the vibrations and instability of a rotating three-layer beam under aerodynamic forces consisting of two functionally graded (FG) composite surfaces and a porous intermediate layer. Linear poroelasticity theory is used for modeling the porous layer while Young’s modulus and density vary along the thickness. Equivalent coefficients of the graphene nanoplatelets-reinforced composite (GPLs) are extracted by modified Halpin-Tsai theory, according to five different configurations. The equations of motion in the GPL-porous multi-layered beam are derived using the Euler–Bernoulli beam (EBB), Timoshenko beam (TB), and Reddy’s higher-order shear deformation theory (HSDT) theories, as well as the energy method and Hamilton’s principle. The most important results show the effect of reinforcements and their different configurations, porosity and its distribution, speed of rotation on natural frequency, critical flutter point, and loss factor for a GPL-porous multi-layered beam. The unique properties of GPL-reinforced porous materials facilitate the design of components capable of effectively managing dynamic loads and resisting aeroelastic instabilities. This results in more efficient, reliable, and durable systems across various industries.

Improved Equations of the Lagrange Top and Examples of Analytical Solutions

Equations of a heavy rotating body with one fixed point can be deduced starting from a variational problem with holonomic constraints. When applying this formalism to the particular case of a Lagrange top, in the formulation with a diagonal inertia tensor the potential energy has a more complicated form as compared with that assumed in the literature on dynamics of a rigid body. This implies the corresponding improvements in equations of motion. Therefore, we revised this case, presenting several examples of analytical solutions to the improved equations. The case of precession without nutation has a surprisingly rich relationship between the rotation and precession rates, which is discussed in detail.

Vertical vibration control of structures with tuned liquid column dampers

Conventional tuned liquid column dampers (TLCDs) can only be used to suppress structural vibrations in horizontal directions. However, in reality, the vibration-induced motion of civil engineering structures normally consists of both vertical and horizontal components. In this study, a TLCD-based vibration control system, which is composed of a pair of inclined U-shaped containers, springs, dashpots, as well as supporting slopes, is proposed to control the vertical vibration of an engineering structure. The proposed control system (denoted as vertical TLCD) successfully extends the application of TLCD-type devices to vertical vibration reduction. First, the general configuration and operating principle of the proposed vertical TLCD are illustrated in detail. Second, the equations of motion are derived based on the Lagrange dynamic equations and further verified by two-way fluid–structure coupling numerical simulations using finite element method and finite volume method. Subsequently, the closed-form solution is derived for the optimized design of the vertical TLCD in the frequency domain. The dynamic characteristics of an optimized vertical TLCD-structure system are revealed. The effects of mass ratio, excitation amplitude and structural damping ratio on the optimization results are elaborated, respectively. Under the same optimized parameters, the vertical TLCD is found to be superior, compared to the vertical tuned mass damper (TMD). Finally, the effectiveness of the proposed system is demonstrated by applying it to the control of a bridge experiencing vertical vibrations under seismic loads. The findings of this study provide a comprehensive understanding of TLCD’s capabilities and identify its potential for vertical vibration control, which contributes to the development of more effective and versatile vibration mitigation strategies for a wide range of structures under different engineering scenarios.

To plasma electrons motion theory in high-frequency fields

The article is dedicated to the kinetic theory of plasma, where the problem of interaction of high-frequency electric fields with weakly non-uniform plasma is investigated using kinetic equations with binary collision integrals for plasma particles. A new methodology for determining the expression of the averaged highfrequency pressure force is proposed based on solving the kinetic equation and using the method of successive approximations (separation of slow motions and fast oscillations) under the limiting conditions. An expression for the averaged quasi-potential force is derived based on the kinetic equation for the electron distribution function in weakly inhomogeneous magnetoactive plasma, taking into account electrons collisions with fixed ions and the presence of a longitudinally inhomogeneous high-frequency electric field using wellknown methods of theoretical and mathematical physics, such as successive approximations, averaging over the effective field oscillation period, and integration over trajectories. The amplitude of the field is a slowly varying function in both time and space coordinates. The obtained expression allows us to estimate the influence of collisions of plasma particles on the Miller force, and under limiting conditions it coincides with the known expressions for the high-frequency pressure force derived from the equation of plasma electrons motion in high-frequency fields. In all calculations, the contribution of the magnetic component of the electromagnetic field is neglected, which is quite justified for the longitudinal electric field.

Instabilities of perturbations in some homogeneous color-electric and -magnetic backgrounds in SU(2) gauge theory

We consider the instabilities of field perturbations around a homogeneous background color-electric and/or -magnetic field in SU(2) pure gauge theory. We investigate a number of distinct cases of background magnetic and electric fields, and we compute the dispersion relations in the linearized theory, identifying stable and unstable momentum modes. In the case of a net homogenous non-Abelian B field, we compute the nonlinear (quadratic and cubic) corrections to the equation of motion and quantify to what extent the instabilities are tempered by these nonlinearities. Published by the American Physical Society 2024

Modeling and vortex-induced vibrations of semi-submersible floating offshore wind turbines

A rigid-flexible coupling dynamic model of a semi-submersible floating offshore wind turbine (FOWT) is established, and the characteristics of vortex-induced vibration (VIV) of the tower are studied. Lagrange’s equation is used to establish the equations of motion of a semi-submersible FOWT, a wake oscillator model is applied to simulate the interaction between tower and wind field, and the semi-empirical parameters in the wake oscillator model are optimized by computational fluid dynamics (CFD) method. The VIV characteristics of the semi-submersible FOWT before and after hoisting are investigated, and the influence of waves on VIV is analyzed. The results show that the VIV of tower bending mode with large amplitude of the semi-submersible FOWT will occur when the wind speed is in certain ranges, and the possibility of VIV of rigid-body modes with large amplitudes of the tower is low. The amplitudes of VIV and corresponding wind speed ranges can be greatly reduced, and the frequency lock-in domains will become narrower after hoisting the rotor system. The action of waves can reduce the amplitudes of VIV and cause the deviation of frequency lock-in domains, which may widen the coverage of wind speed of lock-in domains.