Equations Of Motion For System Research Articles

Null shells and double layers in quadratic gravity

For a singular hypersurface of arbitrary type in quadratic gravity motion equations were obtained using only the least action principle. It turned out that the coefficients in the motion equations are zeroed with a combination corresponding to the Gauss-Bonnet term. Therefore it does not create neither double layers nor thin shells. It has been demonstrated that there is no “external pressure” for any type of null singular hypersurface. It turned out that null spherically symmetric singular hupersurfaces in quadratic gravity cannot be a double layer, and only thin shells are possible. The system of motion equations in this case is reduced to one which is expressed through the invariants of spherical geometry along with the Lichnerowicz conditions. Spherically symmetric null thin shells were investigated for spherically symmetric solutions of conformal gravity as applications, in particular, for various vacua and Vaidya-type solutions.

Structural dynamic responses of building structures with non-viscous dampers under Kanai–Tajimi spectrum excitation

Non-viscous dampers are commonly used in building structures for seismic energy dissipation. In this study, a concise analytical method based on the pseudo-excitation method (PEM) was proposed to obtain closed-form solutions of the structural responses of a non-viscous damper system subjected to the Kanai–Tajimi spectrum excitation. For this purpose, first, the ground motion equation of the non-viscous damper system was decoupled using the complex model method (CMM), and the solution expressions for the dynamic responses were unified using the PEM. Second, the power spectral density function (PSDF) of the Kanai–Tajimi spectrum was quadratically orthogonalised using the residue theorem, which was combined with a quadratically orthogonalised frequency response eigenvalue function (FREF) to obtain a quadratic orthogonalised PSDF (QO-PSDF) of the dynamic responses. Subsequently, a concise closed-form solution of spectral moments and variances of the dynamic responses were successfully obtained using the QO-PSDF. Finally, a comparative study was conducted with three test cases to compare the results of the direct frequency response method, traditional PEM, and proposed method, which proved the applicability of the QO-PSDF and the accuracy of the quadratically orthogonalised PEM (QO-PEM). Moreover, the calculation time required in the proposed method was found to be approximately th one-third at of the traditional PEM.

Analytical and Numerical Investigations of Mechanical Vibration in the Vertical Direction of a Human Body in a Driving Vehicle using Biomechanical Vibration Model

The main reason that affects the discomfort in a driving vehicle is the vibration response. The human body vibration leads to many malfunctions in both comfort and performance in human health. As a result, the human body’s simulation in sitting posture in the driving vehicle has a strategic relationship for all Tires and vehicles manufacturers. The digital process simulation of the human body seat vehicle vibration shows two significant advantages. The first advantage is the prevention of the high-cost modifications in the construction stage of the vehicle, while the second one describes the stability test during the undesirable vibrations. This study modelled the human body’s dynamic characterisations, natural frequency, and mechanical response when seated in the driving vehicle with vibration transmissibility in the vertical direction have been using the biomechanical vibration model. The vertical vibrations and the transmissibility of the human body dynamic response are presented in detail. Exciting results have been obtained, and they are significant for human health, which relates to sitting posture in the driving vehicle. It can assist in understanding the influences of low-frequency vibration on human health, comfort, and performance, and therefore it could be applied for ride comfort evaluation. An analytical solution to derive the general equations of motion for the human system was developed. Then, using the vibration analysis technique and the corresponding equations, the accurate dynamic response of the selected mode is identified. Furthermore, the mathematical modelling for free vibration using the finite element analysis has been performed to determine the appropriate values and set its description. Then, the comparison results of the two techniques have been carried out.

Nonlinear dynamic analysis of the response to the excitation forces of a symmetrical on-board rotor mounted on hydrodynamic bearings using h-p finite elements method

This work aims to study the nonlinear dynamic analysis response of the external excitation forces of an on-board rotor with a mobile non-deformable supports, mounted on hydrodynamic bearings. The modeling of the structure is achieved by the hierarchical polynomial (h-p) version of the finite element method. A theoretical study based on the Euler–Bernoulli beam model made for the establishment of the kinetic energy and the strain energy of the shaft. The Reynolds equation used to determine the hydrodynamic forces of the plain bearings. The motion equations of rotor system are determined by the Lagrange method. Linear results are obtained using the implicit time-step integration scheme of Newmark, and nonlinear results obtained with a complex resolution algorithm combining Newton–Raphson's iterative incremental procedure with the Newmark temporal integration scheme. The calculation steps for the nonlinear analysis of the onboard rotor system dynamic behavior are grouped together in an application created using the MATLAB programming language, and validated with work done previously by the classical version of the finite element method. In this article, we make some examples to study the influence of the excitations of the mobile support on the behavior of the rotor and the hydrodynamic bearings.

Use of Kane’s Method for Multi-Body Dynamic Modelling and Control of Spar-Type Floating Offshore Wind Turbines

This paper demonstrates the use of Kane’s method to derive equations of motion for a spar-type floating offshore wind turbine taking into account the flexibility of the members. The recently emerged Kane’s method reduces the effort required to derive equations of motion for complex multi-body systems, making them simpler to model and more readily solved by computers. Further, the installation procedure of external vibration control devices on the wind turbine using Kane’s method is described, and the ease of using this method has been demonstrated. A tuned mass damper inerter (TMDI) is installed in the tower for illustration. The excellent vibration mitigation properties of the TMDI are also presented in this paper.

Identification of Linear Time-Variant Systems Based on Ensemble Empirical Mode Decomposition and Hilbert Transform

This paper proposes an identification method based on ensemble empirical mode decomposition (EEMD) and Hilbert transform. The main procedures are: First, the system responses of an n degrees of freedom (DOFs) system are decomposed into intrinsic mode functions (IMFs) and residues by EEMD. The obtained ensemble mean IMFs and residues are then analysed by the Hilbert Transform to form the corresponding analytical signals. Second, the analytical signals of the n IMFs and the residue of each system response of each DOF are added to form a new analytical response. Third, the system equation of motion is rewritten into complex matrix equations for parameter identification. Numerical simulations are carried out on multi-DOFs systems with varying system parameters to demonstrate good robustness, effectiveness and accuracy of the proposed method, which attempts to offer a new means to study identification for linear time variant systems.

The dynamic responses of an infinite plate resting on a poroelastic layered half‐space soil medium with imperfect interface to a moving load

AbstractThe present study deals with the dynamic responses of an infinite plate resting on a poroelastic layered half‐space soil medium with imperfect interfaces to a moving load. Kirchhoff thin plate theory is employed to analyze the plate, while Biot's fully dynamic poroelastic theory is used to characterize the poroelastic half‐space. Applying the Fourier transform and vector functions, the completely coupled system of equations of motion for the layered system are reduced to a system of decoupled ordinary differential equations which could be solved semi‐analytically. The propagation relationships for the layered structure with imperfect interfaces are derived by the novel and efficient dual variable and position (DVP) method. In so doing, the calculated expansion coefficients in the transformed domain are utilized to calculate the unknowns in the physical domain by integration at each field point. Numerical analyses validate the obtained results by comparing with the existing work, and soil vertical displacements, accelerations and pore water pressures induced by moving load are calculated. Moreover, the effects of the different factors are investigated. Computed result shows apparent influences of load velocity, intrinsic permeability and imperfect bonded condition of the soil medium on the dynamic responses of the whole system.

The nonlinear dynamic response and vibration of organic solar plate in thermal environment

Renewable energy sources such as wind, solar energy are attracting a lot of attention from governments, investors and especially scientists. In recent years, there are a lot of papers research about organic solar panels, however, there are only a few papers discover the mechanics of organic solar plates. The difference of this paper is using Reddy’s higher-order shear deformation theory (Reddy’s TSDT) and stress function to provide a system of motion equations and deformation compatibility equations including the imperfect factor in the temperature environment to investigate the mechanical treatment of solar plates. Besides, the paper uses Galerkin method and Runge–Kutta method to give an equation in order to calculate the natural oscillation frequency of the solar plate. The numerical and drawing results are shown with the influence of factors such as geometry parameters, elastic foundations, temperature and mechanical load on the oscillation of the solar cell plate.

Heave Compensation Dynamics for Offshore Drilling Operation

In this study, dynamic response analysis of a heave compensation system for offshore drilling operations was conducted based on multibody dynamics. The efficiency of the heave compensation system was computed using simulation techniques and virtually confirmed before being applied to drilling operations. The heave compensation system was installed on a semi-submersible and comprises several interconnected bodies with various joints. Therefore, a dynamics kernel based on multibody dynamics was developed to perform dynamic response analysis. The recursive Newton–Euler formulation was adopted to construct the equations of motion for the multibody system. Functions of the developed dynamics kernel were verified by comparing them with those from other studies. Hydrostatic force, linearized hydrodynamic force, and pneumatic and hydraulic control forces were considered the external forces acting on the platform of the semi-submersible rig and the heave compensation system. The dynamic simulation was performed for the heave compensation system of the semi-submersible rig for drilling operations up to 3600 m water depth. From the results of the simulation, the efficiency of the heave compensation system was evaluated to be approximately 96.7%.

Modeling of lubricated translational joints in rigid-partially flexible multibody systems and its application in two-stroke marine diesel engines

In this paper, a new tribodynamic model for the translational joints with clearance is proposed by coupling the mixed lubrication model and the rigid-flexible multibody system dynamic model. The lubrication behavior is described by the average Reynolds equation and the Greenwood-Tripp asperity contact theory. Under a unified framework of the absolute nodal coordinate formulation (ANCF), motion equations of the rigid-flexible multibody system are established. The assembled equations of tribodynamic model are solved by a stable and effective algorithm. An application example of a two-stroke marine diesel engine is then used to study the tribological and dynamic behaviors of the crosshead-guide translational joint with clearance. The results show that considering the elastic deformation of components will lead to more intense secondary motion and smaller local clearance, resulting in an oil film pressure distribution with great difference from the rigid body modeling.

The Use of Zero-Mass Particles in Analytical and Multi-Body Dynamics: Sphere Rolling on An Arbitrary Surface

Abstract Zero-mass particles are, as a rule, never used in analytical dynamics, because they lead to singular mass matrices. However, recent advances in the development of the explicit equations of motion of constrained mechanical systems with singular mass matrices permit their use under certain circumstances. This paper shows that the use of such particles can be very efficacious in some problems in analytical dynamics that have resisted easy, general formulations, and in obtaining the equations of motion for complex multi-body systems. We explore the ease and simplicity that suitably used zero-mass particles can provide in formulating and simulating the equations of motion of a rigid, non-homogeneous sphere rolling under gravity, without slipping, on an arbitrarily prescribed surface. Computational results comparing the significant difference in the motion of a homogeneous sphere and a non-homogeneous sphere rolling down an asymmetric arbitrarily prescribed surface are obtained, along with measures of the accuracy of the computations. While the paper shows the usefulness of zero-mass particles applied to the classical problem of a rolling sphere, the development given is described in a general enough manner to be applicable to numerous other problems in analytical and multi-body dynamics that may have much greater complexity.

More realistic holographic model of color superconductivity with higher derivative corrections

In this paper, we have constructed a bottom-up holographic model for the color superconductivity (CSC) of the Yang-Mills theory with including the higher derivative corrections which allow us to study the CSC phase with the color number ${N}_{c}\ensuremath{\ge}2$. First, we consider the CSC phase transition in the context of Einstein-Gauss-Bonnet (EGB) gravity. We analyze the Cooper pair condensate in the deconfinement and confinement phases which are dual to the planar GB-RN-AdS black hole and GB-AdS soliton, respectively, where the backreaction of the matter part is taken into account. By examining the breakdown of the Breitenlohner-Freedman bound in the background of the planar GB-RN-AdS black hole, we find that the positive GB coupling parameter $\ensuremath{\alpha}>0$ leads to a lower upper bound of the color number in comparison to Einstein gravity where the CSC phase for ${N}_{c}\ensuremath{\ge}2$ is not realized. But, with the $\ensuremath{\alpha}<0$ case it is possible to observe the Cooper pair condensate for ${N}_{c}\ensuremath{\ge}2$ with the reasonable magnitude of $\ensuremath{\alpha}$. This is confirmed and the corresponding phase diagram is found by solving numerically the equations of motion for the gravitational system. In addition, we show that the CSC phase disappears in the confinement phase for the magnitude of $\ensuremath{\alpha}$ below a certain value which means that beyond that value it might lead to the breakdown region of the EGB gravity in investigating the CSC phase. However, the CSC phase transition occurring with ${N}_{c}\ensuremath{\ge}2$ requires the magnitude of the GB coupling parameter to be rather large. As a result, the GB term would no longer be considered as the correction and it also violates the boundary causality bound. We resolve this problem by including additionally the higher derivative correction for the Maxwell electrodynamics and the nonminimal coupled Maxwell field.

Comparative Analysis and Differential Study on Dynamic Response of Tension Type Mooring and Catenary Mooring System

Mooring system is an important factor affecting the safety and motion law of floating body. It is very important to study the hydrodynamic characteristics of different mooring methods for offshore safety engineering and stability analysis of mooring system. Taking oc4 as a floating body hydrodynamic model and considering the combined action of wind, wave and current, the hydrodynamic characteristics and system stability of taut and catenary mooring mode are compared and analyzed respectively. In addition, the differences and variation rules of dynamic response of floating body under different mooring conditions are explored and summarized. Combined with the time-domain analysis theory and simulation results, the coupled motion equations of taut and catenary mooring system are improved, and the corresponding relationship between slow drift motion and different mooring modes is summarized. It provides an important theoretical basis and support for the research of time domain variation law of mooring system and marine safety control.

On the Stochastic Mechanics Foundation of Quantum Mechanics

Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.

Energy dissipation by friction for sliding blocks subjected to near-fault seismic base motion

The objective of this study was to determine friction ratios that maximize energy dissipation on a seismic damper. The aforementioned friction damper basically consists of mass blocks that are able to slide on a flat surface. To carry out this analysis, a numerical-experimental approach was used. Firstly, the theoretical background and equations of motion for a SDOF system consisting of a mass supported on a flat surface with friction are introduced. Special emphasis is made on the fundamentals of stick-slip motion as well as energy considerations. Secondly, experimental studies carried out on a shaking table with harmonic and seismic records are described. These tests consisted of lead blocks contained on a U-shaped channel type aluminum section with its open end facing upwards. This configuration allowed blocks to slide solely in the direction of the base motion. Five different types of contact interfaces were considered to determine potential friction coefficients for the damper's design. Additionally, computational models based on rigid-body dynamics are presented. Numerical results were satisfactory particularly when comparing model's dissipated energy with empirical results. An analysis was carried out by calculating dissipated energy for the experimentally-calibrated models with varying friction ratios. For this purpose, eight near-fault seismic records were selected. Intervals with friction coefficients that maximize energy dissipation are proposed for each record. Finally, relationships between the computed friction ratios and register's peak ground acceleration (PGA) and root mean square acceleration (RMS) are discussed.

Vibration damping of an offshore wind turbine by optimally calibrated pendulum absorber with shunted electromagnetic transducer

This paper investigates the use of an augmented pendulum absorber for damping of tower vibrations in monopile-supported wind turbines. A shunted electromagnetic transducer is proposed to replace the classic viscous dashpot for pendulum absorbers, in order to improve performance and durability of the absorber. A series RCL (resistive-capacitative-inductive) network is designed as the supplemental shunt for the electromagnetic transducer, which together with the intrinsic resistive-inductive properties of the transducer coil results in an additional resonance. The system equations of motion are established, from which closed-form expressions for optimal calibration are derived using pole-placement. A fully-coupled 14-degree-of-freedom aero-hydro-servo-elastic wind turbine model is established to evaluate the pendulum absorber performance under realistic conditions by non-linear time-domain simulations. Both frequency- and time-domain results show that the calibration procedure results in an optimal absorber that outperforms the classic dashpot-based pendulum absorber with respect to vibration mitigation.

Frictional Impact-Contacts in Multiple Flexible Links

Multiple impacts of 2D (planar) open-loop robotic systems composed of [Formula: see text] elastic links and revolute joints are studied in this paper. The dynamic equations of motion for such systems are derived by the Gibbs-Appell recursive algorithm, while the regularized method is employed to model the impact-contact mechanism. The Timoshenko beam theory is used to model the transverse vibrations of the links. Also, both the structural damping and air damping are considered to enhance the modeling accuracy. The system joints are assumed to be frictionless and slack-free, but friction force is included for the links colliding with the ground. The [Formula: see text]-flexible-link system considered goes through a flight phase and an impact phase during its motion. In the impact phase, new equations of motion are derived by including the terms caused by the viscoelastic forces in the system’s differential equations. Owing to the extremely short acting time of the impact force, the related differential equations can be solved only via special treatment, i.e. by detecting the exact moment of impact. To this end, entering or leaving the impact phase is analyzed and controlled with high precision by a special computational algorithm presented in this work. To demonstrate the efficacy and precision of the algorithm developed, computer simulations are conducted to study the dynamic behavior of a 3-link robotic mechanism. To investigate the effect of mode shape on the elastic deformation of links, four different mode shapes are used in the simulations and their results are compared.

Nonlinear Vibration Analysis of Rotor considering Cogging and Harmonic Effects

This paper aims to investigate nonlinear vibration characteristics of rotor system considering cogging and harmonic effects. Firstly, relative permeance with eccentric was established and then corrected by correction factor caused by the cogging effect. Based on the new formula of relative permeance, the expression of unbalanced magnetic force was obtained, and the coefficient of cogging effect was defined. Motion equations of rotor system were established, and Runge–Kutta method was used to solve the equations. Results showed that errors between finite and analytical results were smaller considering cogging and harmonic effects. When the harmonics were taken into consideration, the vibration of rotor increases sharply. When the cogging and harmonics were taken into consideration simultaneously, the vibration of rotor decreased instead, which means that stator slots have the effect of reducing vibration in rotor system. Rotor vibration was axis symmetry with static eccentricity rather than central symmetry with no eccentricity, and double, four times, and six times supply frequency always existed in the components of main frequency with eccentric.

Chaotic dynamics of active topological defects

ABSTRACT Topological defects are interesting phenomena which can be observed in ordered phases such as oriented active fluids or nematic liquid crystals. Topological defects are determined by the overall structure of the director field in an active fluid in nematic phase and by exerting force to the units of the active particles they can interact or cause motion in the environment. Studying them as particles with dynamical equations, instead of studying the director field of the nematic environment, would provide us the power to study the characteristics of their motion. The equations of motion for multi-defect systems have been previously studied and in this work we focus on the chaotic properties of the dynamics and analyze the effect of activity on the transition of such systems to chaos by numerically estimating their Lyapunov exponents. We show the importance of the defects for this transition (as they are active in the sense of having self-propulsion) and numerically derive critical activity values for multi-defect systems in which this transition occurs.

Non-Hamiltonian dynamics of indirectly coupled classical impurity spins

We discuss the emergence of an effective low-energy theory for the real-time dynamics of two classical impurity spins within the framework of a prototypical and purely classical model of indirect magnetic exchange: Two classical impurity spins are embedded in a host system which consists of a finite number of classical spins localized on the sites of a lattice and interacting via a nearest-neighbor Heisenberg exchange. An effective low-energy theory for the slow impurity-spin dynamics is derived for the regime, where the local exchange coupling between impurity and host spins is weak. To this end we apply the recently developed adiabatic spin dynamics (ASD) theory. Besides the Hamiltonian-like classical spin torques, the ASD additionally accounts for a novel topological spin torque that originates as a holonomy effect in the close-to-adiabatic-dynamics regime. It is shown that the effective low-energy precession dynamics cannot be derived from an effective Hamilton function and is characterized by a non-vanishing precession frequency even if the initial state deviates only slightly from a ground state. The effective theory is compared to the fully numerical solution of the equations of motion for the whole system of impurity and host spins to identify the parameter regime where the adiabatic effective theory applies. Effective theories beyond the adiabatic approximation must necessarily include dynamic host degrees of freedom and go beyond the idea of a simple indirect magnetic exchange. We discuss an example of a generalized constrained spin dynamics which does improve the description but also fails for certain geometrical setups.