Equations Of Motion For System Research Articles

Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints

This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the fractional Lagrange system with nonholonomic constraints are derived. Secondly, by introducing infinitesimal generating functions of space and time, the Noether symmetry of the Herglotz type is defined, along with its criteria, and the conserved quantity of the Herglotz type is given. Finally, to demonstrate how to use this method, two examples are provided.

Approximate Noether theorem and its inverse for nonlinear dynamical systems with approximate nonstandard Lagrangian

The approximate Noether theorem and its inverse theorem for the nonlinear dynamical systems with approximate exponential Lagrangian and approximate power-law Lagrangian are investigated. For each case, the approximate differential equations of motion for the nonlinear dynamical systems with approximate nonstandard Lagrangian are established, the generalized Noether identities are given. The relationship between the approximate Noether symmetries and approximate conserved quantities for the system with approximate nonstandard Lagrangian are established, and the approximate Noether theorems and their inverse theorems are obtained. Two examples are given to illustrate the application of the results.

Quantization of non-Abelian Yang-Mills theories

A non-Abelian theory of fermions interacting with gauge bosons, the constrained system, is studied. The equations of motion for a singular system are obtained as total differential equations in many variables. The integrability conditions are investigated, and the set of equations of motion is integrable. The Senjanovic and the canonical methods are used to quantize the system, and the integration is taken over the canonical phase space coordinates.

Impact-Rubbing Dynamics of Rotor with Hollow Shaft and Offset Discs Based on MDOF Timoshenko Beam Theory

The impact-rubbing dynamic characteristics of the power turbine rotor with the hollow shaft and offset discs for aircraft engine are investigated, and the impact-rubbing analytical method for the complex rotor based on MDOP Timoshenko beam theory is proposed in this paper. Compared with the traditional approach, the novel method can obtain more data to satisfy the need of engineering. The Lagrange equation is adopted to derive the equations of motion for the rotor system, and the Newmark-β method is applied to solve the equations. The diagrams such as the bifurcation, axis trajectory, spectrum, and Poincaré map are obtained to research on the effect of the rotating speed, gap, and eccentricity on the vibration response. The finite element analysis was carried out to validate the correctness of the theoretical modeling method. The research results indicate that the power turbine rotor with the hollow shaft on operation shows the various nonlinear dynamic behaviors including the multiperiod, quasi-period, jumping phenomenon, and chaotic motions; there exists an optimal gap between the rotor and the stator from the perspective of the efficiency and the dynamics; the optimal gap should make system avoid the resulting chaos or the quasi-period motion for the stability and safety of the machinery.

An asymmetric model two-dimensional oscillator

We present a novel 2D oscillator with an asymmetric design and investigate its stable vibration utilizing the Ms-DTM method. Initially, we obtain the equations of motion for the proposed system. Subsequently, by employing Taylor expansion of [Formula: see text] and [Formula: see text], the derived nonlinear equations are transformed into linear ones, which we solve analytically using the eigenvalues-eigenfunctions technique. Additionally, we solve the obtained nonlinear system using the Ms-DTM method. Lastly, we examine the stability of the nonlinear system by visualizing the closed nature of the phase portrait.

Superposition continuum mechanics (hydrodynamics) for systems with spin polarization defined evolution

The class of physical systems with collectivized electrons, in which the influence of spin polarization on the character of evolution crucially matters, is closely scrutinized. The hydrodynamic theory for its description is developed and the grounds of its methodological part are evaluated. The binary factorized representation of the density function serves as a master hypothesis. The system of motion equations has been deduced by means of the stationary action principle. The exponential scaling of its size in relation to the number of spin carriers is identified. We show its fundamental computability in spectral domain of the factorized discrete Fourier transform. The adequacy of description of the physics of electron gas–light interaction with its use has been assessed by the example of localized plasmon excitation in transition metal nanoparticles. It has been demonstrated the equivalence of the proposed model to many body quantum mechanics for the Shrödinger equation (SE) with operator 4–potential. The linearity of superposition mechanics is discussed in parallel. The methodological part of the approach has allowed factoring the many body SE and for the sake of validation has been translated to the approximately solved classically computable model. The results obtained with its use allow interpreting the viscous (drudean) damping in spherical particles as a consequence of maximal decoherence in motion of spin configurations.

An Application of Mechanics to Chemistry: the Dynamic Behaviour of Coupled Chemical Reactions Compared to that of the Two-Body Oscillator

The dynamic behaviour of a chemical system made of two coupled reactions is compared with that of a mechanical system consisting of two oscillating bodies connected by springs. First, the principle of energy departure from equilibrium is employed to derive the motion equations of both systems. Subsequently, the relevant characteristic frequencies and the amplitude parameters are obtained and analysed in terms of “Normal Modes”. The results show that systems belonging to different branches of science can be analysed using the same methodologies. To elucidate the application of Normal Modes to chemistry, the dynamic analysis of a system consisting of a proton transfer reaction coupled to a complex formation reaction is described in the Supporting Information: the procedure enables the evaluation of rate constants, equilibrium constants and reaction enthalpies of a reacting chemical system made of two coupled reactions. The method is then extended to a cycle of three reactions.

On the detection of nonlinear normal mode-related isolated branches of periodic solutions for high-dimensional nonlinear mechanical systems with frictionless contact interfaces

This contribution introduces a methodology for the detection of isolated branches of periodic solutions to the nonlinear mechanical equation of motion for systems featuring frictionless contact interfaces. This methodology relies on a harmonic balance method-based solving procedure combined with the application of the Melnikov energy principle. It is able to predict the location of isolated branches of solutions in the vicinity of families of autonomous periodic solutions, namely nonlinear normal modes. The methodology is first applied on a two-degree-of-freedom phenomenological system in order to illustrate its relevance and accuracy. In particular, for this academic application, the proposed methodology yields an understanding of the discontinuous evolution of the first nonlinear resonance frequency as the forcing amplitude increases. Isolated branches of solutions featuring high amplitudes of vibration are detected far beyond nonlinear resonance frequencies obtained with a typical application of the harmonic balance method coupled with an arc-length continuation algorithm. Demonstration of the applicability of the proposed methodology to high-dimensional industrial finite element models is made with the nonlinear vibration analysis of a transsonic compressor blade, NASA rotor 37, subjected to an harmonic loading and blade-tip/casing structural contacts. The proposed methodology yields numerous isolated branches of solutions, which relevance is assessed by means of time integration simulations. In the end, the presented results underline that typical continuation algorithms may yield a significant underestimation of nonlinear resonance frequencies.

Mechanism study on the effect of floor rotation angle on supported tuned mass damper system of high-rise structure

The bending deformation of the super high-rise structures will cause floor rotation angle, and thus affect the vibration reduction effect of the supported tuned mass damper (TMD) system. This paper proposes a motion equation of the TMD control system with the effect of floor rotation angle. Firstly, a finite element model of the supported TMD control system was established by ETABS. Secondly, the formation and floor rotation angle parameters of the structure were obtained by ETABS modal analysis. Then, the structures ETABS model is simplified into a multi-degree-of-freedom (MDOF) model based on the dynamic characteristics of the structure. This paper also presents an evaluation index of the effect of floor rotation angle on the supported TMD system and a supported TMD control system is selected as a simulation example. The relationship between the floor rotation angle coefficient and the TMD parameters is later parametric analyzed. Finally, experiments were designed to verify the effect of floor rotation angle on the supported TMD control system. The experimental results show that the relationship between the rotation angle of the first formation floor and the generalized displacement of the first formation of the assumed structure is close to linear. Therefore, the effect of floor rotation angle of the structure is highly recommended to be considered as an important factor for the design of the supported TMD control system for super high-rise structures.

Matematyczne modele hydromechaniki przepływu wielofazowego o zmiennej masie

The paper discusses the mathematical model of hydromechanics of multiphase flows with varying mass. A multiphase flow is considered a continuum consisting of a set of a large number of different groups of particles. The derivation of motion equations and similarity criteria are given taking into account both the externally attached (or detached) mass and phase transitions within the medium. The equations of mass, momentum and energy transfer for individual phases and the medium as a whole are derived based on fundamental conservation laws. It was demonstrated that in the absence of sources (or flow-offs) of mass, momentum and energy, the known equations of single- and multi-phase flow hydromechanics follow as a special case from the obtained systems of motion equations and similarity criteria. The obtained motion equations are valid for the description of an ingredient of mixture and the medium as a whole, regardless of their physical and mechanical properties. Thermodynamic and rheological state equations, as well as expressions for heat flow, interfacial mass forces phase transitions, and heat exchange between phases can be used to close them. The implemented models make it possible to simulate both the stationary distribution of parameters along the wellbore during production and non-stationary processes that occur, for example, when the pump shaft speed changes during oil production. The developed approaches were implemented in the DataFlow software tool for analysis of the hydrodynamics of multiphase hydrocarbon flows, taking into account heat exchange with the rocks surrounding the well, and phase transitions in the fluid. Using the software package, test calculations were carried out to demonstrate the performance of the proposed and implemented models.

An efficient Julia framework for hierarchical equations of motion in open quantum systems

The hierarchical equations of motion (HEOM) approach can describe the reduced dynamics of a system simultaneously coupled to multiple bosonic and fermionic environments. The complexity of exactly describing the system-environment interaction with the HEOM method usually results in time-consuming calculations and a large memory cost. Here, we introduce an open-source software package called HierarchicalEOM.jl: a Julia framework integrating the HEOM approach. HierarchicalEOM.jl features a collection of methods to compute bosonic and fermionic spectra, stationary states, and the full dynamics in the extended space of all auxiliary density operators (ADOs). The required handling of the ADOs multi-indexes is achieved through a user-friendly interface. We exemplify the functionalities of the package by analyzing a single impurity Anderson model, and an ultra-strongly coupled charge-cavity system interacting with bosonic and fermionic reservoirs. HierarchicalEOM.jl achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.

Analysis of Train-Track-Bridge Coupling Vibration Characteristics for Heavy-Haul Railway Based on Virtual Work Principle.

This paper introduces an innovative model for heavy-haul train-track-bridge interaction, utilizing a coupling matrix representation based on the virtual work principle. This model establishes the relationship between the wheel-rail contact surface and the bridge-rail interface concerning internal forces and geometric constraints. In this coupled system's motion equation, the degrees of freedom (DOFs) of the wheelsets in a heavy-haul train lacking primary suspension are interdependent. Additionally, the vertical and nodding DOFs of the bogie frame are linked with the rail element. A practical application, a Yellow River Bridge with a heavy-haul railway line, is used to examine the accuracy of the proposed model with regard to discrepancy between the simulated and measured displacement ranging from 1% to 11%. A comprehensive parametric analysis is conducted, exploring the impacts of track irregularities of varying wavelengths, axle load lifting, and the degradation of bridge stiffness and damping on the dynamic responses of the coupled system. The results reveal that the bridge's dynamic responses are particularly sensitive to track irregularities within the wavelength range of 1 to 20 m, especially those within 1 to 10 m. The vertical displacement of the bridge demonstrates a nearly linear increase with heavier axle loads of the heavy-haul trains and the reduction in bridge stiffness. However, there is no significant rise in vertical acceleration under these conditions.

Finite element analyses of rotor/bearing system using second-order journal bearings stiffness and damping coefficients

The rotor-bearing system is a crucial component of rotating machinery, such as turbines, pumps, compressors, and turbogenerators, which are widely used in various advanced engineering fields. This work presented a methodology for studying the behavior of an elastic Jeffcott rotor supported by two similar fluid film journal bearings. A finite element model using consistent matrix formulation was employed to simulate the shaft, including the external load, with four degrees of freedom per node. The small perturbation method was used to evaluate the second-order bearing coefficients of a journal bearing of finite length. These coefficients were further integrated with the finite element model to evaluate the dynamic response of the flexible rotors. Moreover, the system equations of motion were presented in dimensional form. The results of the second-order bearing coefficients analysis agreed with nonlinear analysis when the speed was less than the threshold speed, while there was a pronounced difference in second-order analysis when the speed was above the threshold speed. The behavior of the rotor-bearing system was studied using dynamic response and orbit diagrams, revealing that changes in rotational speed significantly affected the rotor’s stability.

RETRACTED: Analysis of governing equations of multibody dynamic systems through kinematic properties of joints and effects of kinematic chains on closed-form mechanism

In this investigation, the fundamental equations dictating the behavior of dynamic systems were ascertained by means of an innovative approach that amalgamated the kinematic properties of joints and the intricate kinematic chains of multibody systems into a series of governing equations. Subsequently, the equations that govern the behavior of multibody systems were converted into ordinary differential equations (ODEs) utilizing the calculus of matrix-valued functions. This computational technique is adept at swiftly acquiring recursive differential equations of motion for multibody systems. Thus, the expenditure of computational costs associated with the simulation was efficaciously minimized. The mechanisms of Andrew's squeezing and carpet scrapping were employed in conjunction with kinematic constraints to authenticate the validity of the proposed methodology. The findings suggest that, notwithstanding the fact that the root mean square error (RMSE) in the generalized coordinates between the two ODE and differential algebraic equation (DAE) techniques is small in the Andrew's squeezing mechanism (at approximately 0.02) and in the carpet scraping mechanism (at approximately 0.7), there are still discrepancies between the two methods. The degree of convergence in the aforementioned methodologies, despite the slight RMSE difference between them, shows a clear disparity. By way of illustration, when employing the conventional DAE technique, the duration required to resolve the Andrew's squeezing and carpet scraping mechanisms is 5 and 11.34 s, respectively. However, by utilizing the ODE methodology in the aforementioned mechanisms, the time required to determine a solution is truncated to 1.19 and 2.1 s, sequentially.

Dynamic modeling of hard-magnetic soft actuators: Unraveling the role of polymer chain entanglements, crosslinks, and finite extensibility

In recent years, there has been increasing interest in hard-magnetic soft materials (HMSMs) due to their ability to retain high residual magnetization and undergo large deformations under external magnetic loading. The performance of these materials in the dynamic mode of actuation is significantly influenced by internal properties, such as entanglements, crosslinks, and the finite extensibility of polymer chains. This article presents a theoretical framework for modeling the dynamic behavior of a hard-magnetic soft material-based planar actuator. A physics-based nonaffine material model is utilized to consider the inherent properties of polymer chain networks. The governing equation for dynamic motion is derived using Euler–Lagrange’s equation of motion for conservative systems. The devised dynamic model is utilized to examine the dynamic response, stability, periodicity, and resonance properties of a planar hard-magnetic soft actuator for different values of polymer chain entanglements, crosslinks, and finite extensibility parameters. The Poincaré maps and phase-plane plots are presented to analyze the stability and periodicity of the nonlinear vibrations of the actuator. The results reveal that transitions between aperiodic and quasi-periodic oscillations occur when the density of polymer chain entanglements and cross-linking changes. The findings from the present investigation can serve as an initial step towards the design and manufacturing of remotely controlled actuators for various futuristic applications.

Extended dissipaton equation of motion for electronic open quantum systems: Application to the Kondo impurity model.

In this paper, we present an extended dissipaton equation of motion for studying the dynamics of electronic impurity systems. Compared with the original theoretical formalism, the quadratic couplings are introduced into the Hamiltonian accounting for the interaction between the impurity and its surrounding environment. By exploiting the quadratic fermionic dissipaton algebra, the proposed extended dissipaton equation of motion offers a powerful tool for studying the dynamical behaviors of electronic impurity systems, particularly in situations where nonequilibrium and strongly correlated effects play significant roles. Numerical demonstrations are carried out to investigate the temperature dependence of the Kondo resonance in the Kondo impurity model.

ANALYTICAL APPROACH IN THE ANALYSIS AND OPTIMIZATION OF CRANES DYNAMIC LOADS DURING THEIR START-UP: SWINGING THE LOAD

During the operation of the cranes pendulum swings of the load are observed. They cause cranes or cargo carts uneven movement, additional loads on power ele-ments of cranes, create inconvenience during their operation. This must be taken into account during detailed cranes calculations. 
 In bridge, gantry and some other standard types of cranes which move along a rail path, load pendulum oscillations frequency relative to the crane is significantly lower than the elastic oscillations frequency of the crane metal structure and the movement mechanism transmission. Therefore, the load pendulum oscillations can be considered practically independent from the crane elastic oscillations and when cal-culating them, the metal structure and movement mechanism transmission can be taken as absolutely rigid. In this study to determine the dynamic loads acting on the metal structure and the movement mechanism transmission, the law of ropes tension hori-zontal component change, which arises as a result of the load pendulum oscillations, is given in the form of a known time function, which is determined according to the scheme of an absolutely rigid crane. This approach allows to reduce the order of mo-tion equations of the crane dynamic system by two units. In accordance with the above, the calculation of pendulum oscillations of the load on a ropes can be carried out ac-cording to the simplest two-mass scheme. This is exactly the approach implemented in the study. Motion conditions and equations of the “cargo cart − the load on a rope” system analytically determined, at which dynamic loads in the rope/elastic element have the smallest (minimum) values in the starting mode of the bridge-type crane.

Response prediction and fatigue life estimation of thin-walled structures under random excitation using the modified equivalent linearization method

This paper proposes a fast estimation method for the high-cycle fatigue of geometrically nonlinear thin-walled structures under random loading. The core idea of this method is utilizing a modified equivalent linearization method to obtain the statistical properties of the modal displacement response of the thin-walled structures, which is used to generate samples and then transform them into stress samples to determine the peak probability density function of the von Mises stress for fatigue life estimation. There are three critical issues addressed in implementing the method: (1) Using the implicit condensation technique to introduce the membrane softening effect in the regulated equivalent linearization method to improve the computational accuracy of the displacement statistical properties and convert the higher order moments into easily computable second order moments by separating the displacement mean values, so the regulated equations of motion for non-zero mean systems are derived, and fix the problem that the regulated equivalent linearization method does not apply to the prediction of the dynamic response of general thin-walled structures. (2) A nonlinear relationship between modal displacement and stress is derived with the implicit condensation and expansion method to transform the generated displacement samples to stress samples, which are used to determine probability density functions of stress and their derivatives. (3) Simplifying the probability density function of the three-dimensional joint distribution to the probability density function of the two-dimensional joint distribution by taking advantage of the Bayes rule, thus deriving a fast formula for the probability density function of the peak Mises stress.

Nonlinear vibration analysis and stability analysis of rotor systems supported on SFD by combining DQFEM, CMS and IHB methods

In this paper, the differential quadrature finite element method (DQFEM), component mode synthesis (CMS) method and incremental harmonic balance (IHB) method are combined to propose a methodology to perform vibration analysis and stability analysis of rotor systems supported on fluid-film bearings that exhibit localized nonlinearities such as squeeze-film damper (SFD). In multi-degree of freedom systems with SFD, a method for dynamic analysis of the system by reducing the degrees of freedom is first suggested in this paper. DQFEM are used to create the mass matrix, stiffness matrix, and gyroscopic matrix of the system. Then, the system is divided into linear system and nonlinear system, and the motion equation of the linear system reduces the degrees of freedom using the free interface CMS method, and the motion equation of the nonlinear system is written without reducing the degrees of freedom. By synthesizing the equations of motion of each component, the motion equation of the entire system is created. That is, the motion equation of the whole system uses hybrid coordinates. The motion equation of the system is solved using the modified IHB method. To evaluate the effectiveness of the suggested method, the computed solutions are compared with the results presented in the previous literature, and the results are very consistent. Also, stability analysis of the obtained solutions is performed by the Floquet theory. The method suggested in the paper can be effectively used to analyze the dynamic behavior of rotor systems exhibiting localized nonlinearities such as SFD.

Periodic and torus motions of a two-degree-of-freedom dry friction vibration system

Vibration induced by dry friction is ubiquitous in various engineering fields. To explore the vibration characteristics for further studies and/or controls, it is of great theoretical and practical significances to investigate the non-linear dynamic behaviors of the friction systems. This study considers the slight vibration of a two-degree-of-freedom non-linear dry friction excitation system. The differential equations of system motion are established according to Newton’s law of motion. Moreover, the system’s non-linear dynamic is studied when the block velocity is always less than the friction surface velocity. The results indicate that the linearized matrix of the vibration system has a pair of purely imaginary eigenvalues for some critical values of the relevant parameters. The Poincaré-Birkhoff normal forms are utilized to simplify the motion equation under the non-resonant assumption to obtain a simplified equation with only the resonant terms. Furthermore, the truncated part of the simplified equation is analyzed in the case of only linear terms degeneration. Finally, numerical simulations reflect some qualitative conclusions about the system’s local dynamic properties, including equilibrium point, periodic motion, torus motion, and their stability.