Differential Equations Of Motion Research Articles

A Method for Simulating the Positioning Errors of a Robot Gripper

The research is aimed at creating a methodology for increasing the positioning accuracy of an industrial robot and minimizing the vibration of the robot gripper by applying machine learning based on the developed mathematical model for estimating the positioning error. Two components of positioning accuracy are considered: geometric and kinematic errors and elastic static deformations. The dynamic error in the partial system of motion of the robot manipulator links is analyzed. The equation of partial motions is obtained from Lagrange’s differential equation of motion of the II kind. The system of differential equations for the positioning error was solved analytically by Euler’s method. An example of modeling the position and orientation error of the gripper due to temperature deformations of the third link for the manipulator scheme is given. An example of the modeling of static deformations and errors of the manipulator with elastic pliability of the robot links is given. An example of dynamic error modeling in a partial system of motion of the robot links is given. The proposed method of modeling robot gripper positioning errors makes it possible to increase the positioning accuracy of the industrial robot and minimize the vibration of the gripper. Having a mathematical model of positioning errors, it is possible to compensate for the positioning error by changing the speed of movement of the gripper reference point before determining the direct kinematic task.

Analyzing the impact of convective boundary conditions on fluid–particle mixture flow in a ciliated walled horizontal tube filled with Rabinowitsch fluid

This study investigates the impact of convective boundary conditions, thermal radiation, and viscous dissipation on the flow of fluid–particle mixture in a ciliated walled horizontal tube filled with Rabinowitsch fluid. A mathematical model is developed using partial differential equations of continuity, motion, and energy, which are solved using the symbolic software MATHEMATICA 12.0 with convective boundary conditions. Exact solutions for velocity, temperature distribution, pressure gradient, pressure rise, stream function, and heat transfer rate are obtained and analyzed for dilatant, Newtonian, and pseudoplastic fluids. Computational results reveal that the velocity and temperature profiles are highest at the center of the tube for pseudoplastic and Newtonian fluids compared to dilatant fluid. Additionally, pressure gradient fluctuates near the tube walls. The study also explores special cases where the Rabinowitsch fluid model is reduced to dilatant, Newtonian, and pseudoplastic fluids. The findings have potential applications in medical treatments for respiratory system tumors, airway cleaning, physiological fluids, and manufacturing processes involving peristaltic flow. This original research contributes to the understanding of fluid dynamics in complex systems and has implications for various practical applications.

Dynamic Response of Advanced Lightweight Porous Plates Integrated with Nanocomposite Face Sheets Resting on Elastic Substrate

This study centers on the analysis of a scale-dependent microplate configuration characterized by a porous core enveloped by nanocomposite patches embedded with graphene nanoplatelets. The microplate’s behavior is explored within a humid environment to comprehend the effects of moisture variations on its dynamic performance. Additionally, the microplate rests upon a Kerr foundation, a three-parameter elastic substrate. The material properties of all three layers are contingent upon thickness. To enhance precision, an innovative quasi-three-dimensional shear and normal trigonometric theory is employed, elucidating the kinematic interrelations of the microstructure. Notably, this novel theory accommodates the presence of transverse normal strain. For a comprehensive analysis of size influences, the modified couple stress theory is harnessed. This theory integrates a material length-scale parameter to anticipate outcomes at the micro-scale. By invoking Hamilton’s principle, differential motion equations are deduced and subsequently solved analytically. The investigation centers on probing the consequences of varied parameters on the natural frequencies. The findings underscore that the incorporation of GPLs amplifies the microplate’s stiffness, thus elevating its natural frequencies. In contrast, an escalation in the porosity index leads to a reduction in natural frequencies.

Stochastic Analysis for the Embedded Single-Walled Carbon Nanotube Under Random Vibrations

It is inevitable that carbon nanotubes (CNTs) are subjected to random vibrations due to environmental noise, which may impair their mechanical properties. The nonlinear dynamics associated with single-walled carbon nanotubes (SWCNTs) under random noise has been investigated in this work. First, based on the nonlocal theory and the Euler–Bernoulli beam model with fixed supports at both ends, the governed equations and boundary conditions are established according to Hamilton’s principle. Second, the Galerkin method is used to approximate the differential equations of motion and the stochastic averaging method (SAM) is applied to predict the approximate response of the original system. Subsequently, a detailed parametric study is conducted to analyze the nonlinear random vibration response of the SWCNT with nonlocal effects, and the effectiveness of the proposed method is verified by comparing the analytical results with those from the Runge–Kutta method. Finally, by solving the backward Kolmogorov (BK) equation and the generalized Pontryagin (GP) equation simultaneously, the conditional reliability function (CRF) and mean first-passage time (MFPT) for determining the reliability of SWCNT systems are obtained.

Variational Gaussian approximation for the magnetic Schrödinger equation *

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac–Frenkel variational principle. For the approximation we derive ordinary differential equations of motion for the parameters of the variational solution. Moreover, we prove L 2-error bounds and observable error bounds for the approximating Gaussian wave packet.

Dynamical behavior of liquid-filled rotors with axial motion

In the present study, the dynamical behavior of liquid-filled rotors with axial motion is investigated. Based on the differential equations of motion for ideal fluids, the fluid forces exerted on the rotor are obtained. Then, the Euler–Bernoulli beam theory is utilized to establish the governing equations of motion for the rotor system. By applying Hamilton's principle, the governing equation and the corresponding boundary conditions are derived. Furthermore, in order to solve the eigenvalue problem of the system, the extended Galerkin method is applied to discrete equations of motion for the rotor system. As a result, the system stability, divergence, and flutter instability are defined. Moreover, a comparative study is presented to verify the accuracy of the proposed model. Finally, the effects of liquid parameters on the dynamic characteristics of the rotor system are investigated in detail. The results show that for liquid-filled rotors with axial motion, the stability of the system depends on the liquid parameters, axial velocity, and spinning velocity.

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.

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.

Collision Study on New Aluminum Alloy W-Beam Guardrail

Highway guardrails are safety installations set along the sides or center of roads to prevent vehicles from veering off the roadway, thereby protecting pedestrians and vehicles. However, traditional W-beam guardrails have issues such as poor energy absorption, susceptibility to rust, short lifespan, and a high risk of vehicles running off the road during severe collisions. Therefore, this study employs the partial differential equations of the vertical motion of beams to investigate the relationship between the deformation of wave beams and their moment of inertia. The cross-sectional shape of traditional W-beam guardrails was optimized accordingly. Combining this with the high ductility of aluminum alloy Al 6061-T6, a new aluminum alloy W-beam guardrail was proposed. Finite element simulation models of traditional steel guardrails, new steel guardrails, and the new aluminum alloy guardrail were established to evaluate the performance of the three types of guardrails. The results show that the deformation of the wave beam is inversely proportional to the square root of its moment of inertia. Compared to traditional W-beam guardrails, the moment of inertia of the new guardrail’s cross-section increased by 28.6%. In finite element collision simulations, the new guardrail reduced deformation by 12.8%, lowered the risk of vehicles running off the road, and improved safety performance.

Dynamic modeling and analysis of viscoelastic hard-magnetic soft actuators with thermal effects

Hard-magnetic soft materials (HMSMs) are a class of magnetoactive smart polymers capable of sustaining a high residual magnetic flux density and can undergo large actuation strains under an external magnetic excitation. Due to these exceptional characteristics, soft actuators based on HMSMs hold enormous potential for remote-controlled applications. The temperature and viscoelasticity considerably influence the performance of these materials during the operation. This work aims to develop an analytical framework for modeling the dynamic behavior of a planar hard-magnetic soft actuators (HMSA) considering temperature and viscoelastic effects. The constitutive behavior of the viscoelastic HMSA is described by employing an incompressible neo-Hookean model in conjunction with a Zener rheological model and the Rayleigh dissipation function. The dynamic governing differential equations of motion are derived by utilizing the non-conservative form of the Euler–Lagrange equation. This study delves into the collective influence of temperature and viscoelastic properties on the stability, periodicity, and resonance characteristics of nonlinear vibrations exhibited by HMSM-based planar actuator subjected to dynamic magnetic loading, presenting the findings through time-history responses, Poincaré maps, and phase-plane plots. The presented results can help in the efficient and robust design of HMSM-based actuators and can also serve as an initial step toward the development of advanced actuators exposed to dynamic loading under variable temperatures for diverse applications in the fields of engineering and medicine.

Transverse vibration analysis and active disturbance rejection decoupling control of rain erosion whirling arm

The transverse vibration caused by the mass eccentricity of the whirling arm and the unbalanced magnetic pull of the motor rotor is a critical factor that affects the stability and safety of the rain erosion whirling arm. To investigate the transverse vibration characteristics of the rain erosion whirling arm during operation, a four-degree-of-freedom transverse vibration model of the rotor-rain erosion whirling arm is established using the lumped mass method. The differential equation of motion of the rotor-rain erosion whirling arm, considering the influence of mass eccentricity and unbalanced magnetic pull, is then established based on the Lagrange equation. This equation is numerically solved using the Runge-Kutta algorithm to investigate the axis trajectory and amplitude distribution of the rotor-rain erosion whirling arm. To reduce the fatigue damage caused by vibration, the active disturbance rejection decoupling control method is employed. Furthermore, the parameters of the extended state observer are adjusted using pole assignment and bandwidth. The simulation results reveal that the vibration amplitude of the rotating blade can be reduced to within 0.2 mm, referring to the BSS7393 test standard. Additionally, experimental research is conducted to compare the vibration of the rain erosion whirling arm before and after implementing the active disturbance rejection decoupling control. The results demonstrate that the active disturbance rejection decoupling control method effectively suppresses the vibration of the rain erosion whirling arm, illustrating its feasibility and effectiveness for system vibration control.

Theoretical model and parameter sensitivity analysis of concrete slab vibration in concrete-face rockfill dams during underwater nuclear explosions

Large reservoir dams, which are carriers of energy and resources, have historically been the main targets of military conflict and terrorist attacks. A concrete-face rockfill dam (CFRD) is a competitive hydraulic structure widely used in large-scale water conservancy projects. However, the vibration response and damping paths of CFRDs under the influence of underwater explosion shock waves remains insufficiently understood. In this study, we created a theoretical model of slab vibration with four-sided free boundaries using Cole’s formula and the motion differential equation of a slab on an elastic foundation. Furthermore, a global sensitivity analysis of the vibration model parameters on the slab displacement was performed based on the Kriging surrogate model and Sobol method. The results showed that the antiseepage slabs experienced a void phenomenon, resulting in overall instability. The explosive distance and normal stiffness of the slab had the greatest influence on the void displacement of the slab, whereas the dam height and concrete density had insignificant effect. Moreover, increasing the dam height of the CFRD did not increase the risk of slab voids during underwater explosions, which was not the case under strong seismic loads. Therefore, the key factors for active and passive defense against blast resistance in CFRDs were identified, which can provide theoretical support for the blast-resistant design and research on CFRDs. In this study, the reliability of the theoretical model was verified by establishing a numerical analysis model.

SUPPRESSION OF SOMMERFELD EFFECT IN POWER TRANSMISSION SYSTEM EMPLOYING CARDAN SHAFT THROUGH PHASE ANGLE ARRANGEMENT

In a power transmission system with a cardan shaft, the Sommerfeld effect occurs, which is characterized by speed capture and release at the resonance range. Suppression of the Sommerfeld effect is critical for smooth and reliable operation. This study aims to suppress the Sommerfeld effect in a transmission system by compensating the phase angle between the two universal joints installed in the cardan shaft. The differential equations of motion representing the dynamics of the system are derived using the Lagrange equation. The responses are simulated numerically using the Runge–Kutta algorithm for scenarios with constant and gradually varying input torque. To suppress the Sommerfeld effect, the phase angle is set to 25%, 50%, 75% and 100% of the maximum twist angle observed in the subcritical speed range of the in-phase configuration. With the phase angle of 25%, the Sommerfeld effect is damped, where the output speed only deviates by 5% from the estimated value for both input torque scenarios. It is shown how the change of the phase angle attenuates the Sommerfeld effect and the system vibrations, which should be considered in the development and practical implementation.

Formation and development of vortices in power and propulsion systems

A differential equation of motion is proposed that reflects the dynamics of the vortex growth, which makes it possible to establish the boundaries of the existence of vortex flows and determine the vorticity field. Keywords:turbulence, impulse, vortex, Navier—Stokes equation, mass velocity, vorticity swgeorgy@gmail.com, toolgapodymova@yandex.ru

Gauss Principle of Least Compulsion for Relative Motion Dynamics and Differential Equations of Motion

This paper focuses on Gauss principle of least compulsion for relative motion dynamics and derives differential equations of motion from it. Firstly, starting from the dynamic equation of the relative motion of particles, we give the Gauss principle of relative motion dynamics. By constructing a compulsion function of relative motion, we prove that at any instant, its real motion minimizes the compulsion function under Gaussian variation, compared with the possible motions with the same configuration and velocity but different accelerations. Secondly, the formula of acceleration energy and the formula of compulsion function for relative motion are derived because the carried body is rigid and moving in a plane. Thirdly, the Gauss principle we obtained is expressed as Appell, Lagrange, and Nielsen forms in generalized coordinates. Utilizing Gauss principle, the dynamical equations of relative motion are established. Finally, two relative motion examples also verify the results' correctness.

Semiclassical resolution of the black hole singularity inspired in the minimal uncertainty approach

We propose a new lapse function that simplifies the Hamiltonian constraint, describing the interior of the black hole in terms of the Ashtekar-Barbero variables, into a more straightforward form. The new Hamiltonian leads to different equations of motion than those found in the literature, but through a suitable transformation between temporal parameters, it is found that such a choice leads us to the classical solutions of the Schwarzschild metric, still preserving the physical singularity. In order to resolve this singularity, and inspired by the minimal uncertainty approach, we modify the classical algebra between the dynamic variables of the model, imposing an effective dynamics within the black hole. As a consequence, one of the dynamic variables, denoted by pb, acquires a minimum value at the singularity t=0, and on the other hand, the variable related to the radius of the 2-sphere, pc, leads to the resolution of the classical singularity of the black hole by replacing it with a bounce that connects the interior of the black hole with the interior of the white hole. This bounce occurs in the Planck-scale region, where a new event horizon manifests. Upon crossing this horizon, the nature of the interval changes from spatial to temporal outside the white hole.

Perturbation to Symmetries and Adiabatic Invariants for Generalized Birkhoffian Systems on Time Scales

Time scale is a new and powerful tool for dealing with complex dynamics problems. The main result of this study is the exact invariants and adiabatic invariants of the generalized Birkhoffian system and the constrained Birkhoffian system on time scales. Firstly, we establish the differential equations of motion for the above two systems and give the corresponding Noether symmetries and exact invariants. Then, the perturbation to the Noether symmetries and the adiabatic invariants for the systems mentioned above under the action of slight disturbance are investigated, respectively. Finally, two examples are provided to show the practicality of the findings.

A vertical track nonlinear energy sink

Eliminating the effects of gravity and designing nonlinear energy sinks (NESs) that suppress vibration in the vertical direction is a challenging task with numerous damping requirements. In this paper, the dynamic design of a vertical track nonlinear energy sink (VTNES) with zero linear stiffness in the vertical direction is proposed and realized for the first time. The motion differential equations of the VTNES coupled with a linear oscillator (LO) are established. With the strong nonlinearity considered of the VTNES, the steady-state response of the system is analyzed with the harmonic balance method (HBM), and the accuracy of the HBM is verified numerically. On this basis, the VTNES prototype is manufactured, and its nonlinear stiffness is identified. The damping effect and dynamic characteristics of the VTNES are studied theoretically and experimentally. The results show that the VTNES has better damping effects when strong modulation responses (SMRs) occur. Moreover, even for small-amplitude vibration, the VTNES also has a good vibration suppression effect. To sum up, in order to suppress the vertical vibration, an NES is designed and developed, which can suppress the vertical vibration within certain ranges of the resonance frequency and the vibration intensity.

Free flexural vibrations of standard wide-flange H-beams with consideration of the shear effect

The subject of the paper is a simply supported standard wide-flange H-beam. Cross sections of this beam is analytically described as a three-layer structure. The shear effect in its succesive layers is taking into account with consideration of the classical shear stress formula called Zhuravsky shear stress. Based on Hamilton’s principle, two differential equations of motion are obtained. These equations are analytically solved and the fundamental natural frequency of flexural vibration for this beam is derived. Examplary calculations are carried out for selected five I-beams.

Free vibration Analysis of Beams with Functional Gradient Materials with Cracks

The article presents the development of a dynamic model for a functionally graded material (FGM) beam incorporating cracks. Initially, it assumes an exponential distribution of material properties along the thickness direction of the beam and simulates the opening crack using a zero-mass rotational spring model. This approach enables the calculation of bending stiffness and local flexibility at the cracked section. Subsequently, drawing upon Timoshenko beam theory and von Kármán geometric nonlinear theory, the study formulates the energy equation of the beam and establishes the partial differential control equations for the cracked FGM using Hamilton's principle. The method of separation of variables is employed to discretize the partial differential motion equations into ordinary differential motion equations. Beam functions serve as mode functions, whose unknown coefficients are determined according to the boundary and continuity conditions, thereby yielding the natural frequencies and mode shapes of the cracked FGM beam. Numerical analysis is conducted to evaluate the impact of boundary conditions, the relative position of cracks, and the length-to-thickness ratio on the natural frequencies of the cracked FGM beam.