Differential Equations Of Motion Research Articles

A derivation of the Schrödinger equation from Feynman’s path-integral formulation of quantum mechanics

The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.

A semi-analytical methodology for predicting the vibroacoustic response of functionally graded nanoplates under thermal loads

In this investigation, the analysis of the nonlinear vibroacoustic and sound transmission loss behaviors of nanoplates made of functionally graded materials considering the nonlocal strain gradient theory is presented. It is presumed that the properties of the functionally graded nanoplate are in the form of the simple power law scheme and continuous along the thickness, under nonlinear temperature rise and incident oblique acoustic wave as well as the first-order shear deformation theory. For this purpose, applying Hamilton’s principle, the nonlinear partial differential equations of motion are derived by the displacement field function approach and by considering the von Kármán nonlinear strain-displacement relations. To solve the equations, using the Galerkin method, the nonlinear partial differential equations of motion lead to the Duffing equation. Then, applying the homotopy analysis method, the equation of the transverse movement of the nanoplate is solved semi-analytically to obtain the nonlinear frequencies. Lastly, the nonlinear vibration and acoustic response of functionally graded nanoplate are investigated by considering the variation of the significant factors such as material gradient indices, nonlocal parameters, length scale parameters, aspect ratio, dimensionless amplitude, nonlinear temperature changes, and sound characteristics of the functionally graded nanoplate.

Modelling of non-linear elastic constitutive relationship and numerical simulation of rocks based on the Preisach–Mayergoyz space model

SUMMARY The existence of pores, cracks and cleavage in rocks results in significant non-linear elastic phenomena. One important non-linear elastic characteristic is the deviation of the stress–strain curve from the linear path predicted by Hooke's law. To provide a more accurate description of the non-linear elastic characteristics of rocks and to characterize the propagation of non-linear elastic waves, we introduce the Preisach–Mayergoyz space model. This model effectively captures the non-linear mesoscopic elasticity of rocks, allowing us to observe the stress–strain and modulus–stress relationships under different stress protocols. Additionally, we analyse the discrete memory characteristics of rocks subjected to cyclic loading. Based on the Preisach–Mayergoyz space model, we develop a new non-linear elastic constitutive relationship in the form of an exponential function. The new constitutive relationship is validated through copropagating acousto-elastic testing, and the experimental result is highly consistent with the data predicted by the theoretical non-linear elastic constitutive relationship. By combining the new non-linear elastic constitutive relationship with the strain–displacement formula and the differential equation of motion, we derive the non-linear elastic wave equation. We numerically solve the non-linear elastic wave equation with the finite difference method and observe two important deformations during the propagation of non-linear elastic waves: amplitude attenuation and dispersion. We also observe wave front discontinuities and uneven energy distribution in the 2D wavefield snapshot, which are different from those of linear elastic waves. We qualitatively explain these special manifestations of non-linear elastic wave propagation.

E-PINN: A fast physics-informed neural network based on explicit time-domain method for dynamic response prediction of nonlinear structures

The physics-informed neural networks (PINNs), serving as the surrogate models, have emerged in the dynamic response prediction of nonlinear systems. However, in the conventional PINN model, the global differential equation of motion of the nonlinear system is directly integrated into the loss function, which requires the network to capture the intricate global dynamic evolution mechanism of the system and thus poses significant challenges for network learning. In this study, inspired by the explicit time-domain method (ETDM), a novel PINN based on ETDM, called E-PINN, is proposed to address the challenges arising from the network learning of the existing PINNs. A lightweight long short-term memory (LSTM) module is trained to learn the nonlinear evolution mechanism of restoring forces, while a single convolutional layer (SCL) module is utilized to reflect the linear evolution mechanism of the primary structure under the combined action of the external excitations and the nonlinear restoring forces. The weights of the SCL module can be directly retrieved from the discrete convolutional formulation of ETDM, and only the parameters in the LSTM module need to be optimized through a loss function solely based on the nonlinear restoring forces, thereby enabling easy network learning of the proposed E-PINN by decoupling the linear and nonlinear evolution mechanisms embedded in the nonlinear system. Three numerical examples are investigated by the present approach, and the results demonstrate the superior training efficiency and prediction accuracy of E-PINN in dynamic response prediction of nonlinear systems.

Gyroscopic rotor dynamics simulation with anisotropy of elastic and damping characteristics of the support

In the work, the differential equations of motion of the gyroscopic rotor, built taking into account the anisotropy of stiffness and damping of the flexible support, are solved analytically, by the method of harmonic balance, convenient for obtaining separately amplitude-frequency and phase-frequency characteristics in the direction of oscillations. The equations of the non-stationary process are obtained by the method of changing amplitudes. It has been found that when the linear stiffness of the elastic support is different in two orthogonal directions, two critical velocities and the corresponding resonant regions arise. In the area of each critical speed, there are two amplitude-frequency curves of oscillations of the main direction and the direction perpendicular to it, respectively. The geometric nonlinearity of damping suppresses the elevations of these amplitude-frequency curves more significantly than linear damping. If only one of the two directions has a damping nonlinearity, then its effect is on the amplitude-frequency curves of the corresponding critical speed. It is more efficient to control resonant amplitudes for smooth resonant transitions by enhancing the linear damping with geometrically nonlinear damping. The results of the analytical solution of the equations of motion agree well with the results of direct modeling and experimental studies.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

A General Method for Solving Differential Equations of Motion Using Physics-Informed Neural Networks

The physics-informed neural network (PINN) is an effective alternative method for solving differential equations that do not require grid partitioning, making it easy to implement. In this study, using automatic differentiation techniques, the PINN method is employed to solve differential equations by embedding prior physical information, such as boundary and initial conditions, into the loss function. The differential equation solution is obtained by minimizing the loss function. The PINN method is trained using the Adam algorithm, taking the differential equations of motion in structural dynamics as an example. The time sample set generated by the Sobol sequence is used as the input, while the displacement is considered the output. The initial conditions are incorporated into the loss function as penalty terms using automatic differentiation techniques. The effectiveness of the proposed method is validated through the numerical analysis of a two-degree-of-freedom system, a four-story frame structure, and a cantilever beam. The study also explores the impact of the input samples, the activation functions, the weight coefficients of the loss function, and the width and depth of the neural network on the PINN predictions. The results demonstrate that the PINN method effectively solves the differential equations of motion of damped systems. It is a general approach for solving differential equations of motion.

Studies on multi-frequency synchronization of two pairs of counter-rotating exciters in a split drive double-body vibrating system

In this article, the multi-frequency synchronization characteristics and stability of a split drive double-body vibrating system with two rigid frames (RFs), driven by two pairs of counter-rotating exciters symmetrically mounted on two different RFs, are investigated in detail by theory, numeric, and simulation. Based on Lagrange’s equations, the differential equations of motion of the system are deduced. The theoretical criteria for implementing foundation frequency, double-frequency, and triple-frequency synchronization of two pairs of exciters are obtained by applying the asymptotic method. The stability criteria of the synchronous states are derived according to the Routh–Hurwitz criterion. Then, the synchronous and stable regions and stability ability coefficients of the system are analyzed qualitatively by numeric. The simulations are conducted to verify the validity of the theoretical methods used and the correctness of corresponding theoretical results. It is shown that, under the precondition of realizing the multi-frequency synchronization, the ideal working point of the system should be set in Region I to obtain the larger and more stable superposition vibration amplitude between two RFs in engineering. The present work can provide theory guidance for designing new types of vibrating equipment with two different driving frequencies.

D’Alembert–Lagrange Principle in Symmetry of Advanced Dynamics of Systems

The D’Alembert–Lagrange principle is a fundamental concept in analytical mechanics that simplifies the analysis of multi-degree-of-freedom mechanical systems, facilitates the dynamic response prediction of structures under various loads, and enhances the control algorithms in robotics. It is essential for solving complex problems in engineering and robotics. This theoretical study aims to highlight the advantages of using acceleration energy to obtain the differential equations of motion and the generalized driving forces, compared to the classical approach based on the Lagrange equations of the second kind. It was considered a mechanical structure with two degrees of freedom (DOF), namely, a planar robot consisting of two homogeneous rods connected by rotational joints. Both the classical Lagrange approach and the acceleration energy model were applied. It was noticed that while both approaches yielded the same results, using acceleration energy requires only a single differentiation operation, whereas the classical approach involves three such operations to achieve the same results. Thus, applying the acceleration energy method involves fewer mathematical steps and simplifies the calculations. This demonstrates the efficiency and effectiveness of using acceleration energy in dynamic system analysis. By incorporating acceleration energy into the model, enhanced robustness and accuracy in predicting system behavior are achieved.

Synchronization and stability of a vibrating system with two rigid frames driven by two groups of coaxial rotating exciters

This article explores the synchronization, stability and motion characteristics of the generalized dynamical model with two rigid frames (RFs) driven by two groups (even number) of coaxial rotating exciters. In light of this system’s generalized coordinates, we use Lagrange’s equations to derive the generalized differential equation of motion. The responses of absolute and relative motion of the generalized system are obtained using the transfer function method. The synchronization and stability criteria of multiple exciters are derived using the average method and Hamilton’s theory, respectively. Taking a dynamical model driven by two pairs of exciters as an example, the localized studies of generalized results are carried out. The stable synchronous solutions for phase differences and excitation frequency, the stability ability coefficient curves and the response curves are graphically presented considering the effect of two crucial dimensionless parameters on the stable synchronous states. The simulation results of the specific system are obtained using the fourth-order Runge-Kutta algorithms and compared with the numerical qualitative analysis results to reveal the high consistency between them and clarify the used methods’ effectiveness. The strength of this work stems from its use in the field of high power and large scale self-synchronization vibrating machines.

Counter-rotating synchronization theory and experiment of tri-rotor actuated with dual-frequency considering adaptive global sliding mode control strategy

In petroleum drilling engineering, self-synchronous vibration screen with single-frequency actuation is difficult to implement the scheduled synchronous behavior caused by its invariance and instability of dynamic characteristics, which usually leads to reduction of the production efficiency. Hence, to solve the problem mentioned above, a dual-frequency counter-rotating synchronization control method among three exciters is presented by combining adjacent cross-coupling control (ACCC) structure with global sliding mode control (GSMC) theory. Firstly, motion differential equation of the vibration system in each direction is derived according to mathematical model of the tri-rotor vibration machine. Secondly, considering exponential reaching law and adaptive control algorithm, the adjacent cross-coupling controller and the vector controller of each induction motor are investigated to achieve the stable zero phase difference synchronization between double-frequency rotors. Subsequently, Runge-Kutta method is introduced to establish the electromechanical coupling control model and prove validity of the control theory. Meanwhile, it is also discussed that the rotors cannot obtain an ideal self-synchronization state since velocity of high-frequency motor is not strictly equal to twice that of low-frequency motor. Finally, based on the experimental test scheme, an experimental prototype related to controlled multi-rotor synchronization vibration system is designed independently, which further verifies the validity of theory and computer simulation. Research shows that the desirable synchronous state of the vibration system can be accurately controlled via the designed control strategy, and the control system can implement diversity of the dynamical characteristics, due to the existence of strong robustness.

Simulating droplet distribution characteristics in sprinkler irrigation using a modified ballistic model under multifactor coupling

External factors affecting the processes of sprinkler irrigation water flow generation, flight, and landing have not been thoroughly considered in existing ballistic models. This result indicates that ballistic models with better prediction effects under specific conditions are not sufficient for extension to multi-factor coupled scenarios in large-scale farmlands. Therefore, wind, evaporation, surface slope, and tilted sprinkler riser factors were comprehensively considered in this study. Differential equations for jet and droplet motion under the influence of wind, differential equations of droplet evaporation, sprinkler riser deflection angle matrix, and surface slope angle matrix were constructed to establish a droplet distribution model for sprinkler irrigation considering multifactor coupling using MATLAB 2018a software. The results showed that, under different working conditions, the data points of the droplet landing diameter, velocity, and angle were distributed near the 1:1 line. The Nash efficiency coefficients (NSE) for the droplet landing diameter, velocity, and angle varied from 0.821 to 0.932, 0.616 to 0.931, and 0.770 to 0.911, respectively. The increase in slope resulted in droplets with diameters larger than 4.63 mm concentrating on the land in the reverse slope direction. When the ambient temperature increases from 10 to 45 °C and the total evaporation rate increases from 0.45 to 4.37 %, the larger droplets have a larger area of contact with the air, and the higher the temperature, the greater the energy loss to the larger droplet diameters. The higher the wind speed, the more droplets in the downwind direction fall to the ground at a smaller landing angle, which can easily increase the risk of soil shear damage. If the sprinkler riser was tilted east, the droplets on both the east and west sides tended to be distributed centrally; the maximum droplet landing velocity occurred on the east side (tilted side), and the maximum droplet landing angle occurred on the west side. This study considers various factors that may affect the motion of sprinkler irrigation water flow, extends the application scenarios of the theoretical model, and improves the applicability of the theoretical model for sprinkler irrigation droplet motion in more complex and practical agricultural environments.

ε-Nash Equilibrium of Pursuer–Evader–Defender Missile Navigation Dynamic Games

This research is dedicated to developing a min–max robust control strategy for a dynamic game involving pursuers, evaders, and defenders in a multiple-missile scenario. The approach employs neural dynamic programming, utilizing multiple continuous differential neural networks (DNNs). The competitive controller devised addresses the robust optimization of a joint cost function that relies on the trajectories of the pursuer–evader–defender system, accommodating an uncertain mathematical model while adhering to control restrictions. The dynamic programming min–max formulation facilitates robust control by accounting for bounded modeling uncertainties and external disturbances for each game component. The value function of the Hamilton–Jacobi–Bellman (HJB) equation is approximated by a DNN, enabling the estimation of the closed-loop formulation for the joint dynamic game with state restrictions. The controller’s design is grounded in estimating the state trajectory under the worst possible uncertainties and perturbations, providing a robustness factor through the robust neural controller. The learning law class for the time-varying weights in the DNN is generated by studying the HJB partial differential equation for the missile motion for each player in the dynamic game. The controller incorporates the solution of the obtained learning laws and a time-varying Riccati equation, offering an online solution to the control implementation. A recurrent algorithm, based on the Kiefer–Wolfowitz method, adjusts the initial conditions for the weights to satisfy the final condition of the given cost function for the dynamic game. A numerical example is presented to validate the proposed robust control methodology, confirming the optimization solution based on the DNN approximation for Bellman’s value function.

Active vibration control effect of 3D printed cruciform honeycomb laminates based on fiber-reinforced shape memory polymer

Honeycomb structures offer the benefits of being lightweight and having high out-of-face compressive stiffness, making them a popular choice for aerostructures. The present study delineates the design and active vibration control effect for fiber-reinforced cruciform honeycomb laminates (CHLs) characterized by variable parameters. Employing a continuous fiber 3D printer, carbon fiber filament and shape memory polymers (SMPs) were utilized in the fabrication of the specimens. Differential equations of motion for CHLs were deduced from Kirchhoff plate theory. To ascertain their frequency response, these laminates were subjected to a sinusoidal swept frequency excitation. Subsequently, the classical proportional integral derivative (PID) program was employed to scrutinize the efficacy of closed-loop active vibration control on CHLs. The investigation extended to analyzing the impact of active vibration control across laminates with disparate parameters, including an examination of the convergence speed during vibration control procedures. A comparative analysis of simulated versus experimental data revealed a substantial consonance between the two, thereby corroborating the accuracy of the experimental findings. This study offers crucial insights and reference value for the utilization of CHLs in the aerospace industry.

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.