Fractional formulation of Podolsky Lagrangian density

Lagrangians which depend on higher-order derivatives appear frequently in many areas of physics. In this paper, we reformulate Podolsky's Lagrangian in fractional form using left-right Riemann-Liouville fractional derivatives. The equations of motion are obtained using the fractional Euler Lagrange equation. In addition, the energy stress tensor and the Hamiltonian are obtained in fractional form from the Lagrangian density. The resulting equations are very similar to those found in classical field theory.

Research on Virtual Color Restoration of Complex Building System Based on Discrete Wavelet Transform

At present, image restoration has become a research hotspot in computer vision. The purpose of digital image restoration is to restore the lost information of the image or remove redundant objects without destroying the integrity and visual effects of the image. The operation of user interactive color migration is troublesome, resulting in low efficiency. And, when there are many kinds of colors, it is prone to errors. In response to these problems, this paper proposes automatic selection of sample color migration. Considering that the respective gray-scale histograms of the visual source image and the target image are approximately normal distributions, this paper takes the peak point as the mean value of the normal distribution to construct the objective function. We find all the required partitions according to the user’s needs and use the center points in these partitions as the initial clustering centers of the fuzzy C-means (FCM) algorithm to complete the automatic clustering of the two images. This paper selects representative pixels as sample blocks to realize automatic matching of sample blocks in the two images and complete the color migration of the entire image. We introduced the curvature into the energy functional of the p-harmonic model. According to whether there is noise in the image, a new wavelet domain image restoration model is proposed. According to the established model, the Euler–Lagrange equation is derived by the variational method, the corresponding diffusion equation is established, and the model is analyzed and numerically solved in detail to obtain the restored image. The results show that the combination of image sample texture synthesis and segmentation matching method used in this paper can effectively solve the problem of color unevenness. This not only saves the time for mural restoration but also improves the quality of murals, thereby achieving more realistic visual effects and connectivity.

Multi-Gene Genetic Programming-Based Identification of a Dynamic Prediction Model of an Overhead Traveling Crane

This paper proposes a multi-gene genetic programming (MGGP) approach to identifying the dynamic prediction model for an overhead crane. The proposed method does not rely on expert knowledge of the system and therefore does not require a compromise between accuracy and complex, time-consuming modeling of nonlinear dynamics. MGGP is a multi-objective optimization problem, and both the mean square error (MSE) over the entire prediction horizon as well as the function complexity are minimized. In order to minimize the MSE an initial estimate of the gene weights is obtained by using the least squares approach, after which the Levenberg–Marquardt algorithm is used to find the local optimum for a k-step ahead predictor. The method was tested on both a simulation model obtained from the Euler–Lagrange equation with friction and the experimental stand. The simulation and the experimental stand were trained with varying control inputs, rope lengths and payload masses. The resulting predictor model was then validated on a testing set, and the results show the effectiveness of the proposed method.

Dynamic Modeling and Analysis of Nonlinear Compound Planetary System

In the vehicle composite planetary gear transmission system, nonlinear excitations such as time-varying meshing stiffness, backlash and comprehensive error would lead to large vibration and noise, uneven load distribution, unstable operation and other problems. To address these issues, this work focuses on compound planetary gears and develops the bending-torsion coupling nonlinear dynamic model of the system based on the Lagrange equation. There are internal and external multi-source excitations applied to the system. This model is used to study the bending-torsion coupling meshing deformation relationship of each meshing pair along with the translational and torsional directions. The natural frequencies and vibration modal characteristics of the system are extracted from the model, and the influence of rotational inertia and meshing stiffness on the inherent characteristics of the system are studied. The coupling vibration characteristics of the system under operating condition are analyzed in terms of the inherent characteristics and time–frequency characteristics of the system. The simulation results exhibit that the planetary gear system has three modes. The change in natural frequency trajectory has two phenomena: modal transition and trajectory intersection. The main frequencies include engine rotating frequency, meshing frequency and its double frequency, and the rotation frequency and harmonic frequency of the engine have a great influence on the vibration response of the system. Finally, the virtual prototype of the composite planetary system is used to verify the accuracy of the established model from speed, inherent characteristics, meshing force and frequency composition.

A novel electro-hydraulic servo steering system for Ackermann steering of heavy multi-axle vehicle

Ackermann steering is important for the steering performance of heavy multi-axle vehicle. When Ackermann steering condition is not satisfied, it will lead to abnormal tire wear. Traditional trapezoidal mechanism of heavy multi-axle vehicle is a single degree of freedom (DOF) mechanism, which is difficult to completely realize Ackermann steering. In this paper, a new two DOF electro-hydraulic servo steering system (TDEHSSS) by using a variable length tie rod is proposed for solving the issue. First, a complex nonlinear dynamic model of TDEHSSS is established. This model includes the two DOF mechanical model based on a Lagrange equation, the valve-controlled double steering power cylinders model and the valve-controlled tie rod cylinder model. Then, a simulation model is built through MATLAB/Simulink and the simulation results show that TDEHSSS can realize the proposed requirement. At last, a test bench is founded to verify model. It is indicated that the simulation and experimental curves are consistent, showing that mathematical model is in accordance with the experimental system. This research is valuable for analyzing TDEHSSS, designing advanced controllers, and finally realizing Ackermann steering for heavy multi-axle vehicles.

Energy minimizers for an asymptotic MEMS model with heterogeneous dielectric properties

A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This asymptotic model inherits the dielectric properties of the insulating layer. It involves the electrostatic potential in the device and the deformation of the elastic plate defining the geometry of the device. The electrostatic potential is given by an elliptic equation with mixed boundary conditions in the possibly non-Lipschitz region between the two plates. The deformation of the elastic plate is supposed to be a critical point of an energy functional which, in turn, depends on the electrostatic potential due to the force exerted by the latter on the elastic plate. The energy functional is shown to have a minimizer giving the geometry of the device. Moreover, the corresponding Euler–Lagrange equation is computed and the maximal regularity of the electrostatic potential is established.

Study On the Bistable Vibration Behaviour of a Rod Fastened Rotor-Bearing System

Based on the Lagrange equation, the motion equation of a rod fastened rotor-bearing system considering the damping of the contact interface is established. The numerical method is employed for numerical analysis. The bifurcation diagram, time series, frequency waveform, phase spectrum and Poincare map are used to illustrate the nonlinear dynamic behaviour. The transient responses during acceleration and deceleration are calculated to reveal the dynamic behaviour of the system. The numerical results hold that since the oil film is nonlinear, the system presents obvious bistable behaviour and a jumping phenomenon. In addition, a test bench of the rod fastened rotor-bearing system is built. The bistable behaviour and jumping phenomenon are experimentally proven, and the effect of the eccentric distance of the rotor on the bistable behaviour is experimentally explored. The results of this paper can be used for the basic design and fault diagnosis of rod fastened rotors.

Advanced Proportional-Integral-Derivative Control Compensation Based on a Grey Estimated Model in Dynamic Balance of Single-Wheeled Robot

This paper aims to design a one-wheeled robot as regards its pitch freedom and balance control on the one hand and to assess the application feasibility of the GM (1,1) swing estimation controller on the other. System control focuses mainly on one-wheeled robot stability, body swings in position, and speed control. Mathematical modeling and GM (1,1) prediction control are under investigation. The mathematical modeling is firstly conducted through referencing to the Newtonian mechanics and the Lagrange equation, from which the robot transfer function and state-space differential equation are derived. Next, the linear quadratic regulator is applied as the control rule at the balance point. Applying GM (1,1) to assess the robot gyro signal at a dynamic state is a discussion. Next, model reference estimation control is processed, and a mathematical model of the balance control method is completed. Finally, a simulation is conducted to verify the feasibility of the GM (1,1) estimation reference model. The linear quadratic regulator, which is credited with tenacity, can provide pitch swing and balance control of the one-wheeled robot.

Theoretical Study and Analysis of a Device for Pickling and Moving Bulk Materials

Theoretical studies of the operation of a device for pickling and moving bulk materials are discussed. A particular analysis of the Lagrange equation for the motion of a particle in a spiral-helical device is described. The images are presented with all forces and velocity vectors acting on a particle from the side of the spiral screw and the inner surface of the device body during its movement. The expression for the angle θ, which characterizes the geometric parameters of the spiral and cylindrical body, as well as the size of the material particles in the device, has been investigated and transformed.

Modeling and Analysis of Axial Dynamics of Cable Arrangement Device Based on MATLAB

Aiming at the current insufficient dynamic analysis of the cable arrangement device, the axial dynamics analysis of the cable arrangement device is carried out in combination with the force characteristics of the cable arrangement device. The axial dynamic model of cable arrangement device is established by using spring-damping model, and the dynamic equation is established by using Lagrange equation. The influence of system parameters of cable arrangement device on axial first-order natural frequency is analyzed by numerical method. By fitting and loading the external excitation of the cable arrangement device, the axial dynamic response of the cable arrangement device under different axial forces is obtained. Through numerical results, the influence laws of the position and mass of the traveling mechanism and the support stiffness of the lead screw on the axial first-order natural frequency are obtained, It is found that the axial displacement of the cable arrangement device under axial force is very small, and the cable arrangement device has strong retention of dynamic characteristics. The results have certain guiding significance for the structural design and application environment of cable arrangement device.