Principles Of Differential Geometry Research Articles

Study on contact characteristics of double-disk arc-contact lapping for cylindrical rollers based on influence coefficient method

The double-disk arc-contact lapping (DDACL) method is a novel for precision machining of the rolling surface of cylindrical roller. The contact characteristics of DDACL process can affect the lapping quality of the bearing roller. However, there is a limited amount of study on the contact theory of DDACL. Therefore, the objective of this research is to create a semi-analytical modeling approach that can effectively and accurately forecast the contact characteristics of DDACL. In the initial step, the spatial coordinate transformation was employed to establish accurate mathematical models that represents the surfaces of the cylindrical roller and arc groove. Subsequently, the potential contact area was determined and subdivided into discrete elements based on the differential geometry principles. The accurate flexibility of the potential contact area is achieved by utilizing the interpolation theory and considering both the global and local deformations of the roller and groove. By employing the influence coefficient method and the force-deformation coordination relation, the deformation and contact force can be calculated. Finally, the accuracy and efficiency of this calculation is confirmed through a comparison between the finite element method (FEM) and the proposed method.

Discrete differential geometry-based model for nonlinear analysis of axisymmetric shells

In this paper, we propose a novel one-dimensional (1D) discrete differential geometry (DDG)-based numerical method for geometrically nonlinear mechanics analysis (e.g., buckling and snapping) of axisymmetric shell structures. Our numerical model leverages differential geometry principles to accurately capture the complex nonlinear deformation patterns exhibited by axisymmetric shells. By discretizing the axisymmetric shell into interconnected 1D elements along the meridional direction, the in-plane stretching and out-of-bending potentials are formulated based on the geometric principles of 1D nodes and edges under the Kirchhoff–Love hypothesis, and elastic force vector and associated Hessian matrix required by equations of motion are later derived based on symbolic calculation. Through extensive validation with available theoretical solutions and finite element method (FEM) simulations in literature, our model demonstrates high accuracy in predicting the nonlinear behavior of axisymmetric shells. Importantly, compared to the classical theoretical model and three-dimensional (3D) FEM simulation, our model is highly computationally efficient, making it suitable for large-scale real-time simulations of nonlinear problems of shell structures such as instability and snap-through phenomena. Moreover, our framework can easily incorporate complex loading conditions, e.g., boundary nonlinear contact and multi-physics actuation, which play an essential role in the use of engineering applications, such as soft robots and flexible devices. This study demonstrates that the simplicity and effectiveness of the 1D discrete differential geometry-based approach render it a powerful tool for engineers and researchers interested in nonlinear mechanics analysis of axisymmetric shells, with potential applications in various engineering fields.

Investigation of loaded contact characteristics of planetary roller screw mechanism based on influence coefficient method and machine learning

The performance of the planetary roller screw mechanism (PRSM) is directly determined by the mechanical behavior of the loaded contact. The traditional method based on a finite element modeling to describe the loaded contact characteristics (LCCs) is computationally expensive, while the use of Hertzian contact theory and the thread connection model results in a loss of computational accuracy. As a solution, this study aims to develop a semi-analytic modeling technique that can predict the LCCs of PRSM accurately and efficiently. To achieve this, the study first establishes a mathematical model using the spatial coordinate transformation theory to accurately describe the thread surface. The potential contact area is discretely divided using the differential geometry principle. Then, the linear deformation and nonlinear deformation are matched using node interpolation technique to obtain the accurate compliance of the potential contact area. Finally, the contact forces for all activated contact points are derived by a nonlinear iterative algorithm based on the influence coefficient method. The performance of the proposed model is verified by numerical examples. Further, the sensitivity weights of the geometric parameters to the LCCs are identified utilizing machine learning, and a preferred zone is also constructed to optimize these parameters. This study provides a valuable tool for evaluating and improving the LCCs of PRSM in engineering practice.

Subscripto multiplex: A Riemannian symmetric positive definite strategy for offline signature verification

The human handwritten signature is considered to be a significant biometric trait. In the case of offline signatures, the problem is addressed as an image recognition task. On the other hand, the visual representation of symmetric positive definitive matrices, usually by means of the covariance descriptor of the image feature maps, forms a specific Riemannian manifold with a widespread usage and a favorable performance in a plethora of applications. Surprisingly, no records of offline-signature-verification-oriented research in the space of symmetric positive definitive matrix have been found up to now. In this work, we propose, for the first time in offline signature-verification literature, mapping of handwritten signature images in points of the tangent space of a connected symmetric positive definitive manifold for verification purposes. Furthermore, based on the principles of differential geometry, we address the notorious limited training problem of offline signature verification in this manifold by proposing two different feature augmentation methods. The efficiency of the proposed method is evaluated using three popular datasets of Western and Asian origin. Error rates against skilled and random forgery in both baselines as well augmentation scenarios are strong indicators of the informative and highly discriminative nature of symmetric positive definitive manifold oriented representation.

Input-output Feedback Linearization Control for SM-PMSM

This paper introduces a velocity control strategy for Surface-Mounted Permanent Magnet Synchronous Motors SM-PMSM using exact linearization and input-output decoupling techniques, which are rooted in the principles of differential geometry. The primary aim of this control approach is to establish a static state feedback mechanism and to convert the nonlinear PMSM model into a linear, decoupled, and controllable system. Initially, the state model that represents the PMSM dynamics within the d-q reference frame is defined. Subsequently, the process of designing the control through linearization and input-output decoupling is outlined. Lastly, the synthesis of the compensator is grounded in the pole placement method, aiming to drive the direct current towards zero and ensure optimal torque operation. Simulation outcomes conducted on Matlab/Simulink demonstrate the efficacy of the speed control strategy, which is facilitated by a straightforward algorithm for practical implementation. However, it is inadequate against variations in machine parameters and load torque disturbances.

Simulation analysis and experiment of instantaneous temperature field for grinding face gear with a grinding worm

In this paper, a transient temperature calculation model for simulating grinding face gear with a grinding worm is proposed. Firstly, according to the principle of differential geometry, the equations for determining contact length and contact width of the instantaneous grinding area between the worm grinding wheel and the face gear are deduced. Based on the method for the single abrasive particle cutting force and grinding heat, a mathematical model for the grinding force and grinding heat of the face gear is established, and the theoretical calculation formula of the instantaneous maximum temperature for grinding teeth of the face gear is obtained. Secondly, a finite element model of temperature field for face gear grinding with moving heat load of grinding worm wheel is established, and the influence law of grinding process parameters on tooth surface temperature is studied. Finally, the validity of the temperature field simulation model is verified by multi-group temperature testing experiments for face gear grinding.

Aberration characteristic correlation configuration optimization of a conformal dome based on the von Karman surface.

In this paper, the von Karman surface is used in the configuration design of the infrared conformal dome to improve its aerodynamic performance. The principle of differential geometry is used to study the geometric characteristics of the von Karman dome. Additionally, by using ray tracing, the geometric aberrations and wave aberrations of the von Karman dome are analyzed. Further, considering the geometric characteristics and aberration characteristics, an optimization method for the configuration of the von Karman dome is proposed. To prove the effectiveness of the optimization method, the aberrations introduced by the conformal dome after the configuration optimization and the original von Karman dome are compared. The comparison showed that the geometric aberration of the optimized conformal dome is reduced by 43.68%. The optimization method can significantly correct the aberration introduced by the von Karman dome and improve the guidance capability of infrared detection technology.

Analysis of the Torque Loss of High-Speed Transmission Mechanism with a Stacked Roller Set

Two-dimensional pumps have broad application prospects in aerospace. However, the performance of the pump is degraded because of the clearance problem of the current 2D transmission mechanism. In order to eliminate the clearance between the cam rail and the rollers, a high-speed transmission mechanism with a stacked roller set is proposed. The stacked roller set is compressed by the load pressure. The axial inertia force is balanced when the transmission mechanism works at high speed, via the equal acceleration and reverse movement of two cam rail sets. Thus, the transmission mechanism meets the high-speed demand. In this paper, the mathematical model of the transmission mechanism is established based on the enveloping surface theory and the differential geometry principle. Afterwards, numerical analysis of the mathematical model is performed based on MATLAB, combined with the experiment, to study the influence of load pressure and rotational speed on the torque loss. Then, the torque characteristics of the transmission mechanism is obtained. According to a test, the deviation between theoretical data and experimental data is 11.9%; therefore, the mathematical model can predict the torque of the transmission mechanism effectively. It is concluded that the torque loss of the transmission mechanism increases linearly with the load pressure, and the rotational speed has a slight effect on the torque loss.

A novel solder joint diagnosis algorithm using differential geometry approach in high-density flexible integrated circuit substrates

To provide a system with high-density solder joint 3D information acquisition and improve the comprehensive performance of automatic optical inspection systems, a novel diagnosis system for solder joints of high-density flexible integrated circuit substrates is proposed in this paper. The differential geometry principle is introduced to analyze the appearance identity of solder joints, and the diagnosis is handled as a pattern-recognition problem with the Frenet frame-matching approach. First, a high-precision 3D information-acquisition system constituted of two CCD digital cameras, each with a microscope and a three-color LED illumination, is developed. Second, a discrete estimation scheme based on the differential geometry theory is proposed to achieve curvature measurement, and then the shape of the solder joints is characterized by the crest line representation method. Finally, the classification strategy is to establish different models of solder joint types and make a decision based on a set of the predefined diagnosis rules. By inspecting various types of solder joints and comparing with other typical algorithms, the performance of the proposed system is evaluated. The experiment results indicate that the proposed algorithm has advantages of fast inspection speed, low misjudgment rate, which is reduced to 2.2%, and detailed classification.

Network Security Situation Prediction of Improved Lanchester Equation Based on Time Action Factor

Existing network security situational awareness assessment and prediction are not fully considered network defense utility, time factor and other indicators, and the randomness of attacks and the accuracy of the prediction of attack intentions and methods may appear both in active and passive defense this situation leads to uncertainty in attack prediction, which makes it impossible to discover defects in the network system in a timely manner, and there is a lack of effective emergency response strategies for impending or already occurring network attacks. Therefore, this paper proposes a network behavior calculation model based on the Lanchester equation of time action factor is proposed. This paper uses smooth differential manifold homeomorphic transformation to define network behavior, defines the calculation method of behavior utility based on the principle of differential geometry, combines the second linear law of the Lanchester equation and the square law, and uses the time action factor to defend the active defense. The simulation results show that the model can be used to analyze the network offensive and defensive process, and can effectively predict the network offensive and defensive results under both active and passive defense.

Research on three-dimensional modeling and cutting force prediction mathematical model of integral end mill based on differential geometry principle

Research on three-dimensional modeling and cutting force prediction mathematical model of integral end mill based on differential geometry principle

Control strategy design for forced fly-around against the tumbling target based on differential geometric principle

Forced fly-around of spacecraft is the basis for tracking and capturing space tumbling target in active debris removal mission. In this paper, the relative kinematic equation between chaser and target is first obtained with the help of classical differential geometry principle. Then, taking the attitude dynamic model of tumbling target into consideration, the finite time convergence sliding model control (FTCSMC) is designed to enable the forced fly-around in the line of sight rotation coordinate system. Finally, the effectiveness and robustness of the control strategy are verified in two typical cases by numerical simulation.

Kinematic analysis and simulation of a new type of differential micro-feed mechanism with friction.

This article presents a new micro-feed mechanism, whose main transmission component is the nut-rotary ball screw pair. The screw and nut are driven by two motors, and they rotate in the same direction, with their movements enabling micro-feeding. The main contribution of the micro-feed mechanism is to avoid the inevitable low-speed nonlinear creeping phenomenon caused by the inherent properties of traditional electromechanical servo system structure, thus realizing high precision micro-feed. In this study, the motion state of the working ball is analyzed using the principle of differential geometry, the friction at the contact points is calculated, the balance equation for force and moment is established, the influences of the screw and nut on the kinematic parameters of the ball at different velocities and the differences in the motion states of the ball in different drive modes are studied, and the mechanical efficiency of the dual-driven ball screw mechanism is calculated. The potential applications of the new micro-feed mechanism and the results of numerical analysis can be applied to advanced technology fields such as robotics, suspensions, powertrain, national defense, integrated electronics, optoelectronics, medicine, and genetic engineering, so that the new system can have a lower stable speed limit and achieve precise micro-feed control.

The Deep Problems of Symmetry in Mathematical Physics and their Definitive Resolution

Symmetry stands for a central notion in Mathematics with extremely important applications in Theoretical Physics. Mathematician Emmy Noether in 1915 famously axiomatized this notion which resoundingly made its way in fundamental physics in its chapters of Lagrangian mechanics and Hamiltonian mechanics. However there is only one dimension to symmetry in mathematics and mathematical physics: an invariance that is preserved under a particular transformation. Contrary to this approach, there exist many other dimensions to the artifact of symmetry, which, if correctly prospected, could have helped accurately answer the long-running question in mathematical physics concerning the very origin of physical symmetries and the reason for the physical mechanism of spontaneous symmetry-breaking. The ignorance of origin of observable symmetries as well as the existence of other categories of symmetry is accompanied by a misunderstanding of the very states of symmetry, which is reflected in an inexact assessment of the ultra-structures subtending mathematical symmetry. In this writing, we aim to to expose the caveats of symmetry in mathematics, specifically in Geometry, as viewed from the overarching Quanto-Geometric perspective, and to show their resolution with the application of special Differential Geometry principles emanating from the Quanto-Geometric framework.

Path Planning for Spray Painting Robot of Horns Surfaces in Ship Manufacturing

Generating paint gun path for complex free-form surfaces in shipbuilding industry to ensure uniform paint deposition is highly challenging due to their complex geometry, especially for horns surfaces. In this paper, a tool trajectory optimization method was proposed based on dip angle spray in order to solve the problems of poor coating uniformity and material waste for spraying horns surface with curved patches. A dip angle spray model was developed based on the Beta distribution model on the planar surface by principle of differential geometry. The spray path optimization models for dip angle spraying on cylindrical surfaces of three cases were proposed and compared first to define the best case which yields the best painting quality. Then we used the best strategy on a horns surface in a shipyard to evaluate the effectiveness and robustness of designed algorithm. This algorithm can also be extended to other applications.

Displacement vectors control method for orienter ribs during path correction in coiled tubing drilling

The well path easily deviates from the original design trajectory during coiled tubing drilling, which needs to be corrected with an orienter. A three-segment correction trajectory with two arc segments and a line segment is constructed based on the differential geometry principle, and the direction at the target is consistent with the direction of the original borehole. Based on the parameters of the design correction trajectory, a mathematical relationship between trajectory parameters and the displacement vectors of orienter ribs is established by using the vector analysis method. According to the principle of minimum energy, the problem of multisolvability in mathematics is avoided. Then, the control model of the displacement vectors of orienter ribs with the parameters' variation of the designed correction trajectory is established. The similarly periodic variation rule of displacements of orienter ribs with the variation of tool face angle is obtained, and the closed-loop control scheme for correction trajectory while drilling is made. The results of field test show that by using coiled tubing drilling, during the circular arc section of the designed three-segment correction trajectory, the displacement vectors of orienter ribs vary similarly periodically, which is called the directional drilling mode. During drilling in the slanted segment, the displacements of each rib are equal, which is called the keeping drilling mode. The design method of correction trajectory and the displacement vector control method of the orienter ribs are practicable during path correction in coiled tubing drilling, which is helpful to improve the efficiency of coiled tubing drilling.

Physical simulation of environmentally induced thin shell deformation

We present a physically accurate low-order elastic shell model that incorporates active material response to dynamically changing stimuli such as heat, moisture, and growth. Our continuous formulation of the geometrically non-linear elastic energy derives from the principles of differential geometry, and as such naturally incorporates shell thickness, non-zero rest curvature, and physical material properties. By modeling the environmental stimulus as local, dynamic changes in the rest metric of the material, we are able to solve for the corresponding shape changes by integrating the equations of motions given this non-Euclidean rest state. We present models for differential growth and shrinking due to moisture and temperature gradients along and across the surface, and incorporate anisotropic growth by defining an intrinsic machine direction within the material. Comparisons with experiments and volumetric finite elements show that our simulations achieve excellent qualitative and quantitative agreement. By combining the reduced-order shell theory with appropriate physical models, our approach accurately captures all the physical phenomena while avoiding expensive volumetric discretization of the shell volume.

Calculation of the Behavior Utility of a Network System: Conception and Principle

Abstract The service and application of a network is a behavioral process that is oriented toward its operations and tasks, whose metrics and evaluation are still somewhat of a rough comparison. This paper describes scenes of network behavior as differential manifolds. Using the homeomorphic transformation of smooth differential manifolds, we provide a mathematical definition of network behavior and propose a mathematical description of the network behavior path and behavior utility. Based on the principle of differential geometry, this paper puts forward the function of network behavior and a calculation method to determine behavior utility, and establishes the calculation principle of network behavior utility. We also provide a calculation framework for assessment of the network’s attack-defense confrontation on the strength of behavior utility. Therefore, this paper establishes a mathematical foundation for the objective measurement and precise evaluation of network behavior.

Rough Surface Contact

This paper studies the contact of general rough curved surfaces having nearly identical geometries, assuming the contact at each differential area obeys the model proposed by Greenwood and Williamson. In order to account for the most general gross geometry, principles of differential geometry of surface are applied. This method while requires more rigorous mathematical manipulations, the fact that it preserves the original surface geometries thus makes the modeling procedure much more intuitive. For subsequent use, differential geometry of axis-symmetric surface is considered instead of general surface (although this “general case” can be done as well) in Chapter 3.1. The final formulas for contact area, load, and frictional torque are derived in Chapter 3.2.

A GENERAL CAD/CAM MODEL FOR APT-LIKE ROTARY MILLING CUTTERS

This paper presents a general mathematical model for NC-machining an APT-LIKE rotating cutter. The design model of cutting-edge and helical groove is also developed by using the principles of differential geometry and kinematics. Based on the envelope condition, approaches for solving the inverse problems that are related to the manufacturing models are also presented. The velocities of the cutter in the radial and axial directions are then determined based on the gouging avoidance of grooves. The results of the numerical experiment indicate that the proposed systematic design method by using 2-axis NC machine setup is feasible and reliable.