A DIFFERENTIAL GEOMETRIC DESCRIPTION FOR TIME-INDEPENDENT CHETAEV'S NONHOLONOMIC MECHANICAL SYSTEM WITH UNILATERAL CONSTRAINTS

By applying the framework of the tangent bundle geometry to the method of Lagrange multi- pliers,a geometric description of Chetaev's nonholonomic systems subjected to unilateral nonholonomic con- straints trod unilateral holonomic constraints respectively in time-independent circumstances is presented.

The effect of sintering and cooling process on geometry distortion and mechanical properties transition of PTFE/Al reactive materials

In this research,the effect of the sintering and cooling process on geometry distortion and mechanical properties of PTFE/Al reactive material is investigated.Six particularly selected sintering temperatures,three different cooling modes(annealing cooling,normalizing cooling and rapid cooling),three different initial cooling temperature s,as well as six different final cooling temperatures were designed to compare the effects of sintering temperature,cooling rate,initial cooling temperature and final cooling temperature on the properties of reactive materials.Geometry distortion was quantitatively analyzed by a statistic on the dimensional changes of the specimens and microscopic morphology.A mechanical response properties transition from brittle to ductile was found and analyzed.By combining the thermodynamic properties of PTFE and unsteady heat conduction theory,mechanisms of cooling induced morphology change,temperature induced distortion and strength decrease were obtained.The results showed that the cooling rate has the most significant effect on the morphology transformation,while initial cooling temperature has more significant effect on the dimensional distortion than final cooling temperature.As to the mechanical properties transition from brittle to plastic,a more prominent effect of initial cooling temperature than cooling rate and final temperature was revealed.

A geometric framework for time-dependent mechanical systems with unilateral constraints

The description of modern differential geometry for time-dependent Chetaev nonholonomic mechanical systems with unilateral constraints is studied. By using the structure of exact contact manifold, the geometric framework of time- dependent nonholonomic mechanical systems subject to unilateral nonholonomic constraints and unilateral holonomic constraints respectively is presented.

Effect of fractures on mechanical behavior of sand powder 3D printing rock analogue under triaxial compression

In practical engineering applications,rock mass are often found to be subjected to a triaxial stress state.Concurrently,defects like joints and fractures have a notable impact on the mechanical behavior of rock mass.Such defects are identified as crucial contributors to the failure and instability of the surrounding rock,subsequently impacting the engineering stability.The study aimed to investigate the impact of fracture geometry and confining pressure on the deformation,failure characteristics,and strength of specimens using sand powder 3D printing technology and conventional triaxial compression tests.The results indicate that the number of fractures present considerably influences the peak strength,axial peak strain and elastic modulus of the specimens.Confining pressure is an important factor affecting the failure pattern of the specimen,under which the specimen is more prone to shear failure,but the initiation,expansion and penetration processes of secondary cracks in different fracture specimens are different.This study confirmed the feasibility of using sand powder 3D printing specimens as soft rock analogs for triaxial compression research.The insights from this research are deemed essential for a deeper understanding of the mechanical behavior of fractured surrounding rocks when under triaxial stress state.

Type Synthesis of 4-DOF Parallel Kinematic Mechanisms Based on Grassmann Line Geometry and Atlas Method

Many methods are proposed to deal with the type synthesis of parallel kinematic mechanisms（PKMs）, but most of them are less intuitive to some extent. Thus, to propose a concise and intuitive type synthesis method for engineering application is a very challenging issue, which should be further studied in the field. Grassmann line geometry, which can investigate the dimensions of spatial line-clusters in a concise way, is taken as the mathematic foundation. Atlas method is introduced to visually describe the degrees of freedom（DOFs） and constraints of a mechanism, and the dual rule is brought in to realize the mutual conversion of the freedom-space and constraint-space. Consequently, a systematic method based on Grassmann line geometry and Atlas method is generated and the entire type synthesis process is presented. Three type 4-DOF PKMs, i.e., 1T3R, 2T2R and 3T1R（T： translational DOF; R： rotational DOF）, are classified according to the different combinations of the translational DOFs and rotational DOFs. The type synthesis of 4-DOF PKMs is carried out and the possible configurations are thoroughly investigated. Some new PKMs with useful functions are generated during this procedure. The type synthesis method based on Grassmann line geometry and Atlas method is intuitive and concise, and can reduce the complexity of the PKMs＇ type synthesis. Moreover, this method can provide theoretical guidance for other PKMs＇ type synthesis and engineering application. A novel type synthesis method is proposed, which solves the existing methods＇ problems in terms of complicated, not intuitive and unsuitable for practical application.

INVESTIGATION INTO EFFECT OF SPRING PRESSURE ON PERFORMANCE OF BALANCED MECHANICAL SEALS

The loads acting on the sealing elements of balanced mechanical seals are analyzed. When the balance factor approaches the back pressure factor, the spring pressure will become main part of the face pressure. The leakage model of balanced mechanical seals is established on the base of M-B model for rough surface. Several GY-70 type balanced mechanical seals are tested. The influences of the spring pressure both on the leakage rate and on the friction characteristic of balanced mechanical seals are investigated. The research results indicate that as spring pressure increases, both the clear-ance between two end faces and the leakage rate will decrease, and the friction will be more serious because lubrication medium between the rotating ring and the stationary ring reduces, though the increase of the spring pressure may not be enough to change the face friction state of mechanical seals. There exists an optimum spring pressure for mechanical seal operation. Under this spring pres-sure, not only leakage rate is small, but also the seal end surfaces have a fine friction characteristic. Under different operating conditions, identical type mechanical seals may possess different spring pressure. Appropriate selection of spring pressure is valuable to realize long-period and small leakage rate operating of balanced mechanical seals.

Effect of rotary direction and speed on mechanical properties in friction stir spot welding of A6061 sheets e

Friction stir spot welding of A6061 sheets was conducted using a tool with thread pin. The hook geometries, hook formation and mechanical properties of the joints welded with different rotary directions and speeds were investigated. The results show that the hook in the joint welded in clockwise was curved upwards and that in anticlockwise was curved downwards. The hook formation was related to the plastic material flow in the joint. With increasing the rotary speed in clockwise direction, the hook moved upwards and far way from the center of the keyhole, resulting in an increase in the effective weld width and a decrease in the effective sheet thickness. Three types of fractuces were observed and they were affected by the hook geometries. The tensile shear load increased firstly and then decreased when the rotary speed increased in clockwise direction, which was related to the hook geometries.

Geometry of time-dependent PT-symmetric quantum mechanics

A new type of quantum theory known as time-dependent𝒫PT-symmetric quantum mechanics has received much attention recently.It has a conceptually intriguing feature of equipping the Hilbert space of a𝒫PT-symmetric system with a time-varying inner product.In this work,we explore the geometry of time-dependent𝒫𝒯PT-symmetric quantum mechanics.We find that a geometric phase can emerge naturally from the cyclic evolution of a PT-symmetric system,and further formulate a series of related differential-geometry concepts,including connection,curvature,parallel transport,metric tensor,and quantum geometric tensor.These findings constitute a useful,perhaps indispensible,tool to investigate geometric properties of𝒫PT-symmetric systems with time-varying system’s parameters.To exemplify the application of our findings,we show that the unconventional geometric phase[Phys.Rev.Lett.91187902(2003)],which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase,can be expressed as a single geometric phase unveiled in this work.

FULLERENE GEOMETRY PREDICTION WITH MOLECULAR MECHANICS CALCULATIONS

A serial of fullerenes had been built and the optimized geome- tries had been obtained with the energy minimization of molecular mechanics calculations according to the fact that the pentagonal number is exactly 12 in the fullerenes which consist of pentagons and hexagons.The fullerene geometry prediction could facilitate further theoretical and synthetical studies in the near future.

基于SimMechanics的可调整机构设计

在MATLAB及其SimMechanics环境中建立了可调机构的仿真模型。通过曲柄摇杆机构驱动曲柄滑块机构,可在不改变执行机构的结构参数的前提下,调整驱动机构的几何参数来控制并改善执行机构的运动特性。仿真结果证明了该方法的可行性。

A Study on Computer Consciousness on Intuitive Geometry Based on Mathematics Experiments and Statistical Analysis

In this paper, we present our research on building computing machines consciousness about intuitive geometry based on mathematics experiments and statistical inference. The investigation consists of the following five steps. At first, we select a set of geometric configurations and for each configuration we construct a large amount of geometric data as observation data using dynamic geometry programs together with the pseudo-random number generator. Secondly, we refer to the geometric predicates in the algebraic method of machine proof of geometric theorems to construct statistics suitable for measuring the approximate geometric relationships in the observation data. In the third step, we propose a geometric relationship detection method based on the similarity of data distribution, where the search space has been reduced into small batches of data by pre-searching for efficiency, and the hypothetical test of the possible geometric relationships in the search results has be performed. In the fourth step, we explore the integer relation of the line segment lengths in the geometric configuration in addition. At the final step, we do numerical experiments for the pre-selected geometric configurations to verify the effectiveness of our method. The results show that computer equipped with the above procedures can find out the hidden geometric relations from the randomly generated data of related geometric configurations, and in this sense, computing machines can actually attain certain consciousness of intuitive geometry as early civilized humans in ancient Mesopotamia.

Geometry and tectonic deformation of the seismogenic structure for the 8 August 2017 M_S 7.0 Jiuzhaigou earthquake sequence,northern Sichuan, China

To reveal the geometry of the seismogenic structure of the Aug. 8, 2017 M_S 7.0 Jiuzhaigou earthquake in northern Sichuan,data from the regional seismic network from the time of the main event to Oct. 31, 2017 were used to relocate the earthquake sequence by the tomoDD program, and the focal mechanism solutions and centroid depths of the M_L ≥ 3.5 events in the sequence were determined using the CAP waveform inversion method. Further, the segmental tectonic deformation characteristics of the seismogenic faults were analyzed preliminarily by using strain rosettes and areal strains(As). The results indicate:(1) The relocated M_S 7.0 Jiuzhaigou earthquake sequence displays a narrow ～ 38 km long NNW-SSE-trending zone between the NW-striking Tazang Fault and the nearly NSstriking Minjiang Fault, two branches of the East Kunlun Fault Zone. The spatial distribution of the sequence is narrow and deep for the southern segment, and relatively wide and shallow for the northern segment. The initial rupture depth of the mainshock is 12.5 km, the dominant depth range of the aftershock sequence is between 0 and 10 km with an average depth of 6.7 km. The mainshock epicenter is located in the middle of the aftershock region, showing a bilateral rupture behavior. The centroid depths of 32 M_L ≥ 3.5 events range from 3 to 12 km with a mean of about 7.3 km, consistent with the predominant focal depth of the whole sequence.(2) The geometric structure of the seismogenic fault on the southern section of the aftershock area(south of the mainshock) is relatively simple, with overall strike of ～150° and dip angle ～75°, but the dip angle and dip-orientation exhibit some variation along the segment. The seismogenic structure on the northern segment is more complicated; several faults, including the Minjiang Fault, may be responsible for the aftershock activities. The overall strike of this section is ～159° and dip angle is ～59°, illustrating a certain clockwise rotation and a smaller dip angle than the southern segment. The differences between the two segments demonstrate variation of the geometric structure along the seismogenic faults.(3) The focal mechanism solutions of 32 M_L ≥ 3.5 events in the earthquake sequence have obvious segmental characteristics. Strike-slip earthquakes are dominant on the southern segment, while 50% of events on the northern segment are thrusting and oblique thrusting earthquakes, revealing significant differences in the kinematic features of the seismogenic faults between the two segments.(4) The strain rosettes for the mainshock and the entire sequence of 31 M_L ≥ 3.5 aftershocks correspond to strike-slip type with NWW-SEE compressional white lobes and NNE-SSW extensional black lobes of nearly similar size. The strain rosette and As value of the entire sequence of 22 M_L ≥ 3.5 events on the southern segment are the same as those of the M_S 7.0 mainshock,indicating that the tectonic deformation here is strike-slip. However, the strain rosette of the entire sequence of 10 M_L ≥ 3.5 events on the northern segment show prominent white compressional lobes and small black extensional lobes, and the related As value is up to 0.52,indicating that the tectonic deformation of this segment is oblique thrusting with a certain strike-slip component. Differences between the two segments all reveal distinctly obvious segmental characteristics of the tectonic deformation of the seismogenic faults for the Jiuzhaigou earthquake sequence.

Effect of tool geometry on ultraprecision machining of soft-brittle materials:a comprehensive review

Brittle materials are widely used for producing important components in the industry of optics,optoelectronics,and semiconductors.Ultraprecision machining of brittle materials with high surface quality and surface integrity helps improve the functional performance and lifespan of the components.According to their hardness,brittle materials can be roughly divided into hard-brittle and soft-brittle.Although there have been some literature reviews for ultraprecision machining of hard-brittle materials,up to date,very few review papers are available that focus on the processing of soft-brittle materials.Due to the‘soft’and‘brittle’properties,this group of materials has unique machining characteristics.This paper presents a comprehensive overview of recent advances in ultraprecision machining of soft-brittle materials.Critical aspects of machining mechanisms,such as chip formation,surface topography,and subsurface damage for different machining methods,including diamond turning,micro end milling,ultraprecision grinding,and micro/nano burnishing,are compared in terms of tool-workpiece interaction.The effects of tool geometries on the machining characteristics of soft-brittle materials are systematically analyzed,and dominating factors are sorted out.Problems and challenges in the engineering applications are identified,and solutions/guidelines for future R&D are provided.

Modeling the Effects of Tool Shoulder and Probe Profile Geometries on Friction Stirred Aluminum Welds Using Response Surface Methodology

The present paper discusses the modeling of tool geometry effects on the friction stir aluminum welds using response surface methodology. The friction stir welding tools were designed with different shoulder and tool probe geometries based on a design matrix. The matrix for the tool designing was made for three types of tools, based on three types of probes, with three levels each for defining the shoulder surface type and probe profile geometries. Then, the effects of tool shoulder and probe geometries on friction stirred aluminum welds were experimentally investigated with respect to weld strength, weld cross section area, grain size of weld and grain size of thermo-mechanically affected zone. These effects were modeled using multiple and response surface regression analysis. The response surface regression modeling were found to be appropriate for defining the friction stir weldment characteristics.

Fault Geometry and Departure of Precursors From the Epicenter

Different tectonic elements may play different roles and their deformation styles and types of anomalies are also different in the process of deformation.Such differences may be the cause of the phenomena of the departure from the region with obvious anomalies from the coming epicenter,which are called "departure of precursors from epicenter." From some new experimental and numerical modeling results,five types of deformation elements are identified according to their roles in the process of deformation in a region.They are the brake(or locked),slip,yield,sensitive,and valve elements.Spatial distribution of different elements,especially sensitive and slip elements,are discussed by taking the bend fault,extension,compression,and complex en-echelon faults as examples.These elements with different roles may overlap each other in some cases or be separated in other cases.It is obvious that for different fault geometry,the spatial layout of these elements is different and thus the position and style of

NONLINEAR STRAIN COMPONENTS OF GENERAL SHELLS WITH INITIAL GEOMETRIC IMPERFECTIONS

On the basis of nonlinear strain component formulations of three-dimensional continuum, this paper has derived the nonlinear strain component formulations of shells with initial geometric imperfections. The derivation is not confined to a special shell, therefore they possess general properties. These formulations provide the theoretical basis of the strain analysis for geometric nonlinear problems of shells with initial geometric imperfections.

X-ray Crystal Structure and Molecular Mechanics Calculations of (2,6-iso-dipropyl-phcnylamidc) Dimethyl (tetra-methylecyclopenta- dienyl)Silane Titanium Dichloride

Crystal and molecular structure of (2.6-dipropylphenylamidc) dimethyl (tetra-methyl cyclopentadienyl) silane titanium dichloride (I) was fully characterized by X-ray diffraction. The crystal is obtained tyom a mixture of ether/hexane as orthorhombic, with a = 12.658 (3 ) A. b = 16.62 (3) A. c = 11 .760 (2) A. V = 2474,2 (9) A. Z = 4. space group Pnma. R = 0.0399. Compound I compose of the R -bounded ring with its dimethylsilyl-dipropyl phenyl amido group and the two terminal chloride atoms coordinated to central metal to form a so-called constrained geometry catalyst (CGC) structure. The result of molecular mechanics (MM) calculations on compound I shows that bond lengths and bond angles from the MM calculation are comparable to the data obtained from the X-ray diffraction study. The relation of the structure of CGCs and their catalytic activity by MM calculations is also discussed.

Three-dimensional Geometry and Kinematics of the Changning Anticline in the Southern Sichuan Basin

In this study,we aim to clarify the structural characteristics and deformation process of the Changning anticline.We carefully interpret 38 two-dimensional(2D)seismic profiles in the study area and establish three-dimensional(3D)geometric and quantitative kinematic models of the Changning anticline.This study shows that the basement fault controls the formation of the Changning anticline.The fault slope of the main fault in the basement shows’steep in the upper and gentle in the lower’structural characteristics vertically,possessing obvious segmentary characteristics transversely and presents the overall characteristics of’steep in the east and gentle in the west’.Further analysis shows that the Changning anticline proceeds west and terminates at the boundary defined by current surface features but gradually disappears westward across the Mt.Huaying fault zone.Furthermore,we identified that deformation of the Changning anticline began during the early Yanshanian movement period.Under compressional stress from the southeast,the anticline slid forward along the basement fault until the end of the Yanshanian movement period,when the dominant WNW-ESE structure gradually emerged.Since the Himalayan movement period,a series of NE-trending structures have been formed in the anticline,owing to multi-directional compressive stress.

The Geometrical Theory of Science

Classical mechanics and quantum mechanics are the two cornerstones of science. As is well known, classical mechanics, the theory that describes the macrophysical world, has grown and flowered both in experimentation and theorization. The same is not true of quantum mechanics, the theory that describes the microphysical world. While experimentation has shown giant strides, theorization has been essentially static, having not moved appreciably beyond the great achievements of the 1920s. The reason is not difficult to fathom: while theoretical progress in classical mechanics has been intellect-driven, that in quantum mechanics, on the other hand, has been machine-driven! In this paper we describe both classical and quantum systems in an absolute and a common language (geometry). Indeed, we construct the whole of science on the basis of just three numbers, namely, 1, 2, and 3.

Fabrication of three-dimensional islet models by geometry-controlled hanging drop method

Hanging-drop method has been widely used to fabricate three-dimensional (3D) in vitro tissue models due to its advantages such as being easy to perform, inexpensive, and permitting precise control of cell spheroid formation. The geometry of hanging drop may play a critical role on the formation of cell spheroids, which, however, has not been explored. In this study, we developed a modified hanging-drop platform that enables the production of cell spheroids in a high-throughput manner by controlling hanging drop geometry with defined spreading ring. The surface tension force is proportional to the spreading ring and gravitational force is determined by the droplet volume, and the geometry can be determined by the balance between surface tension and gravity.β-TC-6 cell spheroids with optimized diameters were fabricated as 3D in vitro islet models. The models show morphology similar to primary islets and have functionality that more closely resembles primary islets than two-dimensional cell culture. The developed platform holds great potential for engineering well-controlled in vitro tissue models for various applications such as physiological and pathological studies, drug screening, as well as transplantation for treatment purpose.