As for the ultra-precision grinding of the hemispherical fused silica resonator,due to the hard and brittle nature of fused silica,subsurface damage(SSD)is easily generated,which enormously influences the performance ...As for the ultra-precision grinding of the hemispherical fused silica resonator,due to the hard and brittle nature of fused silica,subsurface damage(SSD)is easily generated,which enormously influences the performance of such components.Hence,ultra-precision grinding experiments are carried out to investigate the surface/subsurface quality of the hemispherical resonator machined by the small ball-end fine diamond grinding wheel.The influence of grinding parameters on the surface roughness(SR)and SSD depth of fused silica samples is then analyzed.The experimental results indicate that the SR and SSD depth decreased with the increase of grinding speed and the decrease of feed rate and grinding depth.In addition,based on the material strain rate and the maximum undeformed chip thickness,the effect of grinding parameters on the subsurface damage mechanism of fused silica samples is analyzed.Furthermore,a multi-step ultra-precision grinding technique of the hemispherical resonator is proposed based on the interaction influence between grinding depth and feed rate.Finally,the hemispherical resonator is processed by the proposed grinding technique,and the SR is improved from 454.328 nm to 110.449 nm while the SSD depth is reduced by 94%from 40μm to 2.379μm.The multi-step grinding technique proposed in this paper can guide the fabrication of the hemispherical resonator.展开更多
In order to increase the efficiency of solar air collectors,a new variant with a protrusion is proposed in this study,and its performances are analyzed from two points of view,namely,in terms of optics and thermodynam...In order to increase the efficiency of solar air collectors,a new variant with a protrusion is proposed in this study,and its performances are analyzed from two points of view,namely,in terms of optics and thermodynamics aspects.By comparing and analyzing the light paths of the protrusion and the dimple,it can be concluded that when sunlight shines on the dimple,it is reflected and absorbed multiple times,whereas for the sunlight shining on the protrusion,there is no secondary reflection or absorption of light.When the lighting area and the properties of the surfaces are the same,the absorption rate of the dimple is 10.3 percentage points higher than that of the protrusion.In the range of Reynolds number from 3000 to 11000,numerical simulations about the effects of the relative height(e/Dh=0.033–0.1)and relative spacing(p/e=4.5–8.5)of protrusions on air heat transfer and flow resistance show that,in terms of comprehensive evaluation coefficient(PF),the best relative height is 0.085,when the relative spacing is 5.A correlation of Nu and f with Re,e/Dh and p/e is obtained by linear regression of the results,in order to provide a useful reference for the design and optimization of this kind of solar air collector.展开更多
Examining the contribution of hemispherical photographs in the understanding of Natural forest regeneration is very important in estimating the future forest structure, composition and to enforce conservation regulati...Examining the contribution of hemispherical photographs in the understanding of Natural forest regeneration is very important in estimating the future forest structure, composition and to enforce conservation regulations. This study sets out to examine the interaction between stump sprouting, LAI, site and canopy openness for the entire AKAK forest area and for the logging compartments;2013, 2015 and 2017 respectively. 49 sprouted stump were identified randonly. 20 m × 20 m plots were demarcated along a canopy gaps for each sprouted stump, the plots were established in such a manner that the sprouted stumps will be in the middle. For each of the selected 49 sprouted stump, indirect measurements of canopy cover were performed in the 49 plots of 20 m × 20 m (0.04 ha), giving a total of 1.96 ha of land covered. Galaxy S3 smartphone with a built-in Infinix ZERO 4 fish-eye lens with 198˚ view angle equidistant projection was used to take photos. The fish-eye lens was mounted on the phone camera and photograph were taken at a fixed height of 1.3 m. Results revealed that, the combine Principal Component Factor Analysis (2013, 2015 and 2017) of the correlation matrix for Sprout, Years, LAI 4%, LAI 5%, Canopy and Site openness, shows that factor 1 explained 62.6% of total variance while factor 2 explained 17.9% together explain 80.05% Communalities. For the year 2013, 2015 and 2017 respectively shows that there is a very strong correlation (p p < 0.0005) between LAI4 and LAI5.展开更多
Introduction: Stroke is a worldwide leading cause of mortality and disability, and there are substantial economic costs for post-stroke. A large hemispheric ischemic stroke is a severe form of stroke that may occur in...Introduction: Stroke is a worldwide leading cause of mortality and disability, and there are substantial economic costs for post-stroke. A large hemispheric ischemic stroke is a severe form of stroke that may occur in up to 10% of stroke cases. In these cases, decompressive craniectomy (DC) is an effective strategy to reduce early case fatality. In this study, the authors present their experience in the management of large hemispheric stroke in patients who underwent decompressive craniectomy. Patients and Methods: This study is a retrospective analysis of 9 patients who underwent decompressive craniotomy for large hemispheric stroke from February 2021 to January 2024. Demographic, clinical, and radiologic variables were collected from the hospital archives. All patients underwent head CT scan or MRI showing the involvement of the majority or complete middle cerebral artery distribution area. Hemicraniectomy with duraplasty was the main surgical procedure. The mortality and the outcome using Modified Rankin Scale (mRS) were reported. Results: A total of 9 patients with large hemispheric stroke who underwent DC were reviewed retrospectively. The mean age was 53 years with extremes of 39- and 67-year-old. There were 6 males and 3 females. The main risk factors of stroke were hypertension and diabetes in 5 and 3 cases, respectively. The large craniectomy was made and the dura was opened in the stellate fashion. Duraplasty was performed using epicrane in 7 patients and fascia lata aponeurosis in 2 patients. In the postoperative course, 4 (44%) patients died in 1 month and the mortality rate was 66.6% in 6 months. Conclusion: Decompressive craniectomy is a life-saving procedure used to manage cerebral edema after stroke onset. In our study, the high mortality rate could be explained by many factors, such as patient selection criteria and timing of surgery. Early surgery within the first 48 hours in patients less than 60 years old with large hemispheric stroke is recommended.展开更多
The snow density is a fundamental variable of the snow physical evolution processes,which can reflect the snowpack condition due to the thermal and gravitational compaction.Snow density is a bridge to transfer snow de...The snow density is a fundamental variable of the snow physical evolution processes,which can reflect the snowpack condition due to the thermal and gravitational compaction.Snow density is a bridge to transfer snow depth to snow water equivalent(SWE)for the snow water resources research.Therefore,it is important to understand the spatiotemporal distribution of snow density for the appropriate estimation of SWE.In this study,in situ snow densities from more than 6,000 stations in the Northern Hemisphere were used to analyze the spatial and temporal variations in snow density.The results displayed that snow density varied spatially and temporally in the Northern Hemisphere,with range of below 0.1 to over 0.4 g/cm^(3).The average snow densities in the mountainous regions of western North America,southeastern Canada,and Europe range from approximately 0.24 to 0.26 g/cm^(3),which is significantly greater than the values of 0.16–0.17 g/cm^(3)observed in Siberia,central Canada,the Great Plains of the United States,and China.The seasonal growth rates also present large spatial heterogeneity.The rates are over 0.024 g/cm^(3)per month in Southeastern Canada,the west mountain of North America and Europe,approximately 0.017 g/cm^(3)per month in Siberia,much larger than approximately 0.004 g/cm^(3)per month in other regions.Snow cover duration is a critical factor to determine the snow density.This study endorses the small snow density in China based on meteorological station observations,which results from that the meteorological stations are dominantly distributed in plain areas with relative short snow cover duration and shallow snow.展开更多
This paper is a statistical survey of Southern Hemisphere cold and hot polar cap patches,in relation to the interplanetary magnetic field(IMF)and ionospheric convection geometry.A total of 11,946 patch events were ide...This paper is a statistical survey of Southern Hemisphere cold and hot polar cap patches,in relation to the interplanetary magnetic field(IMF)and ionospheric convection geometry.A total of 11,946 patch events were identified by Defense Meteorological Satellite Program(DMSP)F16 during the years 2011 to 2022.A temperature ratio of ion/electron temperature(T_(i)/T_(e))<0.68 is recommended to define a hot patch in the Southern Hemisphere,otherwise it is defined as a cold patch.The cold and hot patches have different dependencies on IMF clock angle,while their dependencies on IMF cone angle are similar.Both cold and hot patches appear most often on the duskside,and the distribution of cold patches gradually decreases from the dayside to the nightside,while hot patches have a higher occurrence rate near 14 and 21 magnetic local time(MLT).Moreover,we compared the key plasma characteristics of polar cap cold and hot patches in the Southern and Northern Hemispheres.The intensity of the duskside upward field-aligned current of patches in the Southern Hemisphere(SH)is stronger than that in the Northern Hemisphere(SH),which may be due to the discrepancy in conductivities between the two hemispheres,caused by the tilted dipole.In both hemispheres,the downward soft-electron energy flux of the dawnside patches is significantly greater than that of the duskside patches.展开更多
BACKGROUND Currently,adolescent depression is one of the most significant public health concerns,markedly influencing emotional,cognitive,and social maturation.Despite advancements in distinguish the neurobiological s...BACKGROUND Currently,adolescent depression is one of the most significant public health concerns,markedly influencing emotional,cognitive,and social maturation.Despite advancements in distinguish the neurobiological substrates underlying depression,the intricate patterns of disrupted brain network connectivity in adolescents warrant further exploration.AIM To elucidate the neural correlates of adolescent depression by examining brain network connectivity using resting-state functional magnetic resonance imaging(rs-fMRI).METHODS The study cohort comprised 74 depressed adolescents and 59 healthy controls aged 12 to 17 years.Participants underwent rs-fMRI to evaluate functional connectivity within and across critical brain networks,including the visual,default mode network(DMN),dorsal attention,salience,somatomotor,and frontoparietal control networks.RESULTS Analyses revealed pronounced functional disparities within key neural circuits among adolescents with depression.The results demonstrated existence of hemispheric asymmetries characterized by enhanced activity in the left visual network,which contrasted the diminished activity in the right hemisphere.The DMN facilitated increased activity within the left prefrontal cortex and reduced engagement in the right hemisphere,implicating disrupted self-referential and emotional processing mechanisms.Additionally,an overactive right dorsal attention network and a hypoactive salience network were identified,underscoring significant abnormalities in attentional and emotional regulation in adolescent depression.CONCLUSION The findings from this study underscore distinct neural connectivity disruptions in adolescent depression,underscoring the critical role of specific neurobiological markers for precise early diagnosis of adolescent depression.The observed functional asymmetries and network-specific deviations elucidate the complex neurobiological architecture of adolescent depression,supporting the development of targeted therapeutic strategies.展开更多
Spatial confinement has great potential for Laser Induced Breakdown Spectroscopy （LIBS） instruments after it has been proven that it has the ability to enhance the LIBS signal strength and repeatability. In order to...Spatial confinement has great potential for Laser Induced Breakdown Spectroscopy （LIBS） instruments after it has been proven that it has the ability to enhance the LIBS signal strength and repeatability. In order to achieve in-situ measurement of heavy metals in farmland soils by LIBS, a hemispherical spatial confinement device is designed and used to collect plasma spectra, in which the optical fibers directly collect the breakdown spectroscopy of the soil samples. This device could effectively increase the stability of the spectrum intensity of soil. It also has other advantages, such as ease of installation, and its small and compact size. The relationship between the spectrum intensity and the laser pulse energy is studied for this device. It is found that the breakdown threshold is 160 cm-2, and when the laser fluence increases to 250 J/cm2, the spectrum intensity reaches its maximum. Four different kinds of laser pulse energy were set up and in each case the limits of detection of Cd, Cu, Ni, Pb and Zn were calculated. The results show that when the laser pulse fluence was 2.12 GW/cm2, we obtained the smallest limits of detection of these heavy metals, which are all under 10 mg/kg. This device can satisfy the needs of heavy metal in-situ detection, and in the next step it will be integrated into a portable LIBS instrument.展开更多
An analytical solution for the three-dimensional scattering and diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits is developed by employing Fourier-Bessel series expansion te...An analytical solution for the three-dimensional scattering and diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits is developed by employing Fourier-Bessel series expansion technique. Unlike previous studies, in which the saturated soil deposits were simulated with the single-phase elastic theory, in this paper, they are simulated with Biot＇s dynamic theory for saturated porous media, and the half space is assumed as a single-phase elastic medium. The effects of the dimensionless frequency, the incidence angle of P-wave and the porosity of soil deposits on the surface displacement magnifications of the hemispherical alluvial valley are investigated. Numerical results show that the existence of a saturated hemispherical alluvial valley has much influence on the surface displacement magnifications. It is more reasonable to simulate soil deposits with Biot＇s dynamic theory when evaluating the displacement responses of a hemispherical alluvial valley with an incidence of P-waves.展开更多
The mass non-uniformity of hemispherical resonator is one of reasons for frequency split,and frequency split can cause gyroscope to drift.Therefore,it is of great significance to analyze the relationship between mass ...The mass non-uniformity of hemispherical resonator is one of reasons for frequency split,and frequency split can cause gyroscope to drift.Therefore,it is of great significance to analyze the relationship between mass non-uniformity and frequency split,which can provide a theoretical basis for mass balance of imperfect resonator.The starting point of error mechanism analysis for gyroscope is the motion equations of resonator.Firstly,based on the Kirchhoff-Love hypothesis in the elastic thin shell theory,the geometric deformation equations of resonator are deduced.Secondly,the deformation energy equation of resonator is derived according to the vibration mode and relationship between the stress and strain of hemispherical thin shell.Thirdly,the kinetic energy equation of resonator is deduced by the Coriolis theorem.Finally,the motion equations of resonator are established by the Lagrange mechanics principle.The theoretical values of precession factor and natural frequency are calculated by the motion equations,which are substantially consistent with the ones by the finite element method and practical measurement,the errors are within a reasonable range.Simultaneously,the varying trend of natural frequency with respect to the geometrical and physical parameters of resonator by the motion equations is consistent with that by the finite element analysis.The above conclusions prove the correctness and rationality of motion equations.Similarly,the motion equations of resonator with mass non-uniformity are established by the same modeling method in case of ignoring the input angular rate and damping,and the state equations with respect to the velocity and displacement of vibration system are derived,then twonatural frequencies are solved by the characteristic equation.It is concluded that one of reasons for frequency split is the 4 th harmonic of mass non-uniformity,and thus much attention should be paid to minimizing the 4 th harmonic of mass non-uniformity in the course of mass balancing for imperfect resonator.展开更多
Daily Total Column Ozone (TCO) measurements compiled from Total Ozone Mapping Spectrometer (TOMS) and Ozone Monitoring Instruments (OMI) were used to analyze the global and hemispherical TCO interannual variations. Tw...Daily Total Column Ozone (TCO) measurements compiled from Total Ozone Mapping Spectrometer (TOMS) and Ozone Monitoring Instruments (OMI) were used to analyze the global and hemispherical TCO interannual variations. Two periods of TCO measurements were analyzed separately covering full years. For the 1978-1994 period, the TCO showed a global decade decrease rate of 13.45 DU (about -4.3%). For the Northern Hemisphere(NH) the decade decrease rate was of 12.96 DU (-4.0%), while in the Southern Hemisphere (SH) was of 13.57 DU (-4.5%). These decreases in ozone trends, using the totality of TOMS and OMI satellite measurements, are greater than those reported in literature. The 1998-2014 period global TCO decade decrease rate was of 1.56 DU, corresponding 0.94 DU and 0.138 DU for the NH and SH, respectively. The global TCO variations must show a double annual periodicity, the first one with maxima in March due to the Northern Hemisphere (NH) and the second one during September due to the Southern Hemisphere (SH). However, the maxima due to SH TCO interannual variations have gradually vanished. A disturbance in the SH TCO interannual variations has appeared since 1980;graphically the periodicity brakes down and transforms to a double peak from 1985 and on. This effect can be attributed to the hemispheric impact of the ozone hole at the South Pole. Between October 1, 2004 and December 14, 2005 TOMS and OMI have recorded this disturbance unequivocally. We conclude that the disturbance in SH TCO has an irreversible character.展开更多
Abstract An analytical solution to the three-dimen-sional scattering and diffraction of plane SV-waves by a saturated hemispherical alluvial valley in elastic half-space is obtained by using Fourier-Bessel series expa...Abstract An analytical solution to the three-dimen-sional scattering and diffraction of plane SV-waves by a saturated hemispherical alluvial valley in elastic half-space is obtained by using Fourier-Bessel series expan-sion technique. The hemispherical alluvial valley with saturated soil deposits is simulated with Biot＇s dynamic theory for saturated porous media. The following conclusions based on numerical results can be drawn： （1） there are a significant differences in the seismic response simulation between the previous single-phase models and the present two-phase model; （2） the nor-malized displacements on the free surface of the alluvial valley depend mainly on the incident wave angles, the dimensionless frequency of the incident SV waves and the porosity of sediments; （3） with the increase of the incident angle, the displacement distributions become more complicated; and the displacements on the free surface of the alluvial valley increase as the porosity of sediments increases.展开更多
In this paper,the effects of hardening exponent,yield strength and elastic modulus on the deformability of near hemispherical shells are investigated by means of finite element method and orthogonal experiment design....In this paper,the effects of hardening exponent,yield strength and elastic modulus on the deformability of near hemispherical shells are investigated by means of finite element method and orthogonal experiment design.The largest eccentric angle during the deformation process and thickness reduction after the deformation are introduced to estimate the deformability quantitatively according to the deformation characteristics of near hemispherical shells.The results indicate that the hardening exponent is the most influential parameter,followed by elastic modulus and yield strength.The shell exhibits good deformability when the hardening exponent and elastic modulus are in the range of 0.1-0.125 and 70-108 GPa,respectively.展开更多
Hemispherical Resonator Gyroscope(HRG)is a classical high precision Coriolis Vibration Gyroscope(CVG),which performs attitude estimation of carrier by detecting the precession of standing wave of resonator,thus,the dr...Hemispherical Resonator Gyroscope(HRG)is a classical high precision Coriolis Vibration Gyroscope(CVG),which performs attitude estimation of carrier by detecting the precession of standing wave of resonator,thus,the drift of standing wave of resonator has a great influence on the output accuracy of gyroscope,where the quality factor non-uniformity of resonator is one of main error sources.Ring electrode is a classical excitation structure of HRG because the standing wave can precess freely under its excitation,which makes the gyroscope have more accurate scale factor,larger measurement range and better dynamic characteristics.In this paper,the equations of motion of an ideal resonator excited by a ring electrode are derived by the elastic thin shell theory and Lagrange mechanical principle,then the corresponding equivalent mechanical model is established.According to the“average method”,it can be seen that the ideal resonator excited by the ring electrode works in integral mode,and any position in the circumferential direction of resonator can be a working point,which means that the quality factor non-uniformity has a great effect on the drift of standing wave.Therefore,the equations of motion of resonator with quality factor non-uniformity under the ring electrode excitation are deduced by the equivalent mechanical model,and the drift model of standing wave is established by the“average method”,it can be found that both the amplitude of quality factor non-uniformity and angle between the“inherent damping axis”and antinode axis of standing wave can affect the drift rate of standing wave.Moreover,the drift model indicates that if the input angular rate does not reach the threshold,the precession angular rate of standing wave will appear“self-locking”phenomenon,that is,the gyroscope will lose the integral effect.展开更多
In order to investigate the draping behavior of non-crimp fabrics（NCFs）, two types of carbon NCFs with tricot-chain stitches or chain stitches were formed on a hemispherical mould via a stretch forming process. The ...In order to investigate the draping behavior of non-crimp fabrics（NCFs）, two types of carbon NCFs with tricot-chain stitches or chain stitches were formed on a hemispherical mould via a stretch forming process. The shear angle and forming defects of the fabrics were measured on the hemisphere, under different blank holder forces（BHFs）. The results showed that increasing BHF could enhance the shear angle slightly, reduce the asymmetry for the deformation of the fabrics, and change the main type of the process-induced defects. Besides, compression tests were performed on the corresponding composite components. By analyzing the change of fiber volume fraction and structural parameters of the textile reinforcements, the effects of draping behavior of NCFs on the mechanical performance of the composites were studied. The results reveal that draping process has distinguishable impacts on the mechanical properties of the final components, which is closely related to the stitching pattern of the NCFs.展开更多
Maximizing quality factor (Q) is essential to improve the performance of micro hemispherical shell resonators (μHSRs) which can be used in microelectromechanical system (MEMS) gyroscopes to measure angular rotation.S...Maximizing quality factor (Q) is essential to improve the performance of micro hemispherical shell resonators (μHSRs) which can be used in microelectromechanical system (MEMS) gyroscopes to measure angular rotation.Several energy dissipation mechanisms limit Q,where thermoelastic dissipation (TED) is the major one and studied in this paper.Fully coupled thermo-mechanical equations for calculating TED are formulated,and then temperature distribution in a deformed μHSR and its quality factor related to TED (QTED) are obtained by solving the equations through a finite-element method (FEM).It has been found that different fabrication process conditions can obtain various geometrical parameters in our previous studies.In order to provide guidelines for the design and fabrication of μHSRs,the effects of their geometry on resonant frequency (f0) and QTED are studied.The change of anchor height and small enough anchor radius have no effect on both f0 and QTED,but the shell size including its radius,thickness and height has significant impact on f0 and QTED.It is found that whether a μHSR has lower f0 and higher QTED or higher f0 and higher QTED can be achieved by changing these geometrical parameters.The results presented in this paper can also be applied to other similar resonators.展开更多
Ballistic experiments were conducted on thin steel plates that are normally impacted by hemisphericalnosed projectiles at velocities higher than their ballistic limits. The deformation and failure modes of the thin st...Ballistic experiments were conducted on thin steel plates that are normally impacted by hemisphericalnosed projectiles at velocities higher than their ballistic limits. The deformation and failure modes of the thin steel plates were analyzed. A new method was proposed according to the experimental results and the perforation phenomenon of the thin steel plates to determine the radius of the bulging region. In establishing this new method, a dynamic method combined with the plastic wave propagation concept based on the rigid plastic assumption was adopted. The whole perforation process was divided into four consecutive stages, namely, bulging deformation, dishing deformation, ductile hole enlargement, and projectile exit. On the basis of the energy conservation principle, a new model was developed to predict the residual velocities of hemispherical-nosed projectiles that perforate thin steel plates at low velocities.The results obtained from the theoretical calculations by the present model were compared with the experimental results. Theoretical predictions were in good agreement with the experimental results in terms of both the radius of the bulging region and the residual velocity of the projectile when the strain rate effects of the target material during each stage were considered.展开更多
The asymmetric deformation and eccentricity problems of near hemispherical diaphragm under the uniform surface load are quantitatively characterized in the paper.The analysis is based on a 3D finite element analysis (...The asymmetric deformation and eccentricity problems of near hemispherical diaphragm under the uniform surface load are quantitatively characterized in the paper.The analysis is based on a 3D finite element analysis (FEA) model established according to elastic-plasticity and large displacement nonlinear finite element method.Besides,the deformation experiments are taken to validate the reliability of FEA model which shows that the simulation results are in good agreement with the experimental results.Then,three angle parameters,deflection angle β,circumvolving angle θ and distributing angle γ,are introduced and expressed to characterize the asymmetric deformation and eccentricity quantitatively.According to the angle parameters,the inversion processes of uniform thickness diaphragm and varying thickness diaphragm are calculated respectively.The inversion process of varying thickness diaphragm is much steadier than that of uniform thickness diaphragm.The present results show that the asymmetric deformation process can be characterized by curve of three angle parameters (β,θ,γ) exactly,the degrees of eccentricity can be indicated by the final value of deflection angle and the eccentricity position can be characterized by the final values of the three angle parameters.展开更多
This paper makes detailed analyses tor the flexural vibration (frequency) of the hemispherical shell and presents the varying laws of frequency with the rarving boundary angles and the wall thickness of the above shel...This paper makes detailed analyses tor the flexural vibration (frequency) of the hemispherical shell and presents the varying laws of frequency with the rarving boundary angles and the wall thickness of the above shell, It is an important value to develop the instrument, such as hemispherical resonator gyro (HRG), whose sensing component is a hemispherical shell.展开更多
This paper is a study of the gravitational attraction between two uniform hemispherical masses aligned such that the pair is cylindrically symmetric. Three variations are considered: flat side to flat side, curved sid...This paper is a study of the gravitational attraction between two uniform hemispherical masses aligned such that the pair is cylindrically symmetric. Three variations are considered: flat side to flat side, curved side to curved side, and flat side to curved side. Expressions for the second and third variation are derived from the first, with the use of superposition and the well-known gravitational behavior of a spherical mass as equivalent to a point mass at its center. The study covers two masses of equal diameter and of different diameters, such that one is four times that of the other. Calculations are done for separations from zero to fifty times the radius of the larger of the two, which is effectively the asymptotic limit. It is demonstrated that at any separation, the force can be expressed as if the two hemispheres were point masses separated by a certain distance. Expressions for that distance and the location of the (fictitious) point masses within each hemisphere are presented. Unlike the case of two spherical masses, the location within their respective hemisphere is not necessarily the same for each point and both are dependent upon the separation between the two hemispheres. The calculation for the first variation is done in two ways. The first is a “brute force” multi-dimensional integral with the help of Wolfram Mathematica. The second is an axial expansion for the potential modified for off-axis locations by Legendre polynomials. With only a few terms in the expansion, the results of the second method are in extremely good agreement with those of the first. Finally, an interesting application to a split earth is presented.展开更多
基金This work was supported by the National Key Research and Development Program of China(No.2022YFB3403600)the National Natural Science Foundation of China(No.52293403)Self-Planned Task of State Key Laboratory of Robotics and System(HIT)(No.SKLRS202204C).
文摘As for the ultra-precision grinding of the hemispherical fused silica resonator,due to the hard and brittle nature of fused silica,subsurface damage(SSD)is easily generated,which enormously influences the performance of such components.Hence,ultra-precision grinding experiments are carried out to investigate the surface/subsurface quality of the hemispherical resonator machined by the small ball-end fine diamond grinding wheel.The influence of grinding parameters on the surface roughness(SR)and SSD depth of fused silica samples is then analyzed.The experimental results indicate that the SR and SSD depth decreased with the increase of grinding speed and the decrease of feed rate and grinding depth.In addition,based on the material strain rate and the maximum undeformed chip thickness,the effect of grinding parameters on the subsurface damage mechanism of fused silica samples is analyzed.Furthermore,a multi-step ultra-precision grinding technique of the hemispherical resonator is proposed based on the interaction influence between grinding depth and feed rate.Finally,the hemispherical resonator is processed by the proposed grinding technique,and the SR is improved from 454.328 nm to 110.449 nm while the SSD depth is reduced by 94%from 40μm to 2.379μm.The multi-step grinding technique proposed in this paper can guide the fabrication of the hemispherical resonator.
基金the Key Scientific Research Projects of Colleges and Universities in Henan Province(22B480007).
文摘In order to increase the efficiency of solar air collectors,a new variant with a protrusion is proposed in this study,and its performances are analyzed from two points of view,namely,in terms of optics and thermodynamics aspects.By comparing and analyzing the light paths of the protrusion and the dimple,it can be concluded that when sunlight shines on the dimple,it is reflected and absorbed multiple times,whereas for the sunlight shining on the protrusion,there is no secondary reflection or absorption of light.When the lighting area and the properties of the surfaces are the same,the absorption rate of the dimple is 10.3 percentage points higher than that of the protrusion.In the range of Reynolds number from 3000 to 11000,numerical simulations about the effects of the relative height(e/Dh=0.033–0.1)and relative spacing(p/e=4.5–8.5)of protrusions on air heat transfer and flow resistance show that,in terms of comprehensive evaluation coefficient(PF),the best relative height is 0.085,when the relative spacing is 5.A correlation of Nu and f with Re,e/Dh and p/e is obtained by linear regression of the results,in order to provide a useful reference for the design and optimization of this kind of solar air collector.
文摘Examining the contribution of hemispherical photographs in the understanding of Natural forest regeneration is very important in estimating the future forest structure, composition and to enforce conservation regulations. This study sets out to examine the interaction between stump sprouting, LAI, site and canopy openness for the entire AKAK forest area and for the logging compartments;2013, 2015 and 2017 respectively. 49 sprouted stump were identified randonly. 20 m × 20 m plots were demarcated along a canopy gaps for each sprouted stump, the plots were established in such a manner that the sprouted stumps will be in the middle. For each of the selected 49 sprouted stump, indirect measurements of canopy cover were performed in the 49 plots of 20 m × 20 m (0.04 ha), giving a total of 1.96 ha of land covered. Galaxy S3 smartphone with a built-in Infinix ZERO 4 fish-eye lens with 198˚ view angle equidistant projection was used to take photos. The fish-eye lens was mounted on the phone camera and photograph were taken at a fixed height of 1.3 m. Results revealed that, the combine Principal Component Factor Analysis (2013, 2015 and 2017) of the correlation matrix for Sprout, Years, LAI 4%, LAI 5%, Canopy and Site openness, shows that factor 1 explained 62.6% of total variance while factor 2 explained 17.9% together explain 80.05% Communalities. For the year 2013, 2015 and 2017 respectively shows that there is a very strong correlation (p p < 0.0005) between LAI4 and LAI5.
文摘Introduction: Stroke is a worldwide leading cause of mortality and disability, and there are substantial economic costs for post-stroke. A large hemispheric ischemic stroke is a severe form of stroke that may occur in up to 10% of stroke cases. In these cases, decompressive craniectomy (DC) is an effective strategy to reduce early case fatality. In this study, the authors present their experience in the management of large hemispheric stroke in patients who underwent decompressive craniectomy. Patients and Methods: This study is a retrospective analysis of 9 patients who underwent decompressive craniotomy for large hemispheric stroke from February 2021 to January 2024. Demographic, clinical, and radiologic variables were collected from the hospital archives. All patients underwent head CT scan or MRI showing the involvement of the majority or complete middle cerebral artery distribution area. Hemicraniectomy with duraplasty was the main surgical procedure. The mortality and the outcome using Modified Rankin Scale (mRS) were reported. Results: A total of 9 patients with large hemispheric stroke who underwent DC were reviewed retrospectively. The mean age was 53 years with extremes of 39- and 67-year-old. There were 6 males and 3 females. The main risk factors of stroke were hypertension and diabetes in 5 and 3 cases, respectively. The large craniectomy was made and the dura was opened in the stellate fashion. Duraplasty was performed using epicrane in 7 patients and fascia lata aponeurosis in 2 patients. In the postoperative course, 4 (44%) patients died in 1 month and the mortality rate was 66.6% in 6 months. Conclusion: Decompressive craniectomy is a life-saving procedure used to manage cerebral edema after stroke onset. In our study, the high mortality rate could be explained by many factors, such as patient selection criteria and timing of surgery. Early surgery within the first 48 hours in patients less than 60 years old with large hemispheric stroke is recommended.
基金funded by the National Natural Science Foundation of China(Grant Nos:42125604&42171143)the Second Tibetan Plateau Scientific Expedition and Research Program(STEP)(Grant No.2019QZKK0201)。
文摘The snow density is a fundamental variable of the snow physical evolution processes,which can reflect the snowpack condition due to the thermal and gravitational compaction.Snow density is a bridge to transfer snow depth to snow water equivalent(SWE)for the snow water resources research.Therefore,it is important to understand the spatiotemporal distribution of snow density for the appropriate estimation of SWE.In this study,in situ snow densities from more than 6,000 stations in the Northern Hemisphere were used to analyze the spatial and temporal variations in snow density.The results displayed that snow density varied spatially and temporally in the Northern Hemisphere,with range of below 0.1 to over 0.4 g/cm^(3).The average snow densities in the mountainous regions of western North America,southeastern Canada,and Europe range from approximately 0.24 to 0.26 g/cm^(3),which is significantly greater than the values of 0.16–0.17 g/cm^(3)observed in Siberia,central Canada,the Great Plains of the United States,and China.The seasonal growth rates also present large spatial heterogeneity.The rates are over 0.024 g/cm^(3)per month in Southeastern Canada,the west mountain of North America and Europe,approximately 0.017 g/cm^(3)per month in Siberia,much larger than approximately 0.004 g/cm^(3)per month in other regions.Snow cover duration is a critical factor to determine the snow density.This study endorses the small snow density in China based on meteorological station observations,which results from that the meteorological stations are dominantly distributed in plain areas with relative short snow cover duration and shallow snow.
基金supported by the National Natural Science Foundation of China(Grants 42325404,42120104003,42204164,42474219 and U22A2006)the Chinese Meridian Project,the International Partnership Program of Chinese Academy of Sciences(Grant 183311KYSB20200003)+7 种基金Shandong Provincial Natural Science Foundation(Grants ZR2022QD077,ZR2022MD034)the Stable-Support Scientific Project of China Research Institute of Radiowave Propagation(Grant A132312191)the foundation of the National Key Laboratory of Electromagnetic Environment(Grant 6142403180204)the Chongqing Natural Science Foundation(Grants cstc2021ycjh-bgzxm0072,CSTB2023NSCQ-LZX0082)National Program on Key Basic Research Project(Grant 2022173-SD-1)The work in Norway is supported by the Research Council of Norway Grant 326039Work at UCLA has been supported by NSF grant AGS-2055192This research was supported by the International Space Science Institute(ISSI)in Bern and Beijing,through ISSI International Team project#511(Multi-Scale Magnetosphere-Ionosphere-Thermosphere Interaction).
文摘This paper is a statistical survey of Southern Hemisphere cold and hot polar cap patches,in relation to the interplanetary magnetic field(IMF)and ionospheric convection geometry.A total of 11,946 patch events were identified by Defense Meteorological Satellite Program(DMSP)F16 during the years 2011 to 2022.A temperature ratio of ion/electron temperature(T_(i)/T_(e))<0.68 is recommended to define a hot patch in the Southern Hemisphere,otherwise it is defined as a cold patch.The cold and hot patches have different dependencies on IMF clock angle,while their dependencies on IMF cone angle are similar.Both cold and hot patches appear most often on the duskside,and the distribution of cold patches gradually decreases from the dayside to the nightside,while hot patches have a higher occurrence rate near 14 and 21 magnetic local time(MLT).Moreover,we compared the key plasma characteristics of polar cap cold and hot patches in the Southern and Northern Hemispheres.The intensity of the duskside upward field-aligned current of patches in the Southern Hemisphere(SH)is stronger than that in the Northern Hemisphere(SH),which may be due to the discrepancy in conductivities between the two hemispheres,caused by the tilted dipole.In both hemispheres,the downward soft-electron energy flux of the dawnside patches is significantly greater than that of the duskside patches.
基金Supported by the Medical Research Project of the Chongqing Municipal Health Commission,No.2024WSJK110.
文摘BACKGROUND Currently,adolescent depression is one of the most significant public health concerns,markedly influencing emotional,cognitive,and social maturation.Despite advancements in distinguish the neurobiological substrates underlying depression,the intricate patterns of disrupted brain network connectivity in adolescents warrant further exploration.AIM To elucidate the neural correlates of adolescent depression by examining brain network connectivity using resting-state functional magnetic resonance imaging(rs-fMRI).METHODS The study cohort comprised 74 depressed adolescents and 59 healthy controls aged 12 to 17 years.Participants underwent rs-fMRI to evaluate functional connectivity within and across critical brain networks,including the visual,default mode network(DMN),dorsal attention,salience,somatomotor,and frontoparietal control networks.RESULTS Analyses revealed pronounced functional disparities within key neural circuits among adolescents with depression.The results demonstrated existence of hemispheric asymmetries characterized by enhanced activity in the left visual network,which contrasted the diminished activity in the right hemisphere.The DMN facilitated increased activity within the left prefrontal cortex and reduced engagement in the right hemisphere,implicating disrupted self-referential and emotional processing mechanisms.Additionally,an overactive right dorsal attention network and a hypoactive salience network were identified,underscoring significant abnormalities in attentional and emotional regulation in adolescent depression.CONCLUSION The findings from this study underscore distinct neural connectivity disruptions in adolescent depression,underscoring the critical role of specific neurobiological markers for precise early diagnosis of adolescent depression.The observed functional asymmetries and network-specific deviations elucidate the complex neurobiological architecture of adolescent depression,supporting the development of targeted therapeutic strategies.
文摘Spatial confinement has great potential for Laser Induced Breakdown Spectroscopy （LIBS） instruments after it has been proven that it has the ability to enhance the LIBS signal strength and repeatability. In order to achieve in-situ measurement of heavy metals in farmland soils by LIBS, a hemispherical spatial confinement device is designed and used to collect plasma spectra, in which the optical fibers directly collect the breakdown spectroscopy of the soil samples. This device could effectively increase the stability of the spectrum intensity of soil. It also has other advantages, such as ease of installation, and its small and compact size. The relationship between the spectrum intensity and the laser pulse energy is studied for this device. It is found that the breakdown threshold is 160 cm-2, and when the laser fluence increases to 250 J/cm2, the spectrum intensity reaches its maximum. Four different kinds of laser pulse energy were set up and in each case the limits of detection of Cd, Cu, Ni, Pb and Zn were calculated. The results show that when the laser pulse fluence was 2.12 GW/cm2, we obtained the smallest limits of detection of these heavy metals, which are all under 10 mg/kg. This device can satisfy the needs of heavy metal in-situ detection, and in the next step it will be integrated into a portable LIBS instrument.
基金Project supported by the National Natural Science Foundation of China (No. 50478062) and Natural Science Foundation of Beijing (No. 8052015).
文摘An analytical solution for the three-dimensional scattering and diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits is developed by employing Fourier-Bessel series expansion technique. Unlike previous studies, in which the saturated soil deposits were simulated with the single-phase elastic theory, in this paper, they are simulated with Biot＇s dynamic theory for saturated porous media, and the half space is assumed as a single-phase elastic medium. The effects of the dimensionless frequency, the incidence angle of P-wave and the porosity of soil deposits on the surface displacement magnifications of the hemispherical alluvial valley are investigated. Numerical results show that the existence of a saturated hemispherical alluvial valley has much influence on the surface displacement magnifications. It is more reasonable to simulate soil deposits with Biot＇s dynamic theory when evaluating the displacement responses of a hemispherical alluvial valley with an incidence of P-waves.
基金the Pre-Research Fund during the“13th Five-Year Plan” (No. 41417060101)。
文摘The mass non-uniformity of hemispherical resonator is one of reasons for frequency split,and frequency split can cause gyroscope to drift.Therefore,it is of great significance to analyze the relationship between mass non-uniformity and frequency split,which can provide a theoretical basis for mass balance of imperfect resonator.The starting point of error mechanism analysis for gyroscope is the motion equations of resonator.Firstly,based on the Kirchhoff-Love hypothesis in the elastic thin shell theory,the geometric deformation equations of resonator are deduced.Secondly,the deformation energy equation of resonator is derived according to the vibration mode and relationship between the stress and strain of hemispherical thin shell.Thirdly,the kinetic energy equation of resonator is deduced by the Coriolis theorem.Finally,the motion equations of resonator are established by the Lagrange mechanics principle.The theoretical values of precession factor and natural frequency are calculated by the motion equations,which are substantially consistent with the ones by the finite element method and practical measurement,the errors are within a reasonable range.Simultaneously,the varying trend of natural frequency with respect to the geometrical and physical parameters of resonator by the motion equations is consistent with that by the finite element analysis.The above conclusions prove the correctness and rationality of motion equations.Similarly,the motion equations of resonator with mass non-uniformity are established by the same modeling method in case of ignoring the input angular rate and damping,and the state equations with respect to the velocity and displacement of vibration system are derived,then twonatural frequencies are solved by the characteristic equation.It is concluded that one of reasons for frequency split is the 4 th harmonic of mass non-uniformity,and thus much attention should be paid to minimizing the 4 th harmonic of mass non-uniformity in the course of mass balancing for imperfect resonator.
文摘Daily Total Column Ozone (TCO) measurements compiled from Total Ozone Mapping Spectrometer (TOMS) and Ozone Monitoring Instruments (OMI) were used to analyze the global and hemispherical TCO interannual variations. Two periods of TCO measurements were analyzed separately covering full years. For the 1978-1994 period, the TCO showed a global decade decrease rate of 13.45 DU (about -4.3%). For the Northern Hemisphere(NH) the decade decrease rate was of 12.96 DU (-4.0%), while in the Southern Hemisphere (SH) was of 13.57 DU (-4.5%). These decreases in ozone trends, using the totality of TOMS and OMI satellite measurements, are greater than those reported in literature. The 1998-2014 period global TCO decade decrease rate was of 1.56 DU, corresponding 0.94 DU and 0.138 DU for the NH and SH, respectively. The global TCO variations must show a double annual periodicity, the first one with maxima in March due to the Northern Hemisphere (NH) and the second one during September due to the Southern Hemisphere (SH). However, the maxima due to SH TCO interannual variations have gradually vanished. A disturbance in the SH TCO interannual variations has appeared since 1980;graphically the periodicity brakes down and transforms to a double peak from 1985 and on. This effect can be attributed to the hemispheric impact of the ozone hole at the South Pole. Between October 1, 2004 and December 14, 2005 TOMS and OMI have recorded this disturbance unequivocally. We conclude that the disturbance in SH TCO has an irreversible character.
基金The project was supported by the National Natural Science Foundation of China (50478062 and 10532070)Open Fund at the Key Laboratory of Urban Security and Disaster Engineering (Beijing University of Technology)Chinese Ministry of Education.
文摘Abstract An analytical solution to the three-dimen-sional scattering and diffraction of plane SV-waves by a saturated hemispherical alluvial valley in elastic half-space is obtained by using Fourier-Bessel series expan-sion technique. The hemispherical alluvial valley with saturated soil deposits is simulated with Biot＇s dynamic theory for saturated porous media. The following conclusions based on numerical results can be drawn： （1） there are a significant differences in the seismic response simulation between the previous single-phase models and the present two-phase model; （2） the nor-malized displacements on the free surface of the alluvial valley depend mainly on the incident wave angles, the dimensionless frequency of the incident SV waves and the porosity of sediments; （3） with the increase of the incident angle, the displacement distributions become more complicated; and the displacements on the free surface of the alluvial valley increase as the porosity of sediments increases.
基金the National Natural Science Foundation of China (No.50805121)the National Basic Research Program (973) of China (No.2007CB13802)
文摘In this paper,the effects of hardening exponent,yield strength and elastic modulus on the deformability of near hemispherical shells are investigated by means of finite element method and orthogonal experiment design.The largest eccentric angle during the deformation process and thickness reduction after the deformation are introduced to estimate the deformability quantitatively according to the deformation characteristics of near hemispherical shells.The results indicate that the hardening exponent is the most influential parameter,followed by elastic modulus and yield strength.The shell exhibits good deformability when the hardening exponent and elastic modulus are in the range of 0.1-0.125 and 70-108 GPa,respectively.
文摘Hemispherical Resonator Gyroscope(HRG)is a classical high precision Coriolis Vibration Gyroscope(CVG),which performs attitude estimation of carrier by detecting the precession of standing wave of resonator,thus,the drift of standing wave of resonator has a great influence on the output accuracy of gyroscope,where the quality factor non-uniformity of resonator is one of main error sources.Ring electrode is a classical excitation structure of HRG because the standing wave can precess freely under its excitation,which makes the gyroscope have more accurate scale factor,larger measurement range and better dynamic characteristics.In this paper,the equations of motion of an ideal resonator excited by a ring electrode are derived by the elastic thin shell theory and Lagrange mechanical principle,then the corresponding equivalent mechanical model is established.According to the“average method”,it can be seen that the ideal resonator excited by the ring electrode works in integral mode,and any position in the circumferential direction of resonator can be a working point,which means that the quality factor non-uniformity has a great effect on the drift of standing wave.Therefore,the equations of motion of resonator with quality factor non-uniformity under the ring electrode excitation are deduced by the equivalent mechanical model,and the drift model of standing wave is established by the“average method”,it can be found that both the amplitude of quality factor non-uniformity and angle between the“inherent damping axis”and antinode axis of standing wave can affect the drift rate of standing wave.Moreover,the drift model indicates that if the input angular rate does not reach the threshold,the precession angular rate of standing wave will appear“self-locking”phenomenon,that is,the gyroscope will lose the integral effect.
基金Funded by the National Natural Science Foundation of China(No.51203144)
文摘In order to investigate the draping behavior of non-crimp fabrics（NCFs）, two types of carbon NCFs with tricot-chain stitches or chain stitches were formed on a hemispherical mould via a stretch forming process. The shear angle and forming defects of the fabrics were measured on the hemisphere, under different blank holder forces（BHFs）. The results showed that increasing BHF could enhance the shear angle slightly, reduce the asymmetry for the deformation of the fabrics, and change the main type of the process-induced defects. Besides, compression tests were performed on the corresponding composite components. By analyzing the change of fiber volume fraction and structural parameters of the textile reinforcements, the effects of draping behavior of NCFs on the mechanical performance of the composites were studied. The results reveal that draping process has distinguishable impacts on the mechanical properties of the final components, which is closely related to the stitching pattern of the NCFs.
基金the National Natural Science Foundation of China(No.61574093)the National Key Laboratory of Science and Technology on Nano/Micro Fabrication(No.614280504010317)+1 种基金the Aerospace Science and Technology Innovation Fund(No.16GFZJJ01-309),the Space Advanced Technology Joint Research Innovation Fund(No.USCAST2016-5)the Professional Technical Service Platform of Shanghai(No.19DZ2291103)。
文摘Maximizing quality factor (Q) is essential to improve the performance of micro hemispherical shell resonators (μHSRs) which can be used in microelectromechanical system (MEMS) gyroscopes to measure angular rotation.Several energy dissipation mechanisms limit Q,where thermoelastic dissipation (TED) is the major one and studied in this paper.Fully coupled thermo-mechanical equations for calculating TED are formulated,and then temperature distribution in a deformed μHSR and its quality factor related to TED (QTED) are obtained by solving the equations through a finite-element method (FEM).It has been found that different fabrication process conditions can obtain various geometrical parameters in our previous studies.In order to provide guidelines for the design and fabrication of μHSRs,the effects of their geometry on resonant frequency (f0) and QTED are studied.The change of anchor height and small enough anchor radius have no effect on both f0 and QTED,but the shell size including its radius,thickness and height has significant impact on f0 and QTED.It is found that whether a μHSR has lower f0 and higher QTED or higher f0 and higher QTED can be achieved by changing these geometrical parameters.The results presented in this paper can also be applied to other similar resonators.
基金financially supported by the National Security Major Foundation Research Project(973)of China(6133050102)the National Natural Science Foundation of China(Grant No.51409253)
文摘Ballistic experiments were conducted on thin steel plates that are normally impacted by hemisphericalnosed projectiles at velocities higher than their ballistic limits. The deformation and failure modes of the thin steel plates were analyzed. A new method was proposed according to the experimental results and the perforation phenomenon of the thin steel plates to determine the radius of the bulging region. In establishing this new method, a dynamic method combined with the plastic wave propagation concept based on the rigid plastic assumption was adopted. The whole perforation process was divided into four consecutive stages, namely, bulging deformation, dishing deformation, ductile hole enlargement, and projectile exit. On the basis of the energy conservation principle, a new model was developed to predict the residual velocities of hemispherical-nosed projectiles that perforate thin steel plates at low velocities.The results obtained from the theoretical calculations by the present model were compared with the experimental results. Theoretical predictions were in good agreement with the experimental results in terms of both the radius of the bulging region and the residual velocity of the projectile when the strain rate effects of the target material during each stage were considered.
基金the National Basic Research Program(973) of China (No.2007CB613802)the National Natural Science Foundation of China (No.50805121)
文摘The asymmetric deformation and eccentricity problems of near hemispherical diaphragm under the uniform surface load are quantitatively characterized in the paper.The analysis is based on a 3D finite element analysis (FEA) model established according to elastic-plasticity and large displacement nonlinear finite element method.Besides,the deformation experiments are taken to validate the reliability of FEA model which shows that the simulation results are in good agreement with the experimental results.Then,three angle parameters,deflection angle β,circumvolving angle θ and distributing angle γ,are introduced and expressed to characterize the asymmetric deformation and eccentricity quantitatively.According to the angle parameters,the inversion processes of uniform thickness diaphragm and varying thickness diaphragm are calculated respectively.The inversion process of varying thickness diaphragm is much steadier than that of uniform thickness diaphragm.The present results show that the asymmetric deformation process can be characterized by curve of three angle parameters (β,θ,γ) exactly,the degrees of eccentricity can be indicated by the final value of deflection angle and the eccentricity position can be characterized by the final values of the three angle parameters.
基金Projected supported by the National Natural Science Foundation of China
文摘This paper makes detailed analyses tor the flexural vibration (frequency) of the hemispherical shell and presents the varying laws of frequency with the rarving boundary angles and the wall thickness of the above shell, It is an important value to develop the instrument, such as hemispherical resonator gyro (HRG), whose sensing component is a hemispherical shell.
文摘This paper is a study of the gravitational attraction between two uniform hemispherical masses aligned such that the pair is cylindrically symmetric. Three variations are considered: flat side to flat side, curved side to curved side, and flat side to curved side. Expressions for the second and third variation are derived from the first, with the use of superposition and the well-known gravitational behavior of a spherical mass as equivalent to a point mass at its center. The study covers two masses of equal diameter and of different diameters, such that one is four times that of the other. Calculations are done for separations from zero to fifty times the radius of the larger of the two, which is effectively the asymptotic limit. It is demonstrated that at any separation, the force can be expressed as if the two hemispheres were point masses separated by a certain distance. Expressions for that distance and the location of the (fictitious) point masses within each hemisphere are presented. Unlike the case of two spherical masses, the location within their respective hemisphere is not necessarily the same for each point and both are dependent upon the separation between the two hemispheres. The calculation for the first variation is done in two ways. The first is a “brute force” multi-dimensional integral with the help of Wolfram Mathematica. The second is an axial expansion for the potential modified for off-axis locations by Legendre polynomials. With only a few terms in the expansion, the results of the second method are in extremely good agreement with those of the first. Finally, an interesting application to a split earth is presented.
