PENG Xiaoqiang , DAI Yifan , LI Shengyi
2015, 37(6):1-7. DOI: 10.11887/j.cn.201506001
Abstract:Nano-precision optical surfaces are highly demanded in fields of lithography, synchrotron radiation, space observation and inertial confinement fusion. With the performance rising of equipment, the surface accuracy and roughness of optical elements are further strictly required, which is approaching the theoretical physical limitation. The rising demands bring about tougher challenges for optical fabrication, and push the technology to the frontier in the nano-precision fabrication area as well. Magnetorheological polishing and ion beam were figured in this research: the stability of material removal rate in nano-level, controllable compensation of complex asphere and dynamics design of equipment axes. The self-developed innovation polishing equipment was produced and the nano-precision fabrication of typical optical elements was realized, which provides strong support for relative science programs of our country, in terms of fabrication technology.
WAN Wen , DAI Yifan , SHI Feng , PENG Xiaoqiang
2015, 37(6):8-11. DOI: 10.11887/j.cn.201506002
Abstract:In order to improve the laser damage performance of ultraviolet fused silica, a method which combines magneto-rheological finishing (MRF) with HF acid etching process was proposed to reduce crack and contamination defects in the machining process of traditional processing method. MRF can effectively remove the crack defects with its unique shear and removal principle in traditional machining process while not produce new crack defect. HF acid etching process can effectively reduce metal elements contamination. The experimental results show that fused silica sample using the combined process has a better antidamage performance. Damage density reduced from 0.2mm-2 to 0.008mm-2 under the laser flux irradiation of 7J/cm2?3ω and from 1mm-2 to 0.1mm-2 at the irradiation of 8J/cm2?3ω.
TIAN Fujing , YIN Ziqiang , LI Shengyi
2015, 37(6):12-16. DOI: 10.11887/j.cn.201506003
Abstract:A novel fast tool servo system driven by a voice coil motor has been developed,which has the total stroke of 30mm and the maximum acceleration of 920m/s2. The system model was obtained by the experimental process, which was used to design the controller. The micro-lens array, a typical optical surface with complex structure, which was machined, and the experimental results were tested and analyzed. Experimental results indicate that the performances of the developed fast tool servo system satisfy the machining requirements. A good foundation for future application was built in real manufacture.
LU Ying , XIE Xuhui , ZHOU Lin , PENG Wenqiang
2015, 37(6):17-20. DOI: 10.11887/j.cn.201506004
Abstract:In order to solve the problem of ion beam figuring, the design and performance analysis of ultra-precision ion beam polishing tool was studied. The research on ion beam polishing tools design method, the structure design of a 3-grid focused ion optics system and the performance analysis of ion beam was covered. Based on the theoretic calculation and computer simulation, a focused ion optical system was developed. In this ion optical system, an integrated plasma bridge neutralizer was designed to fully neutralize ion beam extracted from ion source. Finally, a 150mm flat optics was figured by two designed focused ion optical systems with 15mm and 10mm output diameter. Figuring result shows that the contour error of this flat is decreased from 15.58nm RMS to 0.79nm RMS. Result further demonstrates that the designed ion beam polishing tool is very efficient to optical ion beam figuring.
GUO Meng , DAI Yifan , PENG Xiaoqiang , TIE Guipeng
2015, 37(6):21-25. DOI: 10.11887/j.cn.201506005
Abstract:How to realize the high-precision measurement of the shape and position errors of the cuboid optical element and how to use the measurement data to correct these errors are the main problem in the manufacturing process. A shape and position errors measurement method of the cuboid optical elements based on the wave-front interference theory was presented and the measurement system with large aperture instrument and high-precision indexing plate was built. With the help of this system, the parallelism and verticality measurement of the cuboid optical element in the accuracy of 1μm/400mm was achieved and the high-accuracy integrated distribution data of shape and position errors were obtained. The high-accuracy machining of this kind of optical element was achieved by using modern optical machining technology as magnetorheological finishing and computer controlled optical surfacing.
TIAN Ye , DAI Yifan , SHI Feng , PENG Xiaoqiang , HAN Kai , ZHU Zhiwu , WAN Wen
2015, 37(6):26-29. DOI: 10.11887/j.cn.201506006
Abstract:In high energy laser systems, the energy absorption rate (EAR) of mono-crystalline silicon reflector is a vital index in the term of elements quality. The substrate process quality affects the EAR of coated optics dramatically. The surface roughness and scratch density for a series of silicon optics were measured. Based on the result, the co-influence and relationship between surface characteristic and EAR were analyzed. It is indicated that, surface roughness level is positively correlated to EAR and decides the global average rate of the entire surface. When the root-mean-square of roughness falls from 0.668nm to 0.345nm, the EAR decreases by 28.0%. In contrast, few scratches do not distinctly raise the EAR, the fluctuation is less than 3.1%. However, the surface scratch may generate laser-induced damage. Furthermore, when the scratch density is large enough, the subsequent EAR will grow constantly, for example, the amplitude reaches 18.3% after the 400s irradiation, compared with 100s irradiation.
DU Hang , LI Shengyi , SONG Ci
2015, 37(6):30-33. DOI: 10.11887/j.cn.201506007
Abstract:CCOS (Computer Controlled Optical Surfacing) is an important technology for off-axis aspheric surface processing. Edge effect of small tool manufacturing restricts the machining precision and efficiency of CCOS technology. After the key parameters of edge effect were obtained, the warping edge effect of CCOS combined with the residual error trace contour path planning was corrected. The full aperture optical measurement data of a stadium off-axis aspheric is achieved by processing, which provides the fundament for subsequent finishing.
SHI Baolu , DAI Yifan , XIE Xuhui , ZHOU Lin
2015, 37(6):34-38. DOI: 10.11887/j.cn.201506008
Abstract:Since the chemical stability of SiC is extremely high, there is a low efficiency of SiC mirrors for conventional inductively coupled plasma (ICP) processing method. The plasma processing experiment reveals that the increase of plasma radio-frequency can enhance the arc discharge effect between plasma and SiC. The enhancing effect of arc can increase the processing efficiency of SiC, so the arc-enhanced plasma (AEP) processing method was developed. In order to research the formation principle of arc, the voltages of ICP and AEP were measured respectively by using the self-made probe. The conventional and the arc-enhanced methods were employed respectively to conduct linear scanning machining experiment on the sintered silicon carbide (S-SiC), which demonstrates the higher processing efficiency of AEP method.
PENG Yanglin , DAI Yifan , SONG Ci , SHI Feng
2015, 37(6):39-42. DOI: 10.11887/j.cn.201506009
Abstract:The differences of wheel shape, machining path, motion axis combinations, and workpiece placed posture, will cause alterations of grinding model, which is the primary problem for the solution to curve generation. A geometric model of disk arc wheel was established; a one-to-one mapping relationship was built between workpiece and grinding wheel through the grinding point normal vector matching. After coordinate transformation, the corresponding machining tool motion trail which can be used for grinding was obtained. The uniform grinding geometry model of curve surface was formed, and the calculating flow of tool path was developed. The model has a wide application range, and can solve the tool path generating problem of multiple type grinding processes, which guarantees a high precision curve surface grinding.
ZHU Dengchao , DAI Yifan , GUAN Chaoliang , WANG Guilin
2015, 37(6):43-47. DOI: 10.11887/j.cn.201506010
Abstract:Deep aspheric optical element can obviously improve the aerodynamic performance of the optical system, thus enhancing and optimizing the comprehensive performance of optical system. The grinding method can be used for primary machining of such elements. However, subsurface damage will be inevitably introduced by this method, and grinding parameters will be changing in real time in the processing of deep aspheric surface, so the depths of subsurface damage of the workpiece are inconsistent. In view of this situation, subsurface damage prediction model was established. Combined with the characteristics of semi-sphere wheels grinding, a theoretical calculation was made to predict distributive regularity of the aspheric grinding subsurface damage depths aspheric characteristics. On the basis of the above work, a simulation experiment on hot-pressing polycrystalline magnesium fluoride plane was carried out. Subsurface damage depths under different conditions of grinding parameters were acquired by the method of magnetorheological finishing spot. Results show that the depth range is between 12.79μm and 20.96μm, which increases from inside to outside along the radius of the workpiece and the results are coincide with previous forecast model.
CHENG Peng , LI Qinglian , ZHANG Xinqiao , KANG Zhongtao , CHEN Huiyuan
2015, 37(6):48-53. DOI: 10.11887/j.cn.201506011
Abstract:The mode transition of combined-cycle engine makes it necessary to throttle the mass flow rate of propellants continuously and stably. Aiming at gaseous propellant, a throttleable sonic nozzle was designed on the basis of normal sonic nozzle. The throttleable sonic nozzle achieves continuous throttling through a plug moving along axis. By adopting the twice-envelope method, the contour of the plug was designed to satisfy the linear characteristic of mass flow rate. Performances of the throttleable sonic nozzle were investigated through the numerical simulation of computational fluid dynamics. It is found that the mass flow rate is independent under the back pressure when the back pressure is lower than the critical back pressure and the critical back pressure ratio increases as mass flow rate decreases. The linear relationship between mass flow rate of linear throttleable sonic nozzle and location of plug was verified and results show that the discharge coefficient is high and hardly be affected by the location of plug.
2015, 37(6):54-60. DOI: 10.11887/j.cn.201506012
Abstract:The fault character of pyramid configuration single gimbal control moment gyroscopes (SGCMG) was analyzed. The calculation approach solving the replanned attitude maneuver path was proposed based on the Legendre pseudo spectral method. Considering the unpredictability of the SGCMG fault, an adaptive steering law was designed. Using this steering law, the fault state of SGCMG was detected and the parameters of the steering law was self-tuned. The simulation results reveal that, the large angle attitude maneuver can be accomplished by using the path replanned method and the adaptive steering law, when the SGCMG encounter partial fault. The proposed method can cope with the partial fault state of the SGCMG during attitude maneuver effectively, thus improving the security and reliability of the space station attitude maneuver mission.
ZHAO Lufeng , LYU Zhenzhou , ZHANG Leigang , WANG Xinwei
2015, 37(6):61-68. DOI: 10.11887/j.cn.201506013
Abstract:Considering the relations among multi-outputs and the mean of single output, the mathematical expectation of single dimensional variable and covariance metric of multi-dimensional variables were introduced into the validation metrics for models. The new metrics of LA-3M and LR-3M were proposed for validating multi-responses at a single validation site, while the metrics of GA-3M and GR3M were proposed to collect data of multiple responses observed at multiple validation sites. These metrics were examined through a numerical test case and an engineering example to illustrate their feasibility and effectiveness. Results show that the proposed metrics are efficient and they can easily measure the differential degree of multiple responses between calculation model and physical experiment.
CHEN Feiqiang , NIE Junwei , SU Yingxue , WANG Feixue
2015, 37(6):69-73,83. DOI: 10.11887/j.cn.201506014
Abstract:An anti-jamming algorithm for satellite navigation antenna array aided by carrier phase was proposed to mitigate the interference of satellite navigation receiver. The blind null steering was achieved through the algorithm and the blind beam steering was realized simultaneously with the aid of the carrier phase of the output signal from each individual antenna, and two steerings could be switched by a control algorithm. The simulation results show that the proposed algorithm performs well at “cold” start when the receiver is initiated in strong interference conditions. It can enhance the satellite signals by steering beams to the satellites under noninterference or weak interference conditions, so the visibility and positioning accuracy of satellite are improved. The proposed algorithm can work normally without the aid of inertial measurement unit, and is robust to the steering vector errors caused by amplitude-phase mismatch. It has less cost than traditional beamforming algorithms for implementation.
PANG Jing , LIU Yingxiang , TANG Xiaomei , OU Gang
2015, 37(6):74-77. DOI: 10.11887/j.cn.201506015
Abstract:The time delay and dispersion effects caused by the ionosphere need to be simulated in the navigation signal simulator. The higher order BOC modulation signal in the new generation of satellite navigation signals has a wider bandwidth than the traditional navigation signal. The simulation method of the traditional navigation signal can lead to the non-negligible modeling error of the ionosphere delay. An improved simulation method of BOC signal based on the bilateral model was presented, and was verified by simulation. The simulation results show that there is an error of meter in the ionosphere delay of the BOC (14, 2) signal, and the improved method can accurately simulate the effect of the ionosphere dispersion.
MA Xiantong , LUO Jingqing , WU Shilong
2015, 37(6):78-83. DOI: 10.11887/j.cn.201506016
Abstract:In multi-station electronic reconnaissance system, TDOA (Time Difference Of Arrival) of the pulse signal received by two stations can be used for signal sorting and positioning. A novel method for joint sorting and location based on TDOA and multiparameter of multi-station was proposed to deal with the problem of being unable to sort the signals with very few pulses caused by weak correlation between pulse information of complex signals in order to obtain final sorting and location results, the pulses of all stations were presorted by the presented method with TDOA window information and the presorting results were integrated according to targets’ positions. Simulation results show that the proposed method is applicable and effective, which can implement the pulse signal sorting and positioning with very few pulses and even a single pulse sorting and positioning.
PENG Shuangchun , ZHU Jianwen , TANG Guojian , CHEN Kejun
2015, 37(6):84-90. DOI: 10.11887/j.cn.201506017
Abstract:A new three-dimensional (3D) nonlinear pseudo optimal guidance law based on twist technology was proposed to resolve the coupling problem between different channels for the guidance of hypersonic vehicle. The basic concepts of angle vector, line-of-sight (LOS) twist and its rate were introduced, and through equivalence certification, the guidance problem was converted into control problem of LOS twist and twist rate, with the conclusion that the LOS twist and its rate control equals to the LOS orientation and LOS angle rate control. A 3D nonlinear guidance model was constructed on the basis of LOS twist and the rate models through twist technology. In addition, the 3D nonlinear model was decoupled into the linear one by introducing pseudo control variable to reduce the complicate resolution of differential Riccati equation. The 3D nonlinear pseudo optimal guidance laws were derived using linear quadratic regulator (LQR) corresponding to no ending constraint and ending constraints respectively. With these novel guidance laws, the decoupling process can be well avoided and the optimal characteristics of guidance parameters were ensured as well. The validity of these guidance laws was also validated through simulation experiments.
2015, 37(6):91-95. DOI: 10.11887/j.cn.201506018
Abstract:Based on Morison equation, a dynamics model of water-exit missile under wave disturbance was built, which takes the coupling between wave and motion of missile into account. The water-exit motion of small submarine-launched missile was simulated and analyzed. The effects of coupling between wave and motion of missile body to wave disturbance, illustrated by change process of attitude of missile, were studied. The influences of sea condition, wave phase, velocity and attitude of missile to the motion of water-exit course were analyzed. The result indicates that the characteristics of submarine-launched, which are small in size and fast in water-exit, reduce the impact of wave to attitude of missile. In addition, the wave disturbance to high-speed water-exit missile is significantly affected by sea condition, duration of water-exit course and wave phase. The dynamic model which takes the coupling between wave and motion of missile into account is more accurate.
2015, 37(6):96-102. DOI: 10.11887/j.cn.201506019
Abstract:Aiming at the problem of automatic planning for agile imaging satellite, the architecture of automatic planning combined with planning, decision making and information feedback was presented. The architecture, components and functions of automatic planning were introduced. The model based on constraint network on timelines was constructed according to the satellite physical constrains and operational constrains. The activity sequences were constructed through connecting various activities. And a heuristic algorithm based on activity sequences was presented. The algorithm was composed of two parts: planning and decision making. The algorithm could plan based on heuristic rules when satellite was performing every activity sequence, and make decision when the activity sequence was completed. Finally, satisfactory results are obtained through experiments, which show that the architecture of automatic planning and the model are reasonable, and the algorithm is efficient.
ZHANG Jiaming , LIU Zhong , SHI Jianmai , HE Yunyue , WANG Yishan , CHEN Chao
2015, 37(6):103-109. DOI: 10.11887/j.cn.201506020
Abstract:Aiming at minimizing the departure time of the last helicopter, a mathematical programming model for multiple helicopters in multiple airports was set up to solve the departure time scheduling problem of multiple helicopters. A heuristic algorithm based on task priority was designed to solve the model, a scheme to deal with the airway intersection was put forward and an algorithm to solve the departure time was also established. Basing on the disaster rescue of the 6.8 magnitude earthquake in Yunnan Ludian, a departure time scheduling case which includes 24 helicopters in 12 airports was designed to validate the model and algorithms. The influence of airway intersection and the treatment measures were deeply discussed. The results demonstrate the efficiency of the model and algorithms in dealing with the scheduling problem of multiple helicopters.
LI Xin , YANG Xuejun , XU Xinhai
2015, 37(6):110-115. DOI: 10.11887/j.cn.201506021
Abstract:While providing big data computing services using Internet resources, there remains a big challenge to researchers, including heterogeneity of Internet resources, dynamics of Internet resources and long latency of Internet communication. Current influent distributed computing models still have some shortage. A novel distributed stream computing model was proposed based on the traditional stream computing model, including the distributed stream programming model and resource management can efficiently support multiple parallel execution modes. The prototype system implemented on the 10 CPU-GPU heterogeneous nodes. Seven different benchmarks used in the simulation experiment. The experimental result shows that the distributed stream architecture can achieve the speedup of at least on average over the local serial computing, with significant potential for applications.
2015, 37(6):116-120 ,134. DOI: 10.11887/j.cn.201506022
Abstract:Robust multi-view triangulation algorithms usually rely on an empirical reprojection error threshold to identify and remove the outliers. The selection of such threshold is critical to both the quantity of successfully reconstructed scene point and its accuracy. Based on the analysis of the noise in feature point localization and the geometry of epipolar transfer, the uncertainty propagation model in epipolar transfer was derived. A novel noise scale estimation approach based on kernel density estimation was proposed and the estimated noise scale was further incorporated into robust state-of-the-art multi-view triangulation algorithm. Experimental results demonstrate that the proposed method is able to obtain accurate estimation of noise scale and to improve the 3D reconstruction quality of multi-view triangulation algorithm significantly.
2015, 37(6):121-128. DOI: 10.11887/j.cn.201506023
Abstract:Ridge regression is an important method in supervised learning. It is wide used in multi-class classification and recognition. An important step in ridge regression is to define a special multivariate label matrix, which is used to encode multi-class samples. By regarding the ridge regression as a supervised learning method based on graph, methods for constructing multivariate label matrix were extended. On the basis of ridge regression, a new method named sparse smooth ridge regression was proposed by considering the global dimension smoothness and the sparseness of the projection matrix. Experiments on several public datasets show that the proposed method performs better than a series of state-of-the-art supervised linear algorithms. Furthermore, experiments show that the proposed label matrix construction methods do not reduce the performance of the original ridge regression. Besides, it can further improve the performance of the proposed sparse smooth ridge regression.
ZHAO Guangyao , CHENG Lei , LI Ruilin , LI Chao , SUN Bing
2015, 37(6):129-134. DOI: 10.11887/j.cn.201506024
Abstract:PUFFIN is a lightweight block cipher, in which the block length is 64 bit while the key size is 128 bit. The integral cryptanalysis resistance ability of PUFFIN was analyzed. The existence of 5 and 6 round integral distinguisher in PUFFIN was constructed and proved. An integral attack on 8 round PUFFIN was mounted by 6 round integral distinguisher to recover 2 round 100 bit round cipher. The data complexity of the attack is 220 chosen plaintexts, the time complexity is about 233 8 round encryptions, and the space complexity is 220. This has been the best integral attack on PUFFIN up to now.
WANG Qiang , WANG Xiaohan , LIU Gang , GENG Huixin
2015, 37(6):135-140. DOI: 10.11887/j.cn.201506025
Abstract:To solve the abstractness and communication difficulties of dynamic fault tree, a distribution method based on Petri Net for system safety index was proposed by utilizing the advantages of Petri Net including intuition, easy and wide application. Failure rate was selected as a safety index through arranging the safety index and related reliability index. In the comparison of the modeling methods between dynamic fault tree and Petri Net, the latter was chosen to establish index distribution models for static and dynamic logic changes. On the basis of this, the hash classification for the refinement of system safety index distribution method was proposed. And through example analysis, a Petri Net hierarchical model for system faults was constructed to distribute index. Distribution results demonstrate that all the distributed values are within the corresponding safety indexes and this method can overcome the defects of non-intuition in dynamic fault tree method and of excessive roughness in equal distribution method, which provides references for the design and evaluation of aircraft safety.
2015, 37(6):141-149. DOI: 10.11887/j.cn.201506026
Abstract:The dynamic characteristic analysis model of damped carbon nanotubes on viscoelastic foundations was built by using Euler-Bernoulli beams. The nonlocal viscoelastic theories, general Maxwell viscoelastic model, velocity-dependent external damping model and viscoelastic foundation model were employed to deduce the governing equation of Euler-Bernoulli beams for dynamic characteristics analysis of carbon nanotubes. On the basis of Kelvin-Voigt model, new general analytical expressions for the natural frequencies of damped carbon nanotubes with no foundation and full foundation were obtained respectively and some typical special cases at full foundation were discussed. Then a transfer function method was developed to obtain a closed-form and uniform solution for the vibration governing equation under arbitrary boundary conditions. Considering a single-walled carbon nanotube as a numerical example, the first four natural frequencies with different boundary conditions were obtained, and the effects of the nonlocal and viscoelastic constants, the foundation stiffness coefficient and length on the natural frequencies and damping factors were analyzed. Results demonstrate the efficiency of the proposed model and the analysis methods in solving dynamic problems of damped carbon nanotubes on viscoelastic foundations.
LI Xiang , TANG Zhiyin , WU Gang , ZHOU Gang , BI Ke , ZHANG Qingzhi
2015, 37(6):150-154. DOI: 10.11887/j.cn.201506027
Abstract:In conduction cooling high temperature superconducting (HTS) magnet system, thermal conductivity of HTS magnetic pancakes and thermal boundary resistance between magnetic pancake and cooling body are the major factors affecting heat transport, and the difficult points for the thermal design of conduction cooling HTS magnet as well. A new method, based on the Levenberg-Marquardt algorithm, to simultaneously identify each heat transport parameter from the temperature distribution on the inspection surface was put forward. A low temperature experiment platform was setup, and a three-dimensional orthotropic heat transfer model of magnetic pancake was built. The orthotropic thermal conductivity and thermal boundary resistance of conduction cooling Bi2223 HTS magnet at 45~72 K were obtained with the method mentioned above. The influence of temperature measurement error on identification results was analyzed. The research result provides a new thought for obtaining heat transport parameters of conduction cooling superconducting magnet.
ZHANG Fangfang , ZHANG Zhenshan , WANG Jinzhong
2015, 37(6):155-161. DOI: 10.11887/j.cn.201506028
Abstract:To study the dynamic characteristics of hydrogen generation reactor, the reaction conversion rate of alloy and water was calculated by chemical reaction dynamic and kinematic analysis of reaction drip, and the mass variation of each substance in reactor was described by the law of mass conservation based on whose molar value was got by the minimum Gibbs free energy method. Based on the law of conservation of energy and a nonlinear moving-boundary model of helical tube, the dynamic process model of this reactor was established. The dynamic process simulation of an underwater hydrogen generation reactor was realized by using the calculation program based on the dynamic model. Results show that the simulation curves can reflect the variation law of parameters during the dynamic process, which proves the validation of the modeling and simulation. The reaction conversion rate determines the variation law of mass and the characteristics of releasing energy in reactor. The established model can be used as the basic foundation for performance analysis and control study of underwater hydrogen generation reactor.
QIAO Chunjie , WANG Yueke , HUANG Zhigang
2015, 37(6):162-166. DOI: 10.11887/j.cn.201506029
Abstract:The velocity of sound is one of the important acoustic parameters in ocean. For realizing rapid and high-precision measurements of seawater sound velocity, a sound velocity measurement method whose system architecture is simple was adopted.The measurements of sound propagation time and velocity value were achieved by continuous wave signal’s parameter estimation method. A pseudo-random sequence was introduced to modulate and handle continuous waves, which overcame the cycle ambiguities and echo interference problems of single-frequency signal in measurement. A sound velocity measurement platform was built to conduct system calibration and measurement experiment. Experiment results show that this method has high accuracy and fast response and is suitable for ocean sound velocity profile measurement on subaqueous mobile platform.
DU Jinfu , FANG Zongde , ZHANG Yongzhen , LI Jianhua
2015, 37(6):167-174. DOI: 10.11887/j.cn.201506030
Abstract:In order to pre-control the meshing performance of cycloid hypoid gears, a conjugate pinion tooth surface was generated by gear theoretical tooth surface. The pinion target tooth surface that meets the preconditions was obtained by modifying the conjugate tooth surface along the contact path and the contact line. The sum of tooth surface normal square errors between pinion theoretical and target tooth surface was calculated. The optimal model was built, setting the modifications of pinion machining parameters as variables and the least sum of square errors between pinion tooth surface and pinion target tooth surface on both sides as object. This optimization model was solved via sequence quadratic program. The validity of this modification method was demonstrated by using a numerical example of a high speed axle gear pair. The results show that the max normal errors on both sides are -4.7μm and -4.67μm, the transmission error deviations are 6.67% and 4%, the max contact path deviations are 0.275 mm and 0.177 mm, the results are found in line with the preconditions.
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