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Modulation of High-Order Harmonic Generation from a Monolayer ZnO by Co-rotating Two-Color Circularly Polarized Laser Fields 被引量:1
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作者 乔月 陈家祺 +3 位作者 周书山 陈基根 蒋士成 杨玉军 《Chinese Physics Letters》 SCIE EI CAS CSCD 2024年第1期34-38,共5页
By numerically solving the two-dimensional semiconductor Bloch equation,we study the high-order harmonic emission of a monolayer ZnO under the driving of co-rotating two-color circularly polarized laser pulses.By chan... By numerically solving the two-dimensional semiconductor Bloch equation,we study the high-order harmonic emission of a monolayer ZnO under the driving of co-rotating two-color circularly polarized laser pulses.By changing the relative phase between the fundamental frequency field and the second one,it is found that the harmonic intensity in the platform region can be significantly modulated.In the higher order,the harmonic intensity can be increased by about one order of magnitude.Through time-frequency analysis,it is demonstrated that the emission trajectory of monolayer ZnO can be controlled by the relative phase,and the harmonic enhancement is caused by the second quantum trajectory with the higher emission probability.In addition,near-circularly polarized harmonics can be generated in the co-rotating two-color circularly polarized fields.With the change of the relative phase,the harmonics in the platform region can be altered from left-handed near-circularly polarization to right-handed one.Our results can obtain high-intensity harmonic radiation with an adjustable ellipticity,which provides an opportunity for syntheses of circularly polarized attosecond pulses. 展开更多
关键词 POLARIZATION ROTATING harmonic
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Harmonics in the Squirrel Cage Induction Motor Analytic Calculation Part III: Influence on the Torque-speed Characteristic 被引量:1
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作者 G.Kovács 《CES Transactions on Electrical Machines and Systems》 EI CSCD 2024年第1期86-102,共17页
The harmonics that appear in the squirrel cage asynchronous machine have been discussed in great detail in the literature for a long time. However, the systematization of the phenomenon is still pending, so we made an... The harmonics that appear in the squirrel cage asynchronous machine have been discussed in great detail in the literature for a long time. However, the systematization of the phenomenon is still pending, so we made an attempt to fill this gap in the previous parts of our study by elaborating formulas for calculation of parasitic torques. It was a general demand among those who work in this field towards the author to verify his formulas with measurements. In the literature, it seems,only one detailed, purposeful series of measurements has been published so far, the purpose of which was to investigate the effect of the number of rotor slots on the torque-speed characteristic curve of the machine. The main goal of this study is to verify the correctness of the formulas by comparing them with the referred series of measurements. Relying on this, the expected synchronous parasitic torques were developed for the frequently used rotor slot numbers-as a design guide for the engineer.Thus, together with our complete table for radial magnetic pull published in our previous work, the designer has all the principles, data and formulas available for the right number of rotor slots for his given machine and for the drive system. This brings this series of papers to an end. 展开更多
关键词 Asynchronous parasitic torques Induction motor Squirrel cage rotor Space harmonics Synchronous parasitic torques
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Bunching enhancement for coherent harmonic generation by using phase merging effects
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作者 Ke Feng Kangnan Jiang +3 位作者 Runshu Hu Shixia Luan Wentao Wang Ruxin Li 《Matter and Radiation at Extremes》 SCIE EI CSCD 2024年第5期10-21,共12页
In this paper,promising but simple schemes are investigated to enhance the micro-bunching of relativistic electron beams for coherent harmonic generation(CHG)by using phase merging effects.In contrast to the standard ... In this paper,promising but simple schemes are investigated to enhance the micro-bunching of relativistic electron beams for coherent harmonic generation(CHG)by using phase merging effects.In contrast to the standard CHG scheme,two specially designed dispersion sections(DSs)are adopted with the DS-modulator–DS configuration.The phase space of the e beam is appropriately coupled in the first DS,and the electrons within one seed wavelength can merge to the same phase with a matched second DS.Micro-bunching of the e beam can thus be enhanced by a large margin with much higher-harmonic components.Taking e beams from laser wakefield accelerators(LWFAs)as an example,start-to-end simulations are performed to show the effectiveness and robustness of the proposed schemes with several configurations.The beam current can be optimized to several tens to hundreds of kiloamperes,and the radiation power reaches hundreds of megawatts in the extreme ultraviolet regime within a 3.5 m-long beamline.The proposed schemes offer new opportunities for future compact free-electron lasers driven by LWFAs and provides prospects for truly compact and widely applicable systems. 展开更多
关键词 MODULATOR harmonic PHASE
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High-order harmonic generation of ZnO crystals in chirped and static electric fields
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作者 张玲玉 何永林 +5 位作者 谢卓璇 高芳艳 徐清芸 葛鑫磊 罗香怡 郭静 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第1期335-343,共9页
High harmonic generation in ZnO crystals under chirped single-color field and static electric field are investigated by solving the semiconductor Bloch equation(SBE). It is found that when the chirp pulse is introduce... High harmonic generation in ZnO crystals under chirped single-color field and static electric field are investigated by solving the semiconductor Bloch equation(SBE). It is found that when the chirp pulse is introduced, the interference structure becomes obvious while the harmonic cutoff is not extended. Furthermore, the harmonic efficiency is improved when the static electric field is included. These phenomena are demonstrated by the classical recollision model in real space affected by the waveform of laser field and inversion symmetry. Specifically, the electron motion in k-space shows that the change of waveform and the destruction of the symmetry of the laser field lead to the incomplete X-structure of the crystal-momentum-resolved(k-resolved) inter-band harmonic spectrum. Furthermore, a pre-acceleration process in the solid four-step model is confirmed. 展开更多
关键词 high-order harmonic generation the semiconductor Bloch equation k-resolved inter-band harmonic spectrum four-step semiclassical model
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Tunable spectral continuous shift of high-order harmonic generation in atoms by a plasmon-assisted shaping pulse
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作者 王源 李玉龙 +7 位作者 乔月 高娜 郭福明 陈洲 赫兰海 杨玉军 赵曦 王俊 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第3期433-440,共8页
We delve into the phenomenon of high-order harmonic generation within a helium atom under the influence of a plasmon-assisted shaping pulse.Our findings reveal an intriguing manipulation of the frequency peak position... We delve into the phenomenon of high-order harmonic generation within a helium atom under the influence of a plasmon-assisted shaping pulse.Our findings reveal an intriguing manipulation of the frequency peak position in the harmonic emission by adjusting the absolute phase parameter within the frequency domain of the shaping pulse.This phenomenon holds potential significance for experimental setups necessitating precisely tuned single harmonics.Notably,we observe a modulated shift in the created harmonic photon energy,spanning an impressive range of 1.2 eV.This frequency peak shift is rooted in the asymmetry exhibited by the rising and falling edges of the laser pulse,directly influencing the position of the peak frequency emission.Our study quantifies the dependence of this tuning range and the asymmetry of the laser pulse,offering valuable insights into the underlying mechanisms driving this phenomenon.Furthermore,our investigation uncovers the emergence of semi-integer order harmonics as the phase parameter is altered.We attribute this discovery to the intricate interference between harmonics generated by the primary and secondary return cores.This observation introduces an innovative approach for generating semi-integer order harmonics,thus expanding our understanding of high-order harmonic generation.Ultimately,our work contributes to the broader comprehension of complex phenomena in laser-matter interactions and provides a foundation for harnessing these effects in various applications,particularly those involving precise spectral control and the generation of unique harmonic patterns. 展开更多
关键词 high-order harmonic semi-integer-order spectra shift inhomogeneous field
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Abnormal transition of the electron energy distribution with excitation of the second harmonic in low-pressure radio-frequency capacitively coupled plasmas
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作者 余乐怡 陆文琪 张丽娜 《Plasma Science and Technology》 SCIE EI CAS CSCD 2024年第8期58-63,共6页
The self-excited second harmonic in radio-frequency capacitively coupled plasma was significantly enhanced by adjusting the external variable capacitor.At a lower pressure of 3 Pa,the excitation of the second harmonic... The self-excited second harmonic in radio-frequency capacitively coupled plasma was significantly enhanced by adjusting the external variable capacitor.At a lower pressure of 3 Pa,the excitation of the second harmonic caused an abnormal transition of the electron energy probability function,resulting in abrupt changes in the electron density and temperature.Such changes in the electron energy probability function as well as the electron density and temperature were not observed at the higher pressure of 16 Pa under similar harmonic changes.The phenomena are related to the influence of the second harmonic on stochastic heating,which is determined by both amplitude and the relative phase of the harmonics.The results suggest that the self-excited high-order harmonics must be considered in practical applications of lowpressure radio-frequency capacitively coupled plasmas. 展开更多
关键词 RADIO-FREQUENCY capacitively coupled plasma harmonicS the electron energy probability function
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Terahertz quasi-perfect vortex beam with integer-order and fractional-order generated by spiral spherical harmonic axicon
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作者 涂思语 刘德峰 +2 位作者 刘劲松 杨振刚 王可嘉 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第1期458-467,共10页
We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spira... We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spiral phase plate is designed and called spiral spherical harmonic axicon. A terahertz Gaussian beam passes through the spiral spherical harmonic axicon to generate a terahertz vortex beam. When only the topological charge number carried by spiral spherical harmonic axicon increases, the ring radius of terahertz vortex beam increases slightly, so the beam is shaped into a terahertz quasi-perfect vortex beam. Importantly, the terahertz quasi-perfect vortex beam can carry not only integer-order topological charge number but also fractional-order topological charge number. This is the first time that vortex beam and quasi-perfect vortex beam with fractional-order have been successfully realized in terahertz domain and experiment. 展开更多
关键词 TERAHERTZ spiral spherical harmonic axicon quasi-perfect vortex beam topological charge number
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THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL
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作者 Han LUO 《Acta Mathematica Scientia》 SCIE CSCD 2024年第4期1189-1199,共11页
In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give ... In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result. 展开更多
关键词 sub-Riemannian manifolds subelliptic harmonic maps with potential gradient estimate Liouville Theorem
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Combinatorial Identities Concerning Harmonic Numbers
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作者 CHEN Yu-lei GUO Dong-wei 《Chinese Quarterly Journal of Mathematics》 2024年第3期307-314,共8页
In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as... In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as binomial coefficients are derived. 展开更多
关键词 harmonic numbers COEFFICIENTS Combinatorial identities
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ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR
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作者 陈洁玲 刘名生 《Acta Mathematica Scientia》 SCIE CSCD 2024年第1期295-310,共16页
In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,... In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,using these results,we present better estimations for the Bloch constants of certain harmonic mappings L(f),where f is a K-quasiregular harmonic or open harmonic.Finally,we establish three versions of BlochLandau type theorem for biharmonic mappings of the form L(f).These results are sharp in some given cases and improve the related results of earlier authors. 展开更多
关键词 Bloch-Landau type theorem Bloch constant linear complex operator harmonic mapping biharmonic mapping UNIVALENT
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Elliptically polarized high-order harmonic generation of Ar atom in an intense laser field
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作者 胡杰 王一琛 +6 位作者 景秋霜 姜威 王革文 赵逸文 肖礴 梁红静 马日 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第5期453-457,共5页
High-order harmonic generation(HHG) of Ar atom in an elliptically polarized intense laser field is experimentally investigated in this work.Interestingly,the anomalous ellipticity dependence on the laser ellipticity(... High-order harmonic generation(HHG) of Ar atom in an elliptically polarized intense laser field is experimentally investigated in this work.Interestingly,the anomalous ellipticity dependence on the laser ellipticity(ε) in the lower-order harmonics is observed,specifically in the 13rd-order,which displays a maximal harmonic intensity at ε ≈ 0.1,rather than at ε = 0 as expected.This contradicts the general trend of harmonic yield,which typically decreases with the increase of laser ellipticity.In this study,we attribute this phenomenon to the disruption of the symmetry of the wave function by the Coulomb effect,leading to the generation of a harmonic with high ellipticity.This finding provides valuable insights into the behavior of elliptically polarized harmonics and opens up a potential way for exploring new applications in ultrafast spectroscopy and light–matter interactions. 展开更多
关键词 high-order harmonic generation Coulomb effect elliptically polarized intense laser field
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Interference of harmonics emitted by different tunneling momentum channels in laser fields
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作者 Ling-Yu Zhang Zhuo-Xuan Xie +2 位作者 Can Wang Xin-Lei Ge Jing Guo 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第9期366-372,共7页
By numerically solving the semiconductor Bloch equation(SBEs),we theoretically study the high-harmonic generation of ZnO crystals driven by one-color and two-color intense laser pulses.The results show the enhancement... By numerically solving the semiconductor Bloch equation(SBEs),we theoretically study the high-harmonic generation of ZnO crystals driven by one-color and two-color intense laser pulses.The results show the enhancement of harmonics and the cut-off remains the same in the two-color field,which can be explained by the recollision trajectories and electron excitation from multi-channels.Based on the quantum path analysis,we investigate contribution of different ranges of the crystal momentum k of ZnO to the harmonic yield,and find that in two-color laser fields,the intensity of the harmonic yield of different ranges from the crystal momentum makes a big difference and the harmonic intensity is depressed from all k channels,which is related to the interferences between harmonics from symmetric k channels. 展开更多
关键词 high-order harmonic generation the semiconductor Bloch equation k-resolvedinter-bandhar-monic spectrum four-step semiclassical model
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Optimization of extreme ultraviolet vortex beam based on high harmonic generation
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作者 肖礴 赵逸文 +6 位作者 程方晶 王革文 姜威 王一琛 胡杰 梁红静 马日 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第5期459-464,共6页
In high harmonic generation(HHG),Laguerre–Gaussian(LG) beams are used to generate extreme ultraviolet(XUV)vortices with well-defined orbital angular momentum(OAM),which have potential applications in fields such as m... In high harmonic generation(HHG),Laguerre–Gaussian(LG) beams are used to generate extreme ultraviolet(XUV)vortices with well-defined orbital angular momentum(OAM),which have potential applications in fields such as microscopy and spectroscopy.An experimental study on the HHG driven by vortex and Gaussian beams is conducted in this work.It is found that the intensity of vortex harmonics is positively correlated with the laser energy and gas pressure.The structure and intensity distribution of the vortex harmonics exhibit significant dependence on the relative position between the gas jet and the laser focus.The ring-like structures observed in the vortex harmonics,and the interference of quantum paths provide an explanation for the distinct structural characteristics.Moreover,by adjusting the relative position between the jet and laser focus,it is possible to discern the contributions from different quantum paths.The optimization of the HH vortex field is applicable to the XUV,which opens up a new way for exploiting the potential in optical spin or manipulating electrons by using the photon with tunable orbital angular momentum. 展开更多
关键词 high harmonic generation phase matching extreme ultraviolet vortex beam quantum path interference
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Harmonic balance simulation of the influence of component uniformity and reliability on the performance of a Josephson traveling wave parametric amplifier
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作者 郑煜臻 熊康林 +1 位作者 冯加贵 杨辉 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第4期339-343,共5页
A Josephson traveling wave parametric amplifier(JTWPA),which is a quantum-limited amplifier with high gain and large bandwidth,is the core device of large-scale measurement and control systems for quantum computing.A ... A Josephson traveling wave parametric amplifier(JTWPA),which is a quantum-limited amplifier with high gain and large bandwidth,is the core device of large-scale measurement and control systems for quantum computing.A typical JTWPA consists of thousands of Josephson junctions connected in series to form a transmission line and hundreds of shunt LC resonators periodically loaded along the line for phase matching.Because the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics,the fabrication of a JTWPA typically necessitates precise processing equipment.To guide the fabrication process and further improve the design for manufacturability,it is necessary to understand how each electronic component affects the amplifier.In this paper,we use the harmonic balance method to conduct a comprehensive study on the impact of nonuniformity and fabrication yield of the electronic components on the performance of a JTWPA.The results provide insightful and scientific guidance for device design and fabrication processes. 展开更多
关键词 Josephson traveling wave parametric amplifier(JTWPA) harmonic balance method YIELDS UNIFORMITY
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Effect of Types and Orders of Electromagnetic Field Finite Element Meshes on Power Communication Harmonic Parameters Calculation Results of Tubular Hydrogenerators
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作者 Fan Zhennan Chen Jie +1 位作者 Zhou Zhiting Yang Yong 《China Communications》 SCIE CSCD 2024年第10期288-300,共13页
In generator design field,waveform total harmonic distortion(THD)and telephone harmonic factor(THF)are parameters commonly used to measure the impact of generator no-load voltage harmonics on the power communication q... In generator design field,waveform total harmonic distortion(THD)and telephone harmonic factor(THF)are parameters commonly used to measure the impact of generator no-load voltage harmonics on the power communication quality.Tubular hydrogenerators are considered the optimal generator for exploiting low-head,high-flow hydro resources,and they have seen increasingly widespread application in China's power systems recent years.However,owing to the compact and constrained internal space of such generators,their internal magnetic-field harmonics are pronounced.Therefore,accurate calculation of their THD and THF is crucial during the analysis and design stages to ensure the quality of power communication.Especially in the electromagnetic field finite element modeling analysis of such generators,the type and order of the finite element meshes may have a significant impact on the THD and THF calculation results,which warrants in-depth research.To address this,this study takes a real 34 MW large tubular hydrogenerator as an example,and establishes its electromagnetic field finite element model under no-load conditions.Two types of meshes,five mesh densities,and two mesh orders are analyzed to reveal the effect of electromagnetic field finite element mesh types and orders on the calculation results of THD and THF for such generators. 展开更多
关键词 calculation results electromagnetic field finite element meshes power communication harmonic parameters tubular hydrogenerator types and orders
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Airgap-harmonic-oriented Partitioned Design Method of PMV Motor with Improved Torque Performances
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作者 Zhaopeng Wu Li Quan +3 位作者 Zixuan Xiang Deyang Fan Tengguang Wang Xiaoyong Zhu 《CES Transactions on Electrical Machines and Systems》 EI CSCD 2024年第1期43-50,共8页
Here,we introduce a partitioned design method that is oriented toward airgap harmonic for permanent magnet vernier(PMV)motors.The method proposes the utilization of airgap flux harmonics as an effective bridge between... Here,we introduce a partitioned design method that is oriented toward airgap harmonic for permanent magnet vernier(PMV)motors.The method proposes the utilization of airgap flux harmonics as an effective bridge between the torque design region and the torque performances.To illustrate the efficacy of this method,a partitioned design PMV motor is presented and compared with the initial design.Firstly,the torque design region of the rotor is artfully divided into the torque enhancement region and ripple reduction region.Meanwhile,the main harmonics that generate output torque are chosen and enhanced,optimization.Moreover,the harmonics that generate torque ripple are selected and reduced based on torque harmonics optimization.Finally,the functions of the partitioned PMV motor torque are assessed based on the finite element method.By the purposeful design of these two regions,the output torque is strengthened while torque ripple is inhibited effectively,verifying the effectiveness and reasonability of the proposed design method. 展开更多
关键词 Permanent magnet vernier(PMV)motor Airgap flux harmonic Partitioned design Low torque ripple
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On the Weyl’s Lemma for Triharmonic Functions
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作者 ZHENG Run-jie ZENG Jia-min FANG Yi 《Chinese Quarterly Journal of Mathematics》 2024年第3期288-294,共7页
In this paper,by choosing some appropriate test functions,we prove the Weyl’s lemma for triharmonic functions based on the new type of mean value formulas.
关键词 Weyl’s lemma harmonic functions Triharmonic functions Test functions Mean value formulas
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Wavelength Dependent Nonlinear Spectroscopic Study of Third Harmonic Generation Probed by Rotational Maker Fringes Method
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作者 Calford Odhiambo Otieno 《Open Journal of Microphysics》 2024年第2期13-23,共11页
Efficient third-order nonlinearities of the Zinc Oxide and Al-doped Zinc Oxide were studied by Third Harmonic Generation (Third Harmonic Generation) Maker fringes to establish the effect Aluminum of Aluminum doping (A... Efficient third-order nonlinearities of the Zinc Oxide and Al-doped Zinc Oxide were studied by Third Harmonic Generation (Third Harmonic Generation) Maker fringes to establish the effect Aluminum of Aluminum doping (Al-doping) on the cubic nonlinearities. Adding the Al-dopant to the Zinc Oxide crystal structure results in changes that affect the optical and nonlinear characteristics. Presented results indicate that the magnitude of X<sup>(3)</sup> was enhanced at single experimental wavelengths;however, across the broadband experimental spectrum, the effect of Al-doping remained relatively constant. The observed enhancement of third-order nonlinearity was purely from the bound electronic response. The observation is attributed to increased charge carriers and spontaneous polarization in the Zinc Oxide and Al-doped Zinc Oxide crystal structure. 展开更多
关键词 Nonlinear Optics harmonic Generation Maker Fringes Zinc Oxide
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Identities on q-Harmonic Numbers
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作者 Mengxiao Zhou Haitao Jin Huanhuan Zheng 《Journal of Applied Mathematics and Physics》 2024年第5期1796-1803,共8页
With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harm... With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given. 展开更多
关键词 harmonic Numbers q-Zeilberger Algorithm Abel’s Lemma
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Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading
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作者 Moussa Sali Fabien Kenmogne +1 位作者 Jean Bertin Nkibeu Abdou Njifenjou 《World Journal of Engineering and Technology》 2024年第2期401-425,共25页
The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation metho... The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%. 展开更多
关键词 Successive Approximations Method Direct Integration Differential Equations Beams of Variable Stiffness Quadratic Parabola Impulse and harmonic Loads
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