There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions...There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions of the crosssection(i.e., the thickness and the width). After reviewing the available equations, two thickness-to-width ratio Independent equations that are symmetric with respect to the two dimensions are obtained for evaluating the maximum torsion stress of rectangular cross-section beams. Based on the resulting equations, outside lamina emergent torsional joints are analyzed and some useful design Insights are obtained. These equations, together with the previous work on symmetric equations for calculating torsional compliance, provide a convenient and effective way for designing and optimizing torsional beams in compliant mechanisms.展开更多
The Fourier series method was extended for the exact analysis of wave propagation in an infinite rectangular beam.Initially,by solving the three-dimensional elastodynamic equations a general analytic solution was deri...The Fourier series method was extended for the exact analysis of wave propagation in an infinite rectangular beam.Initially,by solving the three-dimensional elastodynamic equations a general analytic solution was derived for wave motion within the beam.And then for the beam with stress-free boundaries,the propagation characteristics of elastic waves were presented.This accurate wave propagation model lays a solid foundation of simultaneous control of coupled waves in the beam.展开更多
The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoe...The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoelectric beams is derived by using the general solution of transversely isotropic piezoelasticity and Lur'e method without ad hoc assumptions. Based on the refined theory of piezoelectric beams, the exact equations for the beams without transverse surface loadings are derived, which consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beams under transverse loadings are derived directly from the refined beam theory. As a special case, the governing differential equations for transversely isotropic elastic beams are obtained from the corresponding equations of piezoelectric beams. To illustrate the application of the beam theory developed, a uniformly loaded and simply supported piezoelectric beam is examined.展开更多
A simple nonlinear model is proposed in this paper to study the bending wave in a rectangular piezoelectric laminated beam of infinite length.Based on the constitutive relations for transversely isotropic piezoelectri...A simple nonlinear model is proposed in this paper to study the bending wave in a rectangular piezoelectric laminated beam of infinite length.Based on the constitutive relations for transversely isotropic piezoelectric materials and isotropic elastic materials,combined with some electric conditions,we derive the bending wave equation in a long rectangular piezoelectric laminated beam by using energy method.The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal beam direction.The shock-wave solution,solitary-wave solution and other exact solutions of the bending wave equation are obtained by the extended F-expansion method.And by using the reductive perturbation method we derive the nonlinear Schrodinger(NLS)equation,further more,the bright and dark solitons are obtained.For those soliton solutions,and some parameters derived by the process of solving soliton solutions,some conclusions are drawn by numerical analysis with some fixed conditions.展开更多
The problem of deducing one-dimensional theory from two-dimensional the- ory for a homogeneous isotropic beam is investigated. Based on elasticity theory, the re- fined theory of rectangular beams is derived by using ...The problem of deducing one-dimensional theory from two-dimensional the- ory for a homogeneous isotropic beam is investigated. Based on elasticity theory, the re- fined theory of rectangular beams is derived by using Papkovich-Neuber solution and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. Based on the refined beam theory, the exact equations for the beam without transverse surface loadings are derived and consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beam under transverse loadings are derived directly from the refined beam theory and are almost the same as the governing equations of Timoshenko beam theory. In two ex- amples, it is shown that the new theory provides better results than Levinson’s beam the- ory when compared with those obtained from the linear theory of elasticity.展开更多
Based on elasticity theory, various one-dimensional equations for symmetrical deformation have been deduced systematically and directly from the two-dimensional theory of deep rectangular beams by using the Papkovich-...Based on elasticity theory, various one-dimensional equations for symmetrical deformation have been deduced systematically and directly from the two-dimensional theory of deep rectangular beams by using the Papkovich-Neuber solution and the Lur'e method without ad hoc assumptions, and they construct the refined theory of beams for symmetrical deformation. It is shown that the displacements and stresses of the beam can be represented by the transverse normal strain and displacement of the mid-plane. In the case of homogeneous boundary conditions, the exact solutions for the beam are derived, and the exact equations consist of two governing differential equations: the second-order equation and the transcendental equation. In the case of non-homogeneous boundary conditions, the approximate governing differential equations and solutions for the beam under normal loadings only and shear loadings only are derived directly from the refined beam theory, respectively, and the correctness of the stress assumptions in classic extension or compression problems is revised. Meanwhile, as an example, explicit expressions of analytical solutions are obtained for beams subjected to an exponentially distributed load along the length of beams.展开更多
The electron optical column for the variable rectangular-shaped beam lithographysystem DJ-2 is described,with emphasis on the analysis of the optical configuration and theshaping deflection compensation.In this column...The electron optical column for the variable rectangular-shaped beam lithographysystem DJ-2 is described,with emphasis on the analysis of the optical configuration and theshaping deflection compensation.In this column the variable spot shaping is performed with aminimum number of lenses by a more reasonable optical scheme.A high-sensitivity electrostaticshaping deflector with sequential parallel-plates is implemented for high-speed spot shaping.With a precise linear and rotational approach,the spot current density,the edge resolution aswell as the position of spot origin remain unchanged when the spot size varies.Experiments showthat the spot current density of over 0.4 A/cm^2 is obtained with a tungsten hairpin cathode,andthe edge resolution is better than 0.2μm within a 2×2 mm^2 field size.展开更多
A novel optimal design of sub-wavelength metal rectangular gratings for the polarizing beam splitter （PBS） is proposed. The method is based on effective medium theory and the method of designing single layer antiref...A novel optimal design of sub-wavelength metal rectangular gratings for the polarizing beam splitter （PBS） is proposed. The method is based on effective medium theory and the method of designing single layer antireflection coating. The polarization performance of PBS is discussed by rigorous couple-wave analysis （RCWA） method at a wavelength of 1550 nm. The result shows that sub-wavelength metal rectangular grating is characterized by a high reflectivity, like metal films for TE polarization, and high transmissivity, like dielectric films for TM polarization. The optimal design accords well with the results simulated by RCWA method.展开更多
A three-dimensional model of a dielectric-loaded rectangular Cerenkov maser with a sheet electron beam for the beam-wave interaction is proposed.Based on this model,the hybrid-mode dispersion equation is derived with ...A three-dimensional model of a dielectric-loaded rectangular Cerenkov maser with a sheet electron beam for the beam-wave interaction is proposed.Based on this model,the hybrid-mode dispersion equation is derived with the Borgnis potential function by using the field-matching method.Its approximate solution is obtained under the assumption of a dilute electron beam.By using the Ansoft high frequency structural simulator（HFSS） code,the electromagnetic field distribution in the interaction structure is given.Through numerical calculations,the effects of beam thickness,beam and dielectric-layer gap distance,beam voltage,and current density on the resonant growth rate are analysed in detail.展开更多
A simple rectangular microstrip antenna on low dielectric constant substrate such as air for improved radiation beam performance is theoretically investigated. The conventional patch antenna fabricated on common subst...A simple rectangular microstrip antenna on low dielectric constant substrate such as air for improved radiation beam performance is theoretically investigated. The conventional patch antenna fabricated on common substrates always produces quite broader E plane pattern compared to its H plane. In the present investigation, the same microstrip antenna is designed on air substrate with a view to develop an efficient feed for parabolic reflector antenna, which shows an excellent radiation pattern with symmetrical 3 dB beam widths at its both E and H plane. The present antenna compared to conventional structure to show its excellence in the beam performance is presented. The complete quantitative analysis to explore such radiation beam characteristics for both the structures (conventional and the present one) is also presented in this paper. An easy and handful relationship between the length of patch antenna and its fringing length for different types of substrate is established in the background of 3 dB beam widths. The proposed idea has been verified through a commercial software package for a patch operating in X band and a concrete physical insight into the phenomenon is developed.展开更多
基金Supported by National Science Foundation Research of the United States (Grant No.1663345)National Natural Science Foundation of China(Grant No.51675396)Fundamental Research Fund for the Central Universities(Grant No.12K5051204021)
文摘There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions of the crosssection(i.e., the thickness and the width). After reviewing the available equations, two thickness-to-width ratio Independent equations that are symmetric with respect to the two dimensions are obtained for evaluating the maximum torsion stress of rectangular cross-section beams. Based on the resulting equations, outside lamina emergent torsional joints are analyzed and some useful design Insights are obtained. These equations, together with the previous work on symmetric equations for calculating torsional compliance, provide a convenient and effective way for designing and optimizing torsional beams in compliant mechanisms.
文摘The Fourier series method was extended for the exact analysis of wave propagation in an infinite rectangular beam.Initially,by solving the three-dimensional elastodynamic equations a general analytic solution was derived for wave motion within the beam.And then for the beam with stress-free boundaries,the propagation characteristics of elastic waves were presented.This accurate wave propagation model lays a solid foundation of simultaneous control of coupled waves in the beam.
基金Support from the National Natural Science Foundation of China (Grant No.10372003) is acknowledged.
文摘The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoelectric beams is derived by using the general solution of transversely isotropic piezoelasticity and Lur'e method without ad hoc assumptions. Based on the refined theory of piezoelectric beams, the exact equations for the beams without transverse surface loadings are derived, which consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beams under transverse loadings are derived directly from the refined beam theory. As a special case, the governing differential equations for transversely isotropic elastic beams are obtained from the corresponding equations of piezoelectric beams. To illustrate the application of the beam theory developed, a uniformly loaded and simply supported piezoelectric beam is examined.
文摘A simple nonlinear model is proposed in this paper to study the bending wave in a rectangular piezoelectric laminated beam of infinite length.Based on the constitutive relations for transversely isotropic piezoelectric materials and isotropic elastic materials,combined with some electric conditions,we derive the bending wave equation in a long rectangular piezoelectric laminated beam by using energy method.The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal beam direction.The shock-wave solution,solitary-wave solution and other exact solutions of the bending wave equation are obtained by the extended F-expansion method.And by using the reductive perturbation method we derive the nonlinear Schrodinger(NLS)equation,further more,the bright and dark solitons are obtained.For those soliton solutions,and some parameters derived by the process of solving soliton solutions,some conclusions are drawn by numerical analysis with some fixed conditions.
文摘The problem of deducing one-dimensional theory from two-dimensional the- ory for a homogeneous isotropic beam is investigated. Based on elasticity theory, the re- fined theory of rectangular beams is derived by using Papkovich-Neuber solution and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. Based on the refined beam theory, the exact equations for the beam without transverse surface loadings are derived and consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beam under transverse loadings are derived directly from the refined beam theory and are almost the same as the governing equations of Timoshenko beam theory. In two ex- amples, it is shown that the new theory provides better results than Levinson’s beam the- ory when compared with those obtained from the linear theory of elasticity.
基金Supported by the National Natural Science Foundation of China (Grant Nos.10702077,10672001,and 10602001)the Beijing Natural Science Foundation (Grant No.1083012)the Alexander von Humboldt Foundation in Germany
文摘Based on elasticity theory, various one-dimensional equations for symmetrical deformation have been deduced systematically and directly from the two-dimensional theory of deep rectangular beams by using the Papkovich-Neuber solution and the Lur'e method without ad hoc assumptions, and they construct the refined theory of beams for symmetrical deformation. It is shown that the displacements and stresses of the beam can be represented by the transverse normal strain and displacement of the mid-plane. In the case of homogeneous boundary conditions, the exact solutions for the beam are derived, and the exact equations consist of two governing differential equations: the second-order equation and the transcendental equation. In the case of non-homogeneous boundary conditions, the approximate governing differential equations and solutions for the beam under normal loadings only and shear loadings only are derived directly from the refined beam theory, respectively, and the correctness of the stress assumptions in classic extension or compression problems is revised. Meanwhile, as an example, explicit expressions of analytical solutions are obtained for beams subjected to an exponentially distributed load along the length of beams.
文摘The electron optical column for the variable rectangular-shaped beam lithographysystem DJ-2 is described,with emphasis on the analysis of the optical configuration and theshaping deflection compensation.In this column the variable spot shaping is performed with aminimum number of lenses by a more reasonable optical scheme.A high-sensitivity electrostaticshaping deflector with sequential parallel-plates is implemented for high-speed spot shaping.With a precise linear and rotational approach,the spot current density,the edge resolution aswell as the position of spot origin remain unchanged when the spot size varies.Experiments showthat the spot current density of over 0.4 A/cm^2 is obtained with a tungsten hairpin cathode,andthe edge resolution is better than 0.2μm within a 2×2 mm^2 field size.
基金Project supported by Science Foundation of the Chongqing Committee of Education,China (Grant No KJ071205)
文摘A novel optimal design of sub-wavelength metal rectangular gratings for the polarizing beam splitter （PBS） is proposed. The method is based on effective medium theory and the method of designing single layer antireflection coating. The polarization performance of PBS is discussed by rigorous couple-wave analysis （RCWA） method at a wavelength of 1550 nm. The result shows that sub-wavelength metal rectangular grating is characterized by a high reflectivity, like metal films for TE polarization, and high transmissivity, like dielectric films for TM polarization. The optimal design accords well with the results simulated by RCWA method.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 60801031 and 10905032)the Knowledge Innovation Project of the Chinese Academy of Sciences (Grant No. YYYJ-1123-5)
文摘A three-dimensional model of a dielectric-loaded rectangular Cerenkov maser with a sheet electron beam for the beam-wave interaction is proposed.Based on this model,the hybrid-mode dispersion equation is derived with the Borgnis potential function by using the field-matching method.Its approximate solution is obtained under the assumption of a dilute electron beam.By using the Ansoft high frequency structural simulator（HFSS） code,the electromagnetic field distribution in the interaction structure is given.Through numerical calculations,the effects of beam thickness,beam and dielectric-layer gap distance,beam voltage,and current density on the resonant growth rate are analysed in detail.
文摘A simple rectangular microstrip antenna on low dielectric constant substrate such as air for improved radiation beam performance is theoretically investigated. The conventional patch antenna fabricated on common substrates always produces quite broader E plane pattern compared to its H plane. In the present investigation, the same microstrip antenna is designed on air substrate with a view to develop an efficient feed for parabolic reflector antenna, which shows an excellent radiation pattern with symmetrical 3 dB beam widths at its both E and H plane. The present antenna compared to conventional structure to show its excellence in the beam performance is presented. The complete quantitative analysis to explore such radiation beam characteristics for both the structures (conventional and the present one) is also presented in this paper. An easy and handful relationship between the length of patch antenna and its fringing length for different types of substrate is established in the background of 3 dB beam widths. The proposed idea has been verified through a commercial software package for a patch operating in X band and a concrete physical insight into the phenomenon is developed.
