We demonstrate that,in a simple linearly-polarized plane wave,the optical pulling forces on nanoparticle clusters with gain can be induced by the Fano-like resonance.The numerical results based on the full-wave calcul...We demonstrate that,in a simple linearly-polarized plane wave,the optical pulling forces on nanoparticle clusters with gain can be induced by the Fano-like resonance.The numerical results based on the full-wave calculation show that the optical pulling forces can be attributed to the recoil forces for the nanoparticle clusters composed of dipolar nanoparticles with three different configurations.Interestingly,the recoil forces giving rise to optical pulling forces are exactly dominated by the coupling term between the electric and magnetic dipoles excited in the nanoparticle clusters,while other higherorder terms have a negligible contribution.In addition,the optical pulling force can be tailored by modulating the Fano-like resonance via either the particle size or the gain magnitude,offering an alternative freedom degree for optical manipulations of particle clusters.展开更多
To solve the difficulty of generating an ideal Bessel beam,an simplified annular transducer model is proposed to study the axial acoustic radiation force(ARF)and the corresponding negative ARF(pulling force)exerted on...To solve the difficulty of generating an ideal Bessel beam,an simplified annular transducer model is proposed to study the axial acoustic radiation force(ARF)and the corresponding negative ARF(pulling force)exerted on centered elastic spheres for acoustic-vortex(AV)beams of arbitrary orders.Based on the theory of acoustic scattering,the axial distributions of the velocity potential and the ARF for AV beams of different orders generated by the annular transducers with different physical sizes are simulated.It is proved that the pulling force can be generated by AV beams of arbitrary orders with multiple axial regions.The pulling force is more likely to exert on the sphere with a smaller k0a(product of the wave number and the radius)for the AV beam with a bigger topological charge due to the strengthened off-axis acoustic scattering.The pulling force decreases with the increase of the axial distance for the sphere with a bigger k0a.More pulling force areas with wider axial regions can be formed by AV beams using a bigger-sized annular transducer.The theoretical results demonstrate the feasibility of generating the pulling force along the axes of AV beams using the experimentally applicable circular array of planar transducers,and suggest application potentials for multi-position stable object manipulations in biomedical engineering.展开更多
We investigate optical force on a graphene-coated gain microparticle by adopting the Maxwell's stress tensor method.It is found that there exists a threshold gain in obtaining the Fano-profile optical force which ...We investigate optical force on a graphene-coated gain microparticle by adopting the Maxwell's stress tensor method.It is found that there exists a threshold gain in obtaining the Fano-profile optical force which indicates the reversal of optical pushing and pulling force. And giant pushing/pulling force can be achieved if the gain value of the material is in the proximity of the threshold gain. Our results show that the threshold gain is more sensitive to the relaxation time than to the Fermi energy of the graphene. We further study the optical force on larger microparticle to demonstrate the pulling force occurring at octupole resonance with small gain value and then it will appear at quadrupole resonance by increasing gain value. Our work provides an in-depth insight into the interaction between light and gain material and gives the additional degree of freedom to optical manipulation of microparticle.展开更多
A microtribometer is used to measure and compare pull-off forces and friction forces exerted on (a) micro-dimpled silicon surfaces, (b) bare silicon surfaces, and (c) octadecyltrichlorosilane (OTS) treated sil...A microtribometer is used to measure and compare pull-off forces and friction forces exerted on (a) micro-dimpled silicon surfaces, (b) bare silicon surfaces, and (c) octadecyltrichlorosilane (OTS) treated silicon surfaces at different relative humidity (RH) levels separately. It is found that above a critical RH level, the capillary pull-off force increases abruptly and that the micro-dimple textured surface has a lower critical RH value as well as a higher pull-off force value than the other two surfaces. A micro topography parameter, namely sidewall area ratio, is found to play a major role in controlling the capillary pull-off force. Furthermore, micro-dimpled silicon surface is also proved to be not sensitive to variation in RH level, and can realize a stable and decreased friction coefficient compared with un-textured silicon surfaces. The reservoir-like function of micro dimples is considered to weaken or avoid the breakage effect of liquid bridges at different RH levels, thereby maintaining a stable frictional behaviour.展开更多
Determination of the grouting anchor pullout force is a key step during the design of anchor-pull retaining wall, but it is mostly determined relied on empirical formula at present, and the rationality and the safety ...Determination of the grouting anchor pullout force is a key step during the design of anchor-pull retaining wall, but it is mostly determined relied on empirical formula at present, and the rationality and the safety cannot be effectively guaranteed. Based on the engineering case of the gravity retaining wall of Qinglin Freeway, the model test was designed, and combined with the results of the ABAQUS finite element numerical analysis, it was analyzed that how the anchor axial pulling force distributes. The results showed that the force of the anchor near the wall bolt was large and which far from the wall was small and the ultimate pullout force was proportional to the length, diameter and shear strength. When the end tension of the anchor was small, the top load played a leading role on the anchor tension. This conclusion confirmed the calculation formula of ultimate pullout force was and provided a theoretical basis for anchor-pull retaining wall design and calculation.展开更多
Radial drilling technology, of which the jet bit is the key device, is a research focus in the field of oil drilling and production. This paper establishes mechanical equations for jet bits and analyzes the hydroseal ...Radial drilling technology, of which the jet bit is the key device, is a research focus in the field of oil drilling and production. This paper establishes mechanical equations for jet bits and analyzes the hydroseal of backward jets in bottom holes. Meanwhile this paper establishes a mechanical equation for a high pressure hose and analyzes the axial force distribution. Laboratory experiments indicate that the flow rate, the angle between the backward nozzle axis and the jet bit axis, and the hole diameter are the major influencing factors; the generation of the pulling force is mainly due to the inlet pressure of the jet bit; the backward jets can significantly increase not only the pulling force but also the stability of jet bits. The pulling force would reach 8,376 N under experimental conditions, which can steadily pull the high-pressure hose forward.展开更多
Flher-like particle suspensions are common in both na-ture and industry, but there is little work reported on it.The forces acting on the fiber - like particle in fluid arestudied in this paper, and the Magnus lift, S...Flher-like particle suspensions are common in both na-ture and industry, but there is little work reported on it.The forces acting on the fiber - like particle in fluid arestudied in this paper, and the Magnus lift, Saffman lift,pressure gradient force, and then the dynamics modelhave been received. The numerical study of the simpleshear flow past the cylinders shows that the particles ’motion is controlled by the vortex.展开更多
In the present study, the Volterra series theory is adopted to theoretically investigate the force transmissibility of multiple degrees of freedom (MDOF) structures, in which an isolator with nonlinear anti-symmetri...In the present study, the Volterra series theory is adopted to theoretically investigate the force transmissibility of multiple degrees of freedom (MDOF) structures, in which an isolator with nonlinear anti-symmetric viscous damping is assembled. The results reveal that the anti-symmetric nonlinear viscous damping can significantly reduce the force trans- missibility over all resonance regions for MDOF structures with little effect on the transmissibility over non-resonant and isolation regions. The results indicate that the vibration isolators with an anti-symmetric damping characteristic have great potential to solve the dilemma occurring in the design of linear viscously damped vibration isolators where an increase of the damping level reduces the force transmissibility over resonant frequencies but increases the transmissibility over non-resonant frequency regions. This work is an extension of a previous study in which MDOF structures installed on the mount through an isolator with cubic nonlinear damping are considered. The theoretical analysis results are also verified by simulation studies.展开更多
Nanohairs, which can be found on the epidermis of Tokay gecko's toes, contribute to the adhesion by means of van der Waals force, capillary force, etc. This structure has inspired many researchers to fabricate the at...Nanohairs, which can be found on the epidermis of Tokay gecko's toes, contribute to the adhesion by means of van der Waals force, capillary force, etc. This structure has inspired many researchers to fabricate the attachable nano-scale structures. However, the efficiency of artificial nano-scale structures is not reliable sufficiently. Moreover, the mechanical parameters related to the nano-hair attachment are not yet revealed qualitatively. The mechanical parameters which have influence on the ability of adhesive nano-hairs were investigated through numerical simulation in which only van der Waals force was considered. For the numerical analysis, finite element method was utilized and van der Waals force, assumed as 12-6 Lennard-Jones potential, was implemented as the body force term in the finite element formulation.展开更多
A comprehension of railway dynamic behavior implies the measure of wheel-rail contact forces which are affected by disturbances and errors that are often difficult to be quantified. In this study, a benchmark test cas...A comprehension of railway dynamic behavior implies the measure of wheel-rail contact forces which are affected by disturbances and errors that are often difficult to be quantified. In this study, a benchmark test case is proposed, and a bogie with a layout used on some European locomotives such as SIEMENS El90 is studied. In this layout, an additional shaft on which brake disks are installed is used to transmit the braking torque to the wheelset through a single-stage gearbox. Using a mixed approach based on finite element techniques and statistical considerations, it is possible to evaluate an optimal layout for strain gauge positioning and to optimize the measurement system to diminish the effects of noise and disturbance. We also conducted preliminary evaluations on the precision and frequency response of the proposed system.展开更多
The concept vortex force in aerodynamics is sys- tematically examined based on a new steady vortex-force theory (Wu et al., Vorticity and vortex dynamics, Springer, 2006) which expresses the aerodynamic force (and ...The concept vortex force in aerodynamics is sys- tematically examined based on a new steady vortex-force theory (Wu et al., Vorticity and vortex dynamics, Springer, 2006) which expresses the aerodynamic force (and moment) by the volume and boundary integrals of the Lamb vector. In this paper, the underlying physics of this theory is explored, including the general role of the Lamb vector in non- linear aerodynamics, its initial formation, and its relevance to the total-pressure non-uniformity. As a typical example, the theory is applied to the flow over a slender delta wing at a large angle of attack. The highly localized flow structures with high Lamb-vector peaks are identified in terms of their net contribution to various constituents of the total aerody-namic force. This vortex-force diagnosis sheds new light on the flow control and configuration optimization.展开更多
The aim of the investigation was to develop the use of topographic and nano-adhesion atomic force microscopy(AFM) studies as a means of monitoring the coalescence of latex particles within films produced from a pharma...The aim of the investigation was to develop the use of topographic and nano-adhesion atomic force microscopy(AFM) studies as a means of monitoring the coalescence of latex particles within films produced from a pharmaceutically relevant aqueous dispersion(Eudragit~?NE30 D). Films were prepared via spin coating and analysed using AFM, initially via tapping mode for topographic assessment followed by force-distance measurements which allowed assessment of site-specific adhesion. The results showed that colloidal particles were clearly observed topographically in freshly prepared samples, with coalescence detected on curing via the disappearance of discernible surface features and a decrease in roughness indices. The effects of temperature and humidity on film curing were also studied, with the former having the most pronounced effect. AFM force measurements showed that the variation in adhesive force reduced with increasing curing time, suggesting a novel method of quantifying the rate of film formation upon curing. It was concluded that the AFM methods outlined in this study may be used as a means of qualitatively and quantitatively monitoring the curing of pharmaceutical films as a function of time and other variables, thereby facilitating rational design of curing protocols.展开更多
The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="...The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em></span>), with the commutation relation <img src="Edit_28f5b839-7de4-41e5-9ed8-69dc1bf72c2c.bmp" alt="" />, and using a Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s like equation on these variable, and associating a linear operator to a constant of motion <em>K</em> (<em>x, v, t</em>) of the classical system, The comparison with the quantization in the space (<em>x, p</em>) is done with the usual Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s equation for the Hamiltonian <em>H</em><span style="white-space:normal;">(</span><em style="white-space:normal;">x, p, t</em><span style="white-space:normal;">)</span>, and with the commutation relation <img src="Edit_cca7e318-5b35-4c55-8f09-6089970ce9a2.bmp" alt="" />. It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization has fewer oscillations than the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, the average energy of the system is higher in (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, and the Boltzmann-Shannon entropy on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization is higher than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization.展开更多
基金Project supported by the Natural Science Foundation of Guangxi Province of China (Grant No.2021GXNSFDA196001)the National Natural Science Foundation of China (Grant Nos.12174076,12074084,and 12204117)+1 种基金Guangxi Science and Technology Project (Grant Nos.AD22080042 and AB21220052)Open Project of State Key Laboratory of Surface Physics in Fudan University (Grant No.KF2022_15)。
文摘We demonstrate that,in a simple linearly-polarized plane wave,the optical pulling forces on nanoparticle clusters with gain can be induced by the Fano-like resonance.The numerical results based on the full-wave calculation show that the optical pulling forces can be attributed to the recoil forces for the nanoparticle clusters composed of dipolar nanoparticles with three different configurations.Interestingly,the recoil forces giving rise to optical pulling forces are exactly dominated by the coupling term between the electric and magnetic dipoles excited in the nanoparticle clusters,while other higherorder terms have a negligible contribution.In addition,the optical pulling force can be tailored by modulating the Fano-like resonance via either the particle size or the gain magnitude,offering an alternative freedom degree for optical manipulations of particle clusters.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11934009,11974187,and 11604156).
文摘To solve the difficulty of generating an ideal Bessel beam,an simplified annular transducer model is proposed to study the axial acoustic radiation force(ARF)and the corresponding negative ARF(pulling force)exerted on centered elastic spheres for acoustic-vortex(AV)beams of arbitrary orders.Based on the theory of acoustic scattering,the axial distributions of the velocity potential and the ARF for AV beams of different orders generated by the annular transducers with different physical sizes are simulated.It is proved that the pulling force can be generated by AV beams of arbitrary orders with multiple axial regions.The pulling force is more likely to exert on the sphere with a smaller k0a(product of the wave number and the radius)for the AV beam with a bigger topological charge due to the strengthened off-axis acoustic scattering.The pulling force decreases with the increase of the axial distance for the sphere with a bigger k0a.More pulling force areas with wider axial regions can be formed by AV beams using a bigger-sized annular transducer.The theoretical results demonstrate the feasibility of generating the pulling force along the axes of AV beams using the experimentally applicable circular array of planar transducers,and suggest application potentials for multi-position stable object manipulations in biomedical engineering.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11904184, 11847033, and 11704158)the Natural Science Foundation of Jiangsu Province,China (Grant No. BK20170170)。
文摘We investigate optical force on a graphene-coated gain microparticle by adopting the Maxwell's stress tensor method.It is found that there exists a threshold gain in obtaining the Fano-profile optical force which indicates the reversal of optical pushing and pulling force. And giant pushing/pulling force can be achieved if the gain value of the material is in the proximity of the threshold gain. Our results show that the threshold gain is more sensitive to the relaxation time than to the Fermi energy of the graphene. We further study the optical force on larger microparticle to demonstrate the pulling force occurring at octupole resonance with small gain value and then it will appear at quadrupole resonance by increasing gain value. Our work provides an in-depth insight into the interaction between light and gain material and gives the additional degree of freedom to optical manipulation of microparticle.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 50575123 and 50730007)China Scholarship Council (CSC) and German Research Foundation (DFG)
文摘A microtribometer is used to measure and compare pull-off forces and friction forces exerted on (a) micro-dimpled silicon surfaces, (b) bare silicon surfaces, and (c) octadecyltrichlorosilane (OTS) treated silicon surfaces at different relative humidity (RH) levels separately. It is found that above a critical RH level, the capillary pull-off force increases abruptly and that the micro-dimple textured surface has a lower critical RH value as well as a higher pull-off force value than the other two surfaces. A micro topography parameter, namely sidewall area ratio, is found to play a major role in controlling the capillary pull-off force. Furthermore, micro-dimpled silicon surface is also proved to be not sensitive to variation in RH level, and can realize a stable and decreased friction coefficient compared with un-textured silicon surfaces. The reservoir-like function of micro dimples is considered to weaken or avoid the breakage effect of liquid bridges at different RH levels, thereby maintaining a stable frictional behaviour.
文摘Determination of the grouting anchor pullout force is a key step during the design of anchor-pull retaining wall, but it is mostly determined relied on empirical formula at present, and the rationality and the safety cannot be effectively guaranteed. Based on the engineering case of the gravity retaining wall of Qinglin Freeway, the model test was designed, and combined with the results of the ABAQUS finite element numerical analysis, it was analyzed that how the anchor axial pulling force distributes. The results showed that the force of the anchor near the wall bolt was large and which far from the wall was small and the ultimate pullout force was proportional to the length, diameter and shear strength. When the end tension of the anchor was small, the top load played a leading role on the anchor tension. This conclusion confirmed the calculation formula of ultimate pullout force was and provided a theoretical basis for anchor-pull retaining wall design and calculation.
基金supported by High-tech Research and Development Program of China (No. 2007AA09Z315)Doctoral Foundation of Ministry of Education of China (No. 20070425006)
文摘Radial drilling technology, of which the jet bit is the key device, is a research focus in the field of oil drilling and production. This paper establishes mechanical equations for jet bits and analyzes the hydroseal of backward jets in bottom holes. Meanwhile this paper establishes a mechanical equation for a high pressure hose and analyzes the axial force distribution. Laboratory experiments indicate that the flow rate, the angle between the backward nozzle axis and the jet bit axis, and the hole diameter are the major influencing factors; the generation of the pulling force is mainly due to the inlet pressure of the jet bit; the backward jets can significantly increase not only the pulling force but also the stability of jet bits. The pulling force would reach 8,376 N under experimental conditions, which can steadily pull the high-pressure hose forward.
基金Supported by the financial of the Natural Science Fund for outatending youth of China and the fund fo the state key laboratory of fluid transreiasion and control of Zhejiand University
文摘Flher-like particle suspensions are common in both na-ture and industry, but there is little work reported on it.The forces acting on the fiber - like particle in fluid arestudied in this paper, and the Magnus lift, Saffman lift,pressure gradient force, and then the dynamics modelhave been received. The numerical study of the simpleshear flow past the cylinders shows that the particles ’motion is controlled by the vortex.
基金supported by the EPSRC (UK)the National Science Fund for Distinguished Young Scholars (11125209)the National Natural Science Foundation of China (10902068 and 51121063)
文摘In the present study, the Volterra series theory is adopted to theoretically investigate the force transmissibility of multiple degrees of freedom (MDOF) structures, in which an isolator with nonlinear anti-symmetric viscous damping is assembled. The results reveal that the anti-symmetric nonlinear viscous damping can significantly reduce the force trans- missibility over all resonance regions for MDOF structures with little effect on the transmissibility over non-resonant and isolation regions. The results indicate that the vibration isolators with an anti-symmetric damping characteristic have great potential to solve the dilemma occurring in the design of linear viscously damped vibration isolators where an increase of the damping level reduces the force transmissibility over resonant frequencies but increases the transmissibility over non-resonant frequency regions. This work is an extension of a previous study in which MDOF structures installed on the mount through an isolator with cubic nonlinear damping are considered. The theoretical analysis results are also verified by simulation studies.
文摘Nanohairs, which can be found on the epidermis of Tokay gecko's toes, contribute to the adhesion by means of van der Waals force, capillary force, etc. This structure has inspired many researchers to fabricate the attachable nano-scale structures. However, the efficiency of artificial nano-scale structures is not reliable sufficiently. Moreover, the mechanical parameters related to the nano-hair attachment are not yet revealed qualitatively. The mechanical parameters which have influence on the ability of adhesive nano-hairs were investigated through numerical simulation in which only van der Waals force was considered. For the numerical analysis, finite element method was utilized and van der Waals force, assumed as 12-6 Lennard-Jones potential, was implemented as the body force term in the finite element formulation.
文摘A comprehension of railway dynamic behavior implies the measure of wheel-rail contact forces which are affected by disturbances and errors that are often difficult to be quantified. In this study, a benchmark test case is proposed, and a bogie with a layout used on some European locomotives such as SIEMENS El90 is studied. In this layout, an additional shaft on which brake disks are installed is used to transmit the braking torque to the wheelset through a single-stage gearbox. Using a mixed approach based on finite element techniques and statistical considerations, it is possible to evaluate an optimal layout for strain gauge positioning and to optimize the measurement system to diminish the effects of noise and disturbance. We also conducted preliminary evaluations on the precision and frequency response of the proposed system.
基金the National Natural Science Foundation of China(10572005).
文摘The concept vortex force in aerodynamics is sys- tematically examined based on a new steady vortex-force theory (Wu et al., Vorticity and vortex dynamics, Springer, 2006) which expresses the aerodynamic force (and moment) by the volume and boundary integrals of the Lamb vector. In this paper, the underlying physics of this theory is explored, including the general role of the Lamb vector in non- linear aerodynamics, its initial formation, and its relevance to the total-pressure non-uniformity. As a typical example, the theory is applied to the flow over a slender delta wing at a large angle of attack. The highly localized flow structures with high Lamb-vector peaks are identified in terms of their net contribution to various constituents of the total aerody-namic force. This vortex-force diagnosis sheds new light on the flow control and configuration optimization.
文摘The aim of the investigation was to develop the use of topographic and nano-adhesion atomic force microscopy(AFM) studies as a means of monitoring the coalescence of latex particles within films produced from a pharmaceutically relevant aqueous dispersion(Eudragit~?NE30 D). Films were prepared via spin coating and analysed using AFM, initially via tapping mode for topographic assessment followed by force-distance measurements which allowed assessment of site-specific adhesion. The results showed that colloidal particles were clearly observed topographically in freshly prepared samples, with coalescence detected on curing via the disappearance of discernible surface features and a decrease in roughness indices. The effects of temperature and humidity on film curing were also studied, with the former having the most pronounced effect. AFM force measurements showed that the variation in adhesive force reduced with increasing curing time, suggesting a novel method of quantifying the rate of film formation upon curing. It was concluded that the AFM methods outlined in this study may be used as a means of qualitatively and quantitatively monitoring the curing of pharmaceutical films as a function of time and other variables, thereby facilitating rational design of curing protocols.
文摘The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em></span>), with the commutation relation <img src="Edit_28f5b839-7de4-41e5-9ed8-69dc1bf72c2c.bmp" alt="" />, and using a Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s like equation on these variable, and associating a linear operator to a constant of motion <em>K</em> (<em>x, v, t</em>) of the classical system, The comparison with the quantization in the space (<em>x, p</em>) is done with the usual Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s equation for the Hamiltonian <em>H</em><span style="white-space:normal;">(</span><em style="white-space:normal;">x, p, t</em><span style="white-space:normal;">)</span>, and with the commutation relation <img src="Edit_cca7e318-5b35-4c55-8f09-6089970ce9a2.bmp" alt="" />. It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization has fewer oscillations than the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, the average energy of the system is higher in (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, and the Boltzmann-Shannon entropy on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization is higher than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization.