Internal bond （IB） strength is one of the most important me- chanical properties that indicate particleboard quality. The aim of this study was to find a simple regression model that considers the most important par...Internal bond （IB） strength is one of the most important me- chanical properties that indicate particleboard quality. The aim of this study was to find a simple regression model that considers the most important parameters that can influence on IB strength. In this study, IB strength was predicted by three kinds of equations （linear, quadratic, and exponential） that were based on the percentage of adhesive （8%, 9.5%, and 11%）, particle size （＋5, -5 ＋8, -8 12, and -12 mesh）, and density （0.65, 0.7, and 0.75 g/cm3）. Our analysis of the results （using SHAZAM 9 software） showed that the exponential function best fitted the experi- mental data and predicted the IB strength with 18~,/0 error. In order de- crease the error percentage, the Buckingham Pi theorem was used to build regression models for predicting IB strength based on particle size,展开更多
The Buckingham expansion is important for understanding molecular multipoles and(hyper)polarizabilities.In this study,we give a complete derivation of the Buckingham expansion in the traced form using successive Taylo...The Buckingham expansion is important for understanding molecular multipoles and(hyper)polarizabilities.In this study,we give a complete derivation of the Buckingham expansion in the traced form using successive Taylor series.Based on the derivation results,a general Buckingham expansion in the traced form is proposed,from which highly accurate numerical calculations using the finite field method can be achieved.The transformations from the traced multipoles and multipole-multipole polarizabilities to the corresponding traceless counterparts are realized with an auxiliary traced electric field gradient.The applications of thefinite field method in this study show good agreements with previous theoretical calculations and experimental measurements.展开更多
Based on Buckingham＇s π-Theorem, dimensional analysis has achieved considerable success over the past near-century. Model testing has long been a powerful tool in both scientific studies and engineering applications...Based on Buckingham＇s π-Theorem, dimensional analysis has achieved considerable success over the past near-century. Model testing has long been a powerful tool in both scientific studies and engineering applications. However, the prototype objects are becoming more and more complicated nowadays, and many of the prototype systems can contain several sub-systems. The conventional theories on model-prototype similarity and dimensional analysis have only limited application since the π-Theorem itself does not distinguish between the original system and subsystems. This is particularly true in the field of structural dynamics, where the structure is often modeled as a multi-degree-of-freedom system. In this paper, we attempt to show that, if a system can be decoupled into several nontrivial subsystems, then, in each subsystem, the number of π-terms will be reduced and therefore simplify the model testing. On the other hand, if a system cannot be decoupled into subsystems, then using model testing with reduced π-term analysis, both experimentally and theoretically, may introduce severe errors.展开更多
Purpose: The Pythagorean Comma refers to an ancient Greek musical, mathematical tuning method that defines an integer ratio of exponential coupling constant harmonic law of two frequencies and a virtual frequency. A C...Purpose: The Pythagorean Comma refers to an ancient Greek musical, mathematical tuning method that defines an integer ratio of exponential coupling constant harmonic law of two frequencies and a virtual frequency. A Comma represents a physical harmonic system that is readily observable and can be mathematically simulated. The virtual harmonic is essential and indirectly measurable. The Pythagorean Comma relates to two discrete frequencies but can be generalized to any including infinite harmonics of a fundamental frequency, vF. These power laws encode the physical and mathematical properties of their coupling constant ratio, natural resonance, the maximal resonance of the powers of the frequencies, wave interference, and the beat. The hypothesis is that the Pythagorean power fractions of a fundamental frequency, vF are structured by the same harmonic fraction system seen with standing waves. Methods: The Pythagorean Comma refers to the ratio of (3/2)12 and 27 that is nearly equal to 1. A Comma is related to the physical setting of the maximum resonance of the powers of two frequencies. The powers and the virtual frequency are derived simulating the physical environment utilizing the Buckingham Π theorem, array analysis, and dimensional analysis. The powers and the virtual frequency can be generalized to any two frequencies. The maximum resonance occurs when their dimensionless ratio closest to 1 and the virtual harmonic closest to 1 Hz. The Pythagorean possible power arrays for a vF system or any two different frequencies are evaluated. Results: The generalized Pythagorean harmonic power law for any two different frequencies coupling constant are derived with a form of an infinite number of powers defining a constant power ratio and a single virtual harmonic frequency. This power system has periodic and fractal properties. The Pythagorean power law also encodes the ratio of logs of the frequencies. These must equal or nearly equal the power ratio. When all of the harmonics are powers of a vF the Pythagorean powers are defined by a consecutive integer series structured in the identical form as standard harmonic fractions. The ratio of the powers is rational, and all of the virtual harmonics are 1 Hz. Conclusion: The Pythagorean Comma power law method can be generalized. This is a new isomorphic wave perspective that encompasses all harmonic systems, but with an infinite number of possible powers. It is important since there is new information: powers, power ratio, and a virtual frequency. The Pythagorean relationships are different, yet an isomorphic perspective where the powers demonstrate harmonic patterns. The coupling constants of a vF Pythagorean power law system are related to the vFs raised to the harmonic fraction series which accounts for the parallel organization to the standing wave system. This new perspective accurately defines an alternate valid physical harmonic system.展开更多
The objective of this paper is to derive a tentative mathematical formula of drug permeability of biodegradable membrane for implant use on the basis of Buckingham P i Theorem in principles of dimensional analysis.S...The objective of this paper is to derive a tentative mathematical formula of drug permeability of biodegradable membrane for implant use on the basis of Buckingham P i Theorem in principles of dimensional analysis.Such a theoretical predictive formula would be very useful for not only designing a sustained release system of drug made of biodegradable material for implant use before it is made but also for saving much time and expenses of the experiment. Three basic units were selected to describe the problem due to the invariableness of temperature under the normal physiological condition.They are mass,length and time.At first,a general mathematical formula was derived from Buckingham P i Theorem,in which the relevant parameters were placed into dimensionless Π groups.Secondly,as a special case interested to us,we gave out a predictive formula of permeability of various drugs with different molecular weights and saturated concentrations through Polylactic Acid Membrane with specified characteristics for implant use.For this end,we computed six kinds of drugs.The linear regression analysis of the relationship among these Π groups with drug permeability yielded coefficients for the corresponding Π groups.The obtained results in our paper mainly fitted in with ones obtained by the experiments under in vitro condition.Finally,a computer program for predicting drug permeability was given out in this article.展开更多
文摘Internal bond （IB） strength is one of the most important me- chanical properties that indicate particleboard quality. The aim of this study was to find a simple regression model that considers the most important parameters that can influence on IB strength. In this study, IB strength was predicted by three kinds of equations （linear, quadratic, and exponential） that were based on the percentage of adhesive （8%, 9.5%, and 11%）, particle size （＋5, -5 ＋8, -8 12, and -12 mesh）, and density （0.65, 0.7, and 0.75 g/cm3）. Our analysis of the results （using SHAZAM 9 software） showed that the exponential function best fitted the experi- mental data and predicted the IB strength with 18~,/0 error. In order de- crease the error percentage, the Buckingham Pi theorem was used to build regression models for predicting IB strength based on particle size,
基金the National Natural Science Foundation of China(Grant Nos.21573112 and 21421001)。
文摘The Buckingham expansion is important for understanding molecular multipoles and(hyper)polarizabilities.In this study,we give a complete derivation of the Buckingham expansion in the traced form using successive Taylor series.Based on the derivation results,a general Buckingham expansion in the traced form is proposed,from which highly accurate numerical calculations using the finite field method can be achieved.The transformations from the traced multipoles and multipole-multipole polarizabilities to the corresponding traceless counterparts are realized with an auxiliary traced electric field gradient.The applications of thefinite field method in this study show good agreements with previous theoretical calculations and experimental measurements.
文摘Based on Buckingham＇s π-Theorem, dimensional analysis has achieved considerable success over the past near-century. Model testing has long been a powerful tool in both scientific studies and engineering applications. However, the prototype objects are becoming more and more complicated nowadays, and many of the prototype systems can contain several sub-systems. The conventional theories on model-prototype similarity and dimensional analysis have only limited application since the π-Theorem itself does not distinguish between the original system and subsystems. This is particularly true in the field of structural dynamics, where the structure is often modeled as a multi-degree-of-freedom system. In this paper, we attempt to show that, if a system can be decoupled into several nontrivial subsystems, then, in each subsystem, the number of π-terms will be reduced and therefore simplify the model testing. On the other hand, if a system cannot be decoupled into subsystems, then using model testing with reduced π-term analysis, both experimentally and theoretically, may introduce severe errors.
文摘Purpose: The Pythagorean Comma refers to an ancient Greek musical, mathematical tuning method that defines an integer ratio of exponential coupling constant harmonic law of two frequencies and a virtual frequency. A Comma represents a physical harmonic system that is readily observable and can be mathematically simulated. The virtual harmonic is essential and indirectly measurable. The Pythagorean Comma relates to two discrete frequencies but can be generalized to any including infinite harmonics of a fundamental frequency, vF. These power laws encode the physical and mathematical properties of their coupling constant ratio, natural resonance, the maximal resonance of the powers of the frequencies, wave interference, and the beat. The hypothesis is that the Pythagorean power fractions of a fundamental frequency, vF are structured by the same harmonic fraction system seen with standing waves. Methods: The Pythagorean Comma refers to the ratio of (3/2)12 and 27 that is nearly equal to 1. A Comma is related to the physical setting of the maximum resonance of the powers of two frequencies. The powers and the virtual frequency are derived simulating the physical environment utilizing the Buckingham Π theorem, array analysis, and dimensional analysis. The powers and the virtual frequency can be generalized to any two frequencies. The maximum resonance occurs when their dimensionless ratio closest to 1 and the virtual harmonic closest to 1 Hz. The Pythagorean possible power arrays for a vF system or any two different frequencies are evaluated. Results: The generalized Pythagorean harmonic power law for any two different frequencies coupling constant are derived with a form of an infinite number of powers defining a constant power ratio and a single virtual harmonic frequency. This power system has periodic and fractal properties. The Pythagorean power law also encodes the ratio of logs of the frequencies. These must equal or nearly equal the power ratio. When all of the harmonics are powers of a vF the Pythagorean powers are defined by a consecutive integer series structured in the identical form as standard harmonic fractions. The ratio of the powers is rational, and all of the virtual harmonics are 1 Hz. Conclusion: The Pythagorean Comma power law method can be generalized. This is a new isomorphic wave perspective that encompasses all harmonic systems, but with an infinite number of possible powers. It is important since there is new information: powers, power ratio, and a virtual frequency. The Pythagorean relationships are different, yet an isomorphic perspective where the powers demonstrate harmonic patterns. The coupling constants of a vF Pythagorean power law system are related to the vFs raised to the harmonic fraction series which accounts for the parallel organization to the standing wave system. This new perspective accurately defines an alternate valid physical harmonic system.
文摘The objective of this paper is to derive a tentative mathematical formula of drug permeability of biodegradable membrane for implant use on the basis of Buckingham P i Theorem in principles of dimensional analysis.Such a theoretical predictive formula would be very useful for not only designing a sustained release system of drug made of biodegradable material for implant use before it is made but also for saving much time and expenses of the experiment. Three basic units were selected to describe the problem due to the invariableness of temperature under the normal physiological condition.They are mass,length and time.At first,a general mathematical formula was derived from Buckingham P i Theorem,in which the relevant parameters were placed into dimensionless Π groups.Secondly,as a special case interested to us,we gave out a predictive formula of permeability of various drugs with different molecular weights and saturated concentrations through Polylactic Acid Membrane with specified characteristics for implant use.For this end,we computed six kinds of drugs.The linear regression analysis of the relationship among these Π groups with drug permeability yielded coefficients for the corresponding Π groups.The obtained results in our paper mainly fitted in with ones obtained by the experiments under in vitro condition.Finally,a computer program for predicting drug permeability was given out in this article.
