In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide a...In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.展开更多
In this study,we applied the variational model to fluidization of small spherical particles.Fluidization experiments were carried out for spherical particles with 13 diameters between dp=0.13 and 5.00 mm.We propose a ...In this study,we applied the variational model to fluidization of small spherical particles.Fluidization experiments were carried out for spherical particles with 13 diameters between dp=0.13 and 5.00 mm.We propose a generalized form of our variational model to predict the superficial velocity U and interphase drag coefficientβby introducing an exponent n to describe the different dependences of the drag force Fd on fluid velocity for different particle sizes(different flow regimes).By comparing the predictions with the experimental results,we conclude that n=1 should be used for small particles(dp<1 mm)and n=2 for larger particles(dp>1 mm).This conclusion is generalized by proposing n=1 for particles with Ret<160 and n=2 for particles with Ret>160.The average mean absolute error was 5.49%in calculating superficial velocity for different bed voidages using the modified variational model for all of the particles examined.The calculated values ofβwere compared with values of literature models for particles with dp<1.0 mm.The average mean absolute error of the modified variational model was 8.02%in calculatingβfor different bed voidages for all of the particles examined.展开更多
The study adopts the variational method for analyzing the cantilever tapered beams under a tip load as well as a definite end displacement,and further determining the optimized shapes and materials that can minimize t...The study adopts the variational method for analyzing the cantilever tapered beams under a tip load as well as a definite end displacement,and further determining the optimized shapes and materials that can minimize the weights.Two types of beams are taken into account,i.e.,the Euler-Bernoulli beam without considering shear deformation and the Timoshenko beam with shear deformation.By using the energy theorem and the reference of isoperimetric problem,the width variation curves and the corresponding minimum masses are derived for both beam types.The optimized curve of beam width for the Euler-Bernoulli beam is found to be a linear function,but nonlinear for the Timoshenko beam.It is also found that the optimized curve in the Timoshenko beam case starts from non-zero at the tip end,but its tendency gradually approaches the one of the Euler-Bernoulli beam.The results indicate that with the increase of the Poisson’s ratio,the required minimum mass of the beam will increase no matter how the material changes,suggesting that the optimized mass for the case of Euler-Bernoulli beam is the lower boundary limit which the Timoshenko case cannot go beyond.Furthermore,the ratio r/E(density against Elastic Modulus)of the material should be as small as possible,while the ratio h2/L4 of the beam should be as large as possible in order to minimize the mass for the case of Euler-Bernoulli beam,of which the conclusion is extended to be applicable for the case of Timoshenko beam.In addition,the optimized curves for Euler-Bernoulli beam types are all found to be power functions of length for both tip point load cases and uniform load cases.展开更多
基金supported by FEDER funds through COMPETE - Operational Programme Factors of Competitiveness("Programa Operacional Factores de Competitividade")Portuguese funds through the Center for Research and Development in Mathematics and Applications(University of Aveiro) and the Portuguese Foundation for Science and Technology("FCT - Fundao para a Ciencia e a Tecnologia"),within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690
文摘In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.
基金This work was supported by the Serbian Ministry of Edu-cation,Science and Technological Development(grant number ON172022).
文摘In this study,we applied the variational model to fluidization of small spherical particles.Fluidization experiments were carried out for spherical particles with 13 diameters between dp=0.13 and 5.00 mm.We propose a generalized form of our variational model to predict the superficial velocity U and interphase drag coefficientβby introducing an exponent n to describe the different dependences of the drag force Fd on fluid velocity for different particle sizes(different flow regimes).By comparing the predictions with the experimental results,we conclude that n=1 should be used for small particles(dp<1 mm)and n=2 for larger particles(dp>1 mm).This conclusion is generalized by proposing n=1 for particles with Ret<160 and n=2 for particles with Ret>160.The average mean absolute error was 5.49%in calculating superficial velocity for different bed voidages using the modified variational model for all of the particles examined.The calculated values ofβwere compared with values of literature models for particles with dp<1.0 mm.The average mean absolute error of the modified variational model was 8.02%in calculatingβfor different bed voidages for all of the particles examined.
基金supports from Xi’an Jiaotong–Liverpool University(RDF 14-02-44,RDF 15-01-38,RDF 18-01-23 and PGRS1906002)the Key Program Special Fund at XJTLU(Grant No.KSF-E-19)are gratefully acknowledged.
文摘The study adopts the variational method for analyzing the cantilever tapered beams under a tip load as well as a definite end displacement,and further determining the optimized shapes and materials that can minimize the weights.Two types of beams are taken into account,i.e.,the Euler-Bernoulli beam without considering shear deformation and the Timoshenko beam with shear deformation.By using the energy theorem and the reference of isoperimetric problem,the width variation curves and the corresponding minimum masses are derived for both beam types.The optimized curve of beam width for the Euler-Bernoulli beam is found to be a linear function,but nonlinear for the Timoshenko beam.It is also found that the optimized curve in the Timoshenko beam case starts from non-zero at the tip end,but its tendency gradually approaches the one of the Euler-Bernoulli beam.The results indicate that with the increase of the Poisson’s ratio,the required minimum mass of the beam will increase no matter how the material changes,suggesting that the optimized mass for the case of Euler-Bernoulli beam is the lower boundary limit which the Timoshenko case cannot go beyond.Furthermore,the ratio r/E(density against Elastic Modulus)of the material should be as small as possible,while the ratio h2/L4 of the beam should be as large as possible in order to minimize the mass for the case of Euler-Bernoulli beam,of which the conclusion is extended to be applicable for the case of Timoshenko beam.In addition,the optimized curves for Euler-Bernoulli beam types are all found to be power functions of length for both tip point load cases and uniform load cases.
