Applied and Computational Mathematics, Dec 31, 2022
We analyze the problem of finding the shape of a heavy cantilever that minimizes volume and satis... more We analyze the problem of finding the shape of a heavy cantilever that minimizes volume and satisfies certain condition at the free end that corresponds to the so-called longest reach problem. The Pontryagin's maximum principle with cross-sectional area as "control" is used in the optimization procedure. We point out some specific properties of the problem. Namely, in one formulation we are faced with standard optimization problem and in another case with unconventional optimization problem.
Along with the presentations made during this Round Table, we include here some contributions by ... more Along with the presentations made during this Round Table, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of these discussions was to continue the useful traditions from the first conferences on Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, “how to save and improve the prestige of FC”, to share opinions and try to find ways to resolve them.info:eu-repo/semantics/publishedVersio
As a part of this study, a mathematical model based on two-compartment pharmacokinetic system wit... more As a part of this study, a mathematical model based on two-compartment pharmacokinetic system with general fractional derivatives was developed. Its aim was to describe the release of BisGMA and TEGDMA monomers from composite core resin over three time periods. For this purpose, five Clearfil Photo Core resin samples were prepared and elution was measured using high performance liquid chromatography at 1, 7 and 28 days post-immersion in 75% ethanol. The findings confirmed good model fit to the data related to released monomer quantity.
The analysis presented so far applies to all bodies that could be, with sufficient accuracy, desc... more The analysis presented so far applies to all bodies that could be, with sufficient accuracy, described as continuous bodies. Since we intend to study elastic bodies, we now present an experimental foundation that serves as a basis for mathematical description of elastic bodies. The first experiments on elastic bodies were designed to measure the force needed to break a rod or rope made of a specific material. We mention Galileo’s (1638) apparatus for breaking a beam in bending with an end load, Da Vinci’s (1680) apparatus to measure the force needed to rupture a rope, and Mariotte’s (1700) apparatus for measuring the force needed to break an elastic beam by extension.1
Stability of a compressed fluid conveying pipe is analyzed. It is assumed that the pipe is fixed ... more Stability of a compressed fluid conveying pipe is analyzed. It is assumed that the pipe is fixed at one end and free at the other. At the free end the pipe is loaded with a concentrated force od constant intensity and fixed direction. The constitutive eqautions for the pipe are taken in a form that allows pipe axis extensibility and
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2022
We analyze the classical problem of finding the shape of the column that optimizes certain criter... more We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load F of the column and the length L, find cross‐sectional area A, such that the volume W of the column attains minimal value. This is a classical Clausen problem. However, in this work we shall use the generalized constitutive equations of the column that allows for shear deformation and axis compressibility. This, as well as the novel use of the first integral, are the main novelties of our work. We will formulate a nonlinear boundary value problem for post critical deformation of optimally shaped rod. Finally we show that optimally shaped rod exhibits pitchfork supercritical bifurcation at critical buckling load.
The problem of finding the shape of initially straight elastic cantilever that minimizes volume, ... more The problem of finding the shape of initially straight elastic cantilever that minimizes volume, and satisfies certain condition at the free end is analyzed. The recently analyzed longest reach problem follows as a special case. New results are presented about shape of the cantilever, having prescribed volume and having longest reach. The Pontryagin’s maximum principle with cross-sectional area as ‘control’ is used in the optimization procedure.
We analyze the classical problem of finding the shape of the column that optimizes certain criter... more We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load of the column and the length , find crosssectional area , such that the volume of the column attains minimal value. This is a classical Clausen problem. However, in this work we shall use the generalized constitutive equations of the column that allows for shear deformation and axis compressibility. This, as well as the novel use of the first integral, are the main novelties of our work. We will formulate a nonlinear boundary value problem for post critical deformation of optimally shaped rod. Finally we show that optimally shaped rod exhibits pitchfork supercritical bifurcation at critical buckling load.
Applied and Computational Mathematics, Dec 31, 2022
We analyze the problem of finding the shape of a heavy cantilever that minimizes volume and satis... more We analyze the problem of finding the shape of a heavy cantilever that minimizes volume and satisfies certain condition at the free end that corresponds to the so-called longest reach problem. The Pontryagin's maximum principle with cross-sectional area as "control" is used in the optimization procedure. We point out some specific properties of the problem. Namely, in one formulation we are faced with standard optimization problem and in another case with unconventional optimization problem.
Along with the presentations made during this Round Table, we include here some contributions by ... more Along with the presentations made during this Round Table, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of these discussions was to continue the useful traditions from the first conferences on Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, “how to save and improve the prestige of FC”, to share opinions and try to find ways to resolve them.info:eu-repo/semantics/publishedVersio
As a part of this study, a mathematical model based on two-compartment pharmacokinetic system wit... more As a part of this study, a mathematical model based on two-compartment pharmacokinetic system with general fractional derivatives was developed. Its aim was to describe the release of BisGMA and TEGDMA monomers from composite core resin over three time periods. For this purpose, five Clearfil Photo Core resin samples were prepared and elution was measured using high performance liquid chromatography at 1, 7 and 28 days post-immersion in 75% ethanol. The findings confirmed good model fit to the data related to released monomer quantity.
The analysis presented so far applies to all bodies that could be, with sufficient accuracy, desc... more The analysis presented so far applies to all bodies that could be, with sufficient accuracy, described as continuous bodies. Since we intend to study elastic bodies, we now present an experimental foundation that serves as a basis for mathematical description of elastic bodies. The first experiments on elastic bodies were designed to measure the force needed to break a rod or rope made of a specific material. We mention Galileo’s (1638) apparatus for breaking a beam in bending with an end load, Da Vinci’s (1680) apparatus to measure the force needed to rupture a rope, and Mariotte’s (1700) apparatus for measuring the force needed to break an elastic beam by extension.1
Stability of a compressed fluid conveying pipe is analyzed. It is assumed that the pipe is fixed ... more Stability of a compressed fluid conveying pipe is analyzed. It is assumed that the pipe is fixed at one end and free at the other. At the free end the pipe is loaded with a concentrated force od constant intensity and fixed direction. The constitutive eqautions for the pipe are taken in a form that allows pipe axis extensibility and
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2022
We analyze the classical problem of finding the shape of the column that optimizes certain criter... more We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load F of the column and the length L, find cross‐sectional area A, such that the volume W of the column attains minimal value. This is a classical Clausen problem. However, in this work we shall use the generalized constitutive equations of the column that allows for shear deformation and axis compressibility. This, as well as the novel use of the first integral, are the main novelties of our work. We will formulate a nonlinear boundary value problem for post critical deformation of optimally shaped rod. Finally we show that optimally shaped rod exhibits pitchfork supercritical bifurcation at critical buckling load.
The problem of finding the shape of initially straight elastic cantilever that minimizes volume, ... more The problem of finding the shape of initially straight elastic cantilever that minimizes volume, and satisfies certain condition at the free end is analyzed. The recently analyzed longest reach problem follows as a special case. New results are presented about shape of the cantilever, having prescribed volume and having longest reach. The Pontryagin’s maximum principle with cross-sectional area as ‘control’ is used in the optimization procedure.
We analyze the classical problem of finding the shape of the column that optimizes certain criter... more We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load of the column and the length , find crosssectional area , such that the volume of the column attains minimal value. This is a classical Clausen problem. However, in this work we shall use the generalized constitutive equations of the column that allows for shear deformation and axis compressibility. This, as well as the novel use of the first integral, are the main novelties of our work. We will formulate a nonlinear boundary value problem for post critical deformation of optimally shaped rod. Finally we show that optimally shaped rod exhibits pitchfork supercritical bifurcation at critical buckling load.
Uploads