On the Asymptotic Solution of a Two-Parameter Boundary Value Problem of Chemical Reactor Theory

1974 ◽  
Vol 26 (4) ◽  
pp. 717-729 ◽  
Author(s):  
Jye Chen ◽  
R. E. O’Malley, Jr.
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Boundary Value ◽  
Chemical Reactor ◽  
Two Parameter

scholarly journals COMPARISON OF SOLUTIONS TO THE BENDING PROBLEMS OF THREE-LAYER PLATES ON THE WINKLER AND PASTERNAK FOUNDATIONS

2021 ◽  
Vol 1 (54) ◽  
pp. 30-37
Author(s):  
Anastasiya G. KOZEL ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Median Plane ◽  
Winkler Foundation ◽  
Numerical Comparison ◽  
Relative Shear ◽  
Cylindrical Coordinate ◽  
Pasternak Model ◽  
Bending Problems ◽  
Two Parameter

Solutions of problems on axisymmetric bending of an elastic three-layer circular plate on the Winkler and Pasternak foundations are given. The bearing layers are taken as isotropic, for which Kirchhoff’s hypotheses are fulfilled. In a sufficiently thick lightweight, incompressible in thickness aggregate, the Timoshenko model is valid. The cylindrical coordinate system, in which the statements and solutions of boundary value problems are carried out, is connected with the median plane of the filler. On the plate contour, it is assumed that there is a rigid diaphragm that prevents the relative shear of the layers. The system of differential equations of equilibrium is obtained by the variational method. Three types of boundary conditions are formulated. One- and two-parameter Winkler and Pasternak models are used to describe the reaction of an elastic foundation. The solution to the boundary value problem is reduced to finding three desired functions, plate deflection, shear, and radial displacement in the filler. The general analytical solution to the boundary value problem is written out in the case of the Pasternak model in Bessel functions. At the Winkler foundation, the known solution is given in Kelvin functions. A numerical comparison of the displacements and stresses obtained by both models with a uniformly distributed load and rigid sealing of the plate contour is carried out.

scholarly journals Asymptotic Solution of a Boundary Value Problem for a Spring–Mass Model of Legged Locomotion

2020 ◽  
Vol 30 (6) ◽  
pp. 2971-2988
Author(s):  
Hanna Okrasińska-Płociniczak ◽  
Łukasz Płociniczak
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Asymptotic Expansions ◽  
Boundary Value ◽  
Legged Locomotion ◽  
Asymptotic Formulas ◽  
Numerical Verification ◽  
Mass Model ◽  
Leg Stiffness ◽  
Elastic Pendulum

Abstract Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value problem arises where one has to determine the leg stiffness. In this paper, we find asymptotic expansions of the stiffness. These are conducted perturbatively: once with respect to small angles of attack, and once for large velocities. Our findings are in agreement with previous results and numerical simulations. In particular, we show that the leg stiffness is inversely proportional to the square of the attack angle for its small values, and proportional to the velocity for large speeds. We give exact asymptotic formulas to several orders and conclude the paper with a numerical verification.

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