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.

Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations

2020 ◽  
pp. 163-181
Author(s):  
Anar Adiloğlu-Nabiev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Boundary Value ◽  
Asymptotic Formulas ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Sturm Liouville ◽  
Complex Valued

A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n>1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.

Stable Running with Asymmetric Legs: A Bifurcation Approach

2015 ◽  
Vol 25 (11) ◽  
pp. 1550152 ◽  
Author(s):  
Andreas Merker ◽  
Dieter Kaiser ◽  
Andre Seyfarth ◽  
Martin Hermann
Keyword(s):  
Boundary Value Problem ◽  
System Dynamics ◽  
Boundary Value ◽  
Mass Model ◽  
Animal Locomotion ◽  
Bifurcation Points ◽  
Stable Solutions ◽  
The Stability

The spring-mass model is a frequently used gait template to describe human and animal locomotion. In this study, we transform the spring-mass model for running into a boundary value problem and use it for the computation of bifurcation points. We show that the analysis of the region of stable solutions can be reduced to the calculation of its boundaries. Using the new bifurcation approach, we investigate the influence of asymmetric leg parameters on the stability of running. Like previously found in walking, leg asymmetry does not necessarily restrict the range of stable running and may even provide benefits for system dynamics.

Boundary-Layer Similarity Near the Edge of a Rotating Disk

1975 ◽  
Vol 42 (3) ◽  
pp. 584-590 ◽  
Author(s):  
R. J. Bodonyi ◽  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Flow Field ◽  
Boundary Value Problem ◽  
Asymptotic Expansions ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Rotating Disk ◽  
Angular Speed ◽  
Rotating Fluid

Numerical solutions of the similarity equations governing the flow near the edge of a finite rotating disk are found to be possible only for −2.06626 ≤ α ≤ 1, where α is the ratio of the disk’s angular speed to that of the rigidly rotating fluid far from the disk. Furthermore, for α ≤ −1 the solutions of the boundary-value problem are not unique, and along one of the solution branches a singular structure of the flow field is approached as α → −1. Using the method of matched asymptotic expansions an approximate solution is found along the singular branch which explains some of the problems encountered in finding numerical solutions.

Lifting three-dimensional wings in transonic flow

1979 ◽  
Vol 95 (2) ◽  
pp. 223-240 ◽  
Author(s):  
M. S. Cramer
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Expansions ◽  
Transonic Flow ◽  
Near Field ◽  
Boundary Value ◽  
Three Dimensional ◽  
Matched Asymptotic Expansions ◽  
Far Field ◽  
First Order ◽  
Order Contribution

The far field of a lifting three-dimensional wing in transonic flow is analysed. The boundary-value problem governing the flow far from the wing is derived by the method of matched asymptotic expansions. The main result is to show that corrections which are second order in the near field make a first-order contribution to the far field. The present study corrects and simplifies the work of Cheng & Hafez (1975) and Barnwell (1975).

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