The Motion Equation of a Spring-Magnet-Mass System Placed in Nonlinear Magnetic Field. An Analytical Solution of Elliptic Sine form Functions

2016 ◽  
Vol 12 (4) ◽  
pp. 1-15
Author(s):  
Nicusor Nistor ◽  
Constantin Gheorghies ◽  
Nelu Cazacu
Keyword(s):  
Magnetic Field ◽  
Analytical Solution ◽  
Motion Equation ◽  
Mass System ◽  
Magnet Mass

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

AN ANALYTICAL SOLUTION OF THE COSMIC RAYS TRANSPORT EQUATION IN THE PRESENCE OF THE GEOMAGNETIC FIELD

2002 ◽  
Vol 17 (12n13) ◽  
pp. 1645-1653
Author(s):  
MARINA GIBILISCO
Keyword(s):  
Magnetic Field ◽  
Cosmic Rays ◽  
Analytical Solution ◽  
Transport Equation ◽  
Geomagnetic Field ◽  
The Earth ◽  
Factorization Technique ◽  
Diffusion Motion ◽  
Geomagnetic Fields ◽  
The Magnetic Field

In this work, I study the propagation of cosmic rays inside the magnetic field of the Earth, at distances d ≤ 500 Km from its surface; at these distances, the geomagnetic field deeply influences the diffusion motion of the particles. I compare the different effects of the interplanetary and of the geomagnetic fields, by also discussing their role inside the cosmic rays transport equation; finally, I present an analytical method to solve such an equation through a factorization technique.

scholarly journals Estimation of leg stiffness using an approximation to the planar spring–mass system in high-speed running

2020 ◽  
Vol 17 (1) ◽  
pp. 172988141989071
Author(s):  
Wei Guo ◽  
Changrong Cai ◽  
Mantian Li ◽  
Fusheng Zha ◽  
Pengfei Wang ◽  
...  
Keyword(s):  
Analytical Solution ◽  
High Speed ◽  
Stance Phase ◽  
Center Of Mass ◽  
Critical Role ◽  
Line Center ◽  
Motion Model ◽  
Mass System ◽  
Leg Stiffness ◽  
Wide Range

Leg stiffness plays a critical role in legged robots’ speed regulation. However, the analytic solutions to the differential equations of the stance phase do not exist, of course not for the exact analytical solution of stiffness. In view of the challenge in dealing with every circumstance by numerical methods, which have been adopted to tabulate approximate answers, the “harmonic motion model” was used as approximation of the stance phase. However, the wide range leg sweep angles and small fluctuations of the “center of mass” in fast movement were overlooked. In this article, we raise a “triangle motion model” with uniform forward speed, symmetric movement, and straight-line center of mass trajectory. The characters are then shifted to a quadratic equation by Taylor expansion and obtain an approximate analytical solution. Both the numerical simulation and ADAMS-Matlab co-simulation of the control system show the accuracy of the triangle motion model method in predicting leg stiffness even in the ultra-high-speed case, and it is also adaptable to low-speed cases. The study illuminates the relationship between leg stiffness and speed, and the approximation model of the planar spring–mass system may serve as an analytical tool for leg stiffness estimation in high-speed locomotion.

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