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One-dimensional finite strain consolidation
Published in Jian-Hua Yin, Guofu Zhu, Consolidation Analyses of Soils, 2020
Infinitesimal strain theories of consolidation assume that the thickness of the compressible layer is constant; the deformation of the layer, during consolidation, is assumed to be small compared to its thickness. There is no need to distinguish an Eulerian system and a Lagrangian system. If the deformations are large compared to the thickness of the compressible layer, the situation is different. For example, a piezometer fixed in space but within the zone of settlement, i.e. near the boundary of the clay layer, may be outside of the layer after some time. A real measuring system is one which convects with the material particles. The piezometer would always be surrounded by the same material points and would measure the pore pressure of this part of the skeleton as a function of its momentary position and time. This type of system is termed as a convective coordinate system.
One-Dimensional Unsteady Motions of a Gas
Published in L. I. Sedov, A. G. Volkovets, in Mechanics, 2018
Let the initial coordinate of the gas particles in a Lagrangian system of coordinates be denoted by r0. It is obvious that for every particle coordinate r0 is equal to the radius of the shock wave r2 at the instant it passes through the particle. From the conditions on the shock wave and from the adiabatic equation behind the wave front it follows that () pργ=p2ρ2γ(r0)=8(γ−1)γ(v+2)2(γ+1)(γ+1)Eρ1γ1r0v.
Introduction to Systems of ODEs
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
Many interesting models originate from classical mechanical problems. The most general way to derive the corresponding systems of differential equations describing these models is the Euler–Lagrange equations. The method originated in works by Leonhard Euler in 1733; later in 1750, his student Joseph-Louis Lagrange substantially improved this method. There are two very important reasons for working with Lagrange's equations rather than Newton's. The first is that Lagrange's equations hold in any coordinate system, while Newton's is restricted to an inertial frame. The second is the ease with which we can deal with constraints in the Lagrangian system.
Modeling of the dynamics of plane functionally graded waveguides based on the different formulations of the plate theory of I. N. Vekua type
Published in Mechanics of Advanced Materials and Structures, 2021
Olga V. Egorova, Alexey S. Kurbatov, Lev N. Rabinskiy, Sergey I. Zhavoronok
The treatment of the plate model as a constrained Lagrangian system secures the exact satisfaction of the boundary conditions on the faces. On the other hand, it leads to more complex equations for higher-order plate theories, therefore this method can be interpreted as a second approximation if the good convergence of higher frequencies and mode forms is required by the application. For more detailed convergence analysis see also [30] and [61, 62, 66–70] and finally [77] where the use of the extended plate theory satisfying the boundary conditions on the plate faces is shown.
How AD can help solve differential-algebraic equations
Published in Optimization Methods and Software, 2018
John D. Pryce, Nedialko S. Nedialkov, Guangning Tan, Xiao Li
If , i.e. constraints are present, (15), (16) is termed a Lagrangian system of the first kind. It is a DAE system, of index 3 in the classical sense or index 2 as defined in (5), since two t-differentiations of each are needed. If the coordinates are chosen so that , it is of the second kind and is an ODE system.
Coordination motion of Lagrangian systems with multiple oscillatory leaders under diverse interaction topologies
Published in International Journal of Systems Science, 2019
Liyun Zhao, Juan Wang, Jianfeng Lv, Rui Wang
Every Lagrangian system is linearly parameterisable, that is, where is a regressor matrix of known function about the vectors and , is a vector unknown but containing constant Lagrangian system parameters.