scholarly journals A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs

2020 ◽  
Vol 32 (4) ◽  
pp. 455-487
Author(s):  
R. Borsche ◽  
D. Kocoglu ◽  
S. Trenn
Keyword(s):  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Electric Power ◽  
Unique Solvability ◽  
Differential Algebraic Equations ◽  
Hyperbolic Partial Differential Equations ◽  
Power Lines ◽  
Algebraic Equations ◽  
Distributional Solution ◽  
Telegraph Equations

AbstractA distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modelled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.

An electronic analogue for supersonic flow

1957 ◽  
Vol 2 (4) ◽  
pp. 383-396
Author(s):  
Leslie S. G. Kovásznay
Keyword(s):  
Partial Differential Equations ◽  
Supersonic Flow ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Characteristic Lines ◽  
Analogue Method ◽  
Cathode Ray Tube

An analogue method for solving certain quasi-linear hyperbolic partial differential equations is presented. The characteristic lines are formed by scanning electronically the screen of a cathoderay tube. The boundary conditions are introduced in the form of an opaque mask. The solution appears as a picture on the screen of a second cathode-ray tube. The experiments demonstrate the feasibility of the approach, but the development of the machine has not been carried beyond this stage.

Dynamic characteristics of a 3-RPR planar parallel manipulator with flexible intermediate links

Robotica ◽  
2014 ◽  
Vol 33 (9) ◽  
pp. 1909-1925 ◽  
Author(s):  
Amirhossein Eshaghiyeh Firoozabadi ◽  
Saeed Ebrahimi ◽  
Ghasem Amirian
Keyword(s):  
Boundary Conditions ◽  
Dynamic Modeling ◽  
Parallel Manipulator ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
End Effector ◽  
Assumed Mode Method ◽  
Planar Parallel Manipulator

SUMMARYThis paper presents the dynamic modeling of a 3-RPR planar parallel manipulator with three flexible intermediate links in order to investigate the effects of the intermediate links flexibility on the undesired vibrations of the end-effector. For this purpose, the intermediate links are modeled as Euler--Bernoulli beams with two types of fixed-pinned and fixed-free boundary conditions based on the assumed mode method (AMM). The equations of motion of the 3-RPR manipulator are formulated using the augmented Lagrange multipliers method in the form of differential algebraic equations (DAEs) by incorporating the elastic and rigid coordinates in the set of generalized coordinates. After defining the initial conditions and imposing external forces to the manipulator, the equations are then solved numerically using the Modified Extended Backward-Differentiation Formula Implicit (MEBDFI) approach. Comparison of the simulation results for two different boundary conditions shows clearly the effects of flexibility of the intermediate links on the vibration of the end-effector trajectory. Results of this work can be used for the dynamic modeling of other manipulators or to design a controller for reducing the undesired vibrations.

A Numerical Method for Solving Second-Order Linear Partial Differential Equations Under Dirichlet, Neumann and Robin Boundary Conditions

2017 ◽  
Vol 14 (02) ◽  
pp. 1750015 ◽  
Author(s):  
Şuayip Yüzbaşı
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Second Order ◽  
Robin Boundary Conditions ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Robin Boundary

The aim of this paper is to give a collocation method to solve second-order partial differential equations with variable coefficients under Dirichlet, Neumann and Robin boundary conditions. By using the Bessel functions of the first kind, the matrix operations and the collocation points, the method is constructed and it transforms the partial differential equation problem into a system of algebraic equations. The unknown coefficients of the assuming solution are determined by solving this system. The algorithm of the proposed method is presented. Also, error estimation technique is introduced and the approximate solutions are improved by means of it. To show the validity and applicability of the presented method, we solve numerical examples and give the comparison of solutions and comparisons of the errors (actual and estimation).

scholarly journals Co-Simulation of Algebraically Coupled Dynamic Subsystems

2001 ◽  
Author(s):  
Bei Gu ◽  
H. Harry Asada
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Discrete Time ◽  
Sliding Mode ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Sliding Manifold ◽  
Sliding Manifolds ◽  
Algebraic Constraints

Abstract This paper analyzes the problem of Co-Simulation. The term Co-Simulation is used to describe a large dynamic system that is simulated by running a group of independently coded subsystem simulators. Very commonly, the Co-Simulation of subsystems faces incompatible boundary conditions, i.e., causal conflicts. These causal conflicts cannot be directly resolved, due to the nonlinearity and/or difficulties in modification of coded subsystem simulators. Causal conflicts result in algebraic constraints. Boundary Condition Coordinators (BCCs) are designed to calculate boundary conditions based on subsystem models and their algebraic constraints. The Co-Simulation, which is modeled as Differential Algebraic Equations, then relies on BCC to provide compatible boundary conditions for subsystem simulators. The high index constraint is reduced to index one by defining a sliding manifold. Different ways of enforcing the sliding manifold are discussed: A new Discrete-Time Sliding Mode (DTSM) controller is devised to serve as a BCC, enforcing sliding manifolds and providing boundary conditions. The multi-rate scheme can guarantee Co-Simulation stability at any given step size of all subsystem simulators, provided the subsystem simulators are tested stable at that step size. An example is given to demonstrate the DTSM method. Advantages and possible future improvements are discussed.

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