Multi-Integral Method for Solving the Forward Dynamics of Stiff Multibody Systems

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Numerical Solution ◽  
State Vector ◽  
Taylor Series Expansion ◽  
High Order ◽  
The State ◽  
System State ◽  
Time Step ◽  
Entire Interval ◽  
Free Parameters ◽  
Derivatives Of

The Hermite–Obreshkov–Padé (HOP) procedure is an implicit method for the numerical solution of a system of ordinary differential equations (ODEs) applicable to stiff dynamical systems. This procedure applies an Obreshkov condition to multiple derivatives of the system state vector, both at the start and end of a time step in the numerical solution. That condition is shown to be satisfied by the Hermite interpolating polynomial that matches the state vector and its derivatives, also at the start and end of a time step. The Hermite polynomial, in turn, can be specified in terms of the system state and its derivatives at the start of a step together with a collection of free parameters. Adjusting these free parameters to minimize magnitudes of the ODE residual and its derivatives at the end of a step serves as a proxy for matching the system state and its derivatives. A high-order Taylor expansion at the start of a time step interval models the residual and its derivatives over the entire interval. A variant of this procedure adjusts those parameters to match integrals of the system state over the duration of that interval. This is done by minimizing magnitudes of integrals of the ODE residual calculated from the extrapolating Taylor-series expansion, a process that avoids the need to determine integration constants for multiple integrals of the state. This alternative method eliminates the calculation of high-order derivatives of the system state and hence avoids loss in accuracy from floating-point round off. Numerical performance is evaluated on a dynamically unbalanced constant-velocity (CV) coupling having a high spring rate constraining shaft deflection.

A Tabular Taylor Series Expansion Method for Fast Calculation of Steam Properties

1997 ◽  
Vol 119 (2) ◽  
pp. 485-491 ◽  
Author(s):  
K. Miyagawa ◽  
P. G. Hill
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
International Association ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
The State ◽  
Steam Power ◽  
Direct Use ◽  
Derivatives Of ◽  
Plane Grid

A new method is proposed for rapid and accurate calculation of steam properties in the regions of the state plane of greatest importance to the steam power industry. The method makes direct use of the derivatives of that Helmholtz function that is the best available wide-ranging scientific formulation of the properties of steam. It is rapid because, with a six-term Taylor series expansion, it uses property values and derivatives evaluated once and for all from the Helmholtz function and stored in tables pertaining to an optimized state plane grid configuration. The method eliminates the need for iterative property calculations and is amenable to any region of the state plane. For properties in the ranges of temperature from 0 to 800°C and pressure from 0 to 100 MPa the core memory requirement for three functions of any given pair of independent properties is less than 1 Mb. With this memory allocation it is possible everywhere in the stated range to satisfy the specific volume and enthalpy tolerances specified by the International Association for the Properties of Water and Steam. An optimized formulation of the method is demonstrated in this paper for enthalpy, entropy, and volume functions of pressure and temperature in the superheat region.

Application of the Extended Kalman Filter to Control of a Shape Memory Alloy Arm

2003 ◽  
Author(s):  
Mohammad H. Elahinia ◽  
Hashem Ashrafiuon ◽  
Mehdi Ahmadian ◽  
Daniel J. Inman
Keyword(s):  
Kalman Filter ◽  
Shape Memory Alloy ◽  
Shape Memory ◽  
Extended Kalman Filter ◽  
State Vector ◽  
Angular Position ◽  
The State ◽  
State Variables ◽  
Time Step ◽  
State Variable

This paper presents a robust nonlinear control that uses a state variable estimator for control of a single degree of freedom rotary manipulator actuated by Shape Memory Alloy (SMA) wire. A model for SMA actuated manipulator is presented. The model includes nonlinear dynamics of the manipulator, a constitutive model of the Shape Memory Alloy, and the electrical and heat transfer behavior of SMA wire. The current experimental setup allows for the measurement of only one state variable which is the angular position of the arm. Due to measurement difficulties, the other three state variables, arm angular velocity and SMA wire stress and temperature, cannot be directly measured. A model-based state estimator that works with noisy measurements is presented based on the Extended Kalman Filter (EKF). This estimator predicts the state vector at each time step and corrects its prediction based on the angular position measurements. The estimator is then used in a nonlinear and robust control algorithm based on Variable Structure Control (VSC). The VSC algorithm is a control gain switching technique based on the arm angular position (and velocity) feedback and EKF estimated SMA wire stress and temperature. The state vector estimates help reduce or avoid the undesirable and inefficient overshoot problem in SMA one-way actuation control.

scholarly journals A Temperature-based Controller for a Shape Memory Alloy Actuator

2005 ◽  
Vol 127 (3) ◽  
pp. 285-291 ◽  
Author(s):  
Mohammad H. Elahinia ◽  
Hashem Ashrafiuon ◽  
Mehdi Ahmadian ◽  
Hanghao Tan
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory ◽  
State Vector ◽  
Angular Position ◽  
Variable Structure ◽  
The State ◽  
State Variables ◽  
Time Step ◽  
State Variable ◽  
Control Gain

This paper presents a robust nonlinear control that uses a state variable estimator for control of a single degree of freedom rotary manipulator actuated by shape memory alloy (SMA) wire. A model for SMA actuated manipulator is presented. The model includes nonlinear dynamics of the manipulator, a constitutive model of the shape memory alloy, and the electrical and heat transfer behavior of SMA wire. The current experimental setup allows for the measurement of only one state variable which is the angular position of the arm. Due to measurement difficulties, the other three state variables, arm angular velocity and SMA wire stress and temperature, cannot be directly measured. A model-based state estimator that works with noisy measurements is presented based on the extended Kalman filter (EKF). This estimator estimates the state vector at each time step and corrects its estimation based on the angular position measurements. The estimator is then used in a nonlinear and robust control algorithm based on variable structure control (VSC). The VSC algorithm is a control gain switching technique based on the arm angular position (and velocity) feedback and EKF estimated SMA wire stress and temperature. Using simulation it is shown that the state vector estimates help reduce or avoid the undesirable and inefficient overshoot problem in SMA one-way actuation control.

scholarly journals Integral estimates of the solutions of fractional-like equations of perturbed motion

2018 ◽  
Vol 24 (1) ◽  
pp. 138-149 ◽  
Author(s):  
Anatoliy Martynyuk ◽  
Gani Stamov ◽  
Ivanka Stamova
Keyword(s):  
Nonlinear Systems ◽  
State Vector ◽  
The State ◽  
Integral Inequalities ◽  
Scalar Equation ◽  
Derivatives Of ◽  
Rassias Stability

In this paper, the results of the analysis of nonlinear systems with fractional-like derivatives of the state vector are presented. Using the method of integral inequalities, some estimates of the solutions are obtained, and criteria for Heyers–Ulam–Rassias stability of a fractional-like scalar equation are established.

scholarly journals Extending high order derivatives for special differential equations of the form \(y' = f(y)\) by using monotonically labeled rooted trees.

2015 ◽  
Vol 4 (4) ◽  
pp. 513
Author(s):  
Hossein Hassani ◽  
Mohammad Shafie Dahaghin
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Initial Value Problems ◽  
High Order ◽  
Rooted Trees ◽  
Initial Value ◽  
Derivatives Of ◽  
Special Case

<p>This paper presents a review of the role played by labeled rooted trees to obtain derivatives for numerical solution of initial value problems in special case \(y' = f(y), y(x_0) = y_0\). We extend a process to find successive derivatives according to monotonically labeled rooted trees, and prove some relevant lemmas and theorems. In this regard, the  derivatives, of the monotonically labeled rooted trees with n vertices are presented by using the monotonically labeled rooted trees with k + n vertices. Eventually, this process is applied to trees without labeling.</p>

Space–time adaptive numerical methods for geophysical applications

2009 ◽  
Vol 367 (1907) ◽  
pp. 4613-4631 ◽  
Author(s):  
C. E. Castro ◽  
M. Käser ◽  
E. F. Toro
Keyword(s):  
Wave Propagation ◽  
Large Scale ◽  
Computational Cost ◽  
Taylor Series Expansion ◽  
Space Time ◽  
High Order ◽  
Optimal Time ◽  
Mathematical Framework ◽  
Time Step ◽  
Tsunami Waves

In this paper we present high-order formulations of the finite volume and discontinuous Galerkin finite-element methods for wave propagation problems with a space–time adaptation technique using unstructured meshes in order to reduce computational cost without reducing accuracy. Both methods can be derived in a similar mathematical framework and are identical in their first-order version. In their extension to higher order accuracy in space and time, both methods use spatial polynomials of higher degree inside each element, a high-order solution of the generalized Riemann problem and a high-order time integration method based on the Taylor series expansion. The static adaptation strategy uses locally refined high-resolution meshes in areas with low wave speeds to improve the approximation quality. Furthermore, the time step length is chosen locally adaptive such that the solution is evolved explicitly in time by an optimal time step determined by a local stability criterion. After validating the numerical approach, both schemes are applied to geophysical wave propagation problems such as tsunami waves and seismic waves comparing the new approach with the classical global time-stepping technique. The problem of mesh partitioning for large-scale applications on multi-processor architectures is discussed and a new mesh partition approach is proposed and tested to further reduce computational cost.

scholarly journals On the numerical solution of chaotic dynamical systems using extend precision floating point arithmetic and very high order numerical methods

2011 ◽  
Vol 16 (3) ◽  
pp. 340-352 ◽  
Author(s):  
Scott A. Sarra ◽  
Clyde Meador
Keyword(s):  
Dynamical Systems ◽  
Numerical Solution ◽  
Numerical Method ◽  
High Order ◽  
Floating Point ◽  
Time Step ◽  
Floating Point Arithmetic ◽  
Chaotic Dynamical Systems ◽  
Point Arithmetic ◽  
Very High

Multiple results in the literature exist that indicate that all computed solutions to chaotic dynamical systems are time-step dependent. That is, solutions with small but different time steps will decouple from each other after a certain (small) finite amount of simulation time. When using double precision floating point arithmetic time step independent solutions have been impossible to compute, no matter how accurate the numerical method. Taking the well-known Lorenz equations as an example, we examine the numerical solution of chaotic dynamical systems using very high order methods as well as extended precision floating point number systems. Time step independent solutions are obtained over a finite period of time. However even with a sixteenth order numerical method and with quad-double floating point numbers, there is a limit to this approach.

Operators and meaning of wave function

2016 ◽  
pp. 4039-4042
Author(s):  
Viliam Malcher
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
State Vector ◽  
The State ◽  
Physical Significance ◽  
Central Problem ◽  
Quantum Mechanical ◽  
Mechanical Description ◽  
Quantum Mechanical Description ◽  
Interpretation Of Quantum Theory

The interpretation problems of quantum theory are considered. In the formalism of quantum theory the possible states of a system are described by a state vector. The state vector, which will be represented as |ψ> in Dirac notation, is the most general form of the quantum mechanical description. The central problem of the interpretation of quantum theory is to explain the physical significance of the |ψ>. In this paper we have shown that one of the best way to make of interpretation of wave function is to take the wave function as an operator.

The analysis is geometrically stable orbits of spacecraft remote sensing of the Еarth

2018 ◽  
Vol 15 (1) ◽  
pp. 12-22
Author(s):  
V. M. Artyushenko ◽  
D. Y. Vinogradov
Keyword(s):  
Remote Sensing ◽  
Gravitational Field ◽  
State Vector ◽  
Semimajor Axis ◽  
The State ◽  
Zonal Harmonics ◽  
The Earth ◽  
Conditions Of Stability

The article reviewed and analyzed the class of geometrically stable orbits (GUO). The conditions of stability in the model of the geopotential, taking into account the zonal harmonics. The sequence of calculation of the state vector of GUO in the osculating value of the argument of the latitude with the famous Ascoli-royski longitude of the ascending node, inclination and semimajor axis. The simulation is obtained the altitude profiles of SEE regarding the all-earth ellipsoid model of the gravitational field of the Earth given 7 and 32 zonal harmonics.

scholarly journals Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1799
Author(s):  
Irene Gómez-Bueno ◽  
Manuel Jesús Castro Díaz ◽  
Carlos Parés ◽  
Giovanni Russo
Keyword(s):  
High Order ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Balance Laws ◽  
Source Terms ◽  
Time Step ◽  
Systems Of Balance Laws ◽  
Reconstruction Operator ◽  
Non Linear ◽  
Linear Problems

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

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