scholarly journals Analytical Solution of a Nonlinear Index-Three DAEs System Modelling a Slider-Crank Mechanism

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Brahim Benhammouda ◽  
Hector Vazquez-Leal
Keyword(s):  
Analytical Solution ◽  
Series Solution ◽  
Perturbation Parameter ◽  
Differential Algebraic Equations ◽  
Modern Technology ◽  
Power Series Solution ◽  
System Modelling ◽  
Algebraic Equations ◽  
Transform Method ◽  
Crank Mechanism

The slider-crank mechanism (SCM) is one of the most important mechanisms in modern technology. It appears in most combustion engines including those of automobiles, trucks, and other small engines. The SCM model considered here is an index-three nonlinear system of differential-algebraic equations (DAEs), and therefore difficult to integrate numerically. In this work, we present the application of the differential transform method (DTM) to obtain an approximate analytical solution of the SCM model in convergent series form. In addition, we propose a posttreatment of the power series solution with the Padé resummation method to extend the domain of convergence of the approximate series solution. The main advantage of the proposed technique is that it does not require an index reduction and does not generate secular terms or depend on a perturbation parameter.

scholarly journals Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Brahim Benhammouda ◽  
Hector Vazquez-Leal ◽  
Arturo Sarmiento-Reyes
Keyword(s):  
Perturbation Parameter ◽  
Convergent Series ◽  
Differential Algebraic Equations ◽  
Differential Transform Method ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Partial Differential Algebraic Equations ◽  
Differential Transform ◽  
Reduced Differential Transform Method

This work presents the application of the reduced differential transform method (RDTM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-two and index-three are solved to show that RDTM can provide analytical solutions for PDAEs in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful technique to find exact solutions. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend on a perturbation parameter.

An analytical solution of the Gibbs–Dehem equation in multicomponent systems

1970 ◽  
Vol 48 (5) ◽  
pp. 752-763 ◽  
Author(s):  
A. D. Pelton
Keyword(s):  
Analytical Solution ◽  
Power Series ◽  
Series Solution ◽  
Power Series Solution ◽  
Multicomponent Systems ◽  
Duhem Equation ◽  
Number Of Components ◽  
Analytical Power

A general analytical power-series solution of the Gibbs–Duhem equation in multicomponent systems of any number of components has been developed. The simplicity and usefulness of the solution is made possible through the choice of a special set of composition variables.

Automating the Derivation of the Equations of Motion of a Multibody Dynamic System With Uncertainty Using Polynomial Chaos Theory and Variational Work

2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Paul S. Ryan ◽  
Sarah C. Baxter ◽  
Philip A. Voglewede
Keyword(s):  
Chaos Theory ◽  
Polynomial Chaos ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Multibody Dynamic ◽  
Mass Spring ◽  
Polynomial Chaos Theory ◽  
Multibody Dynamic System ◽  
Crank Mechanism

Abstract Understanding how variation impacts a multibody dynamic (MBD) system's response is important to ensure the robustness of a system. However, how the variation propagates into the MBD system is complicated because MBD systems are typically governed by a system of large differential algebraic equations. This paper presents a novel process, variational work, along with the polynomial chaos multibody dynamics (PCMBoD) automation process for utilizing polynomial chaos theory (PCT) in the analysis of uncertainties in an MBD system. Variational work allows the complexity of the traditional PCT approach to be reduced. With variational work and the constrained Lagrangian formulation, the equations of motion of an MBD PCT system can be constructed using the PCMBoD automated process. To demonstrate the PCMBoD process, two examples, a mass-spring-damper and a two link slider–crank mechanism, are shown.

Variational Analysis of a Two Link Slider-Crank Mechanism Using Polynomial Chaos Theory

2017 ◽  
Author(s):  
Paul S. Ryan ◽  
Sarah C. Baxter ◽  
Philip A. Voglewede
Keyword(s):  
Chaos Theory ◽  
Closed Loop ◽  
Polynomial Chaos ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Advantages And Disadvantages ◽  
Multi Body ◽  
Polynomial Chaos Theory ◽  
Body Dynamic ◽  
Crank Mechanism

Variation occurs in many closed loop multi-body dynamic (MBD) systems in the geometry, mass, or forces. Understanding how MBD systems respond to variation is imperative for the design of a robust system. However, simulation of how variation propagates into the solution is complicated as most MBD systems cannot be simplified into to a system of ordinary differential equations (ODE). This paper investigates polynomial chaos theory (PCT) as a means of quantifying the effects of uncertainty in a closed loop MBD system governed by differential algebraic equations (DAE). To demonstrate how PCT could be used, the motion of a two link slider-crank mechanism is simulated with variation in the link lengths. To validate and show the advantages and disadvantages of PCT in closed loop MBD systems, the PCT approach is compared to Monte Carlo simulations.

scholarly journals Stability and Bifurcation Analysis of a Singular Delayed Predator-Prey Bioeconomic Model with Stochastic Fluctuations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yue Zhang ◽  
Qingling Zhang
Keyword(s):  
Deterministic Model ◽  
Gaussian White Noise ◽  
Differential Algebraic Equations ◽  
Bioeconomic Model ◽  
Algebraic Equations ◽  
Stability Switches ◽  
Transform Method ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Stochastic Fluctuations

This study investigates a singular delayed predator-prey bioeconomic model with stochastic fluctuations, which is described by differential-algebraic equations because of economic factors. The interior equilibrium of the singular delayed predator-prey bioeconomic model switches from being stable to unstable and then back to being stable, with the increase in time delay. The critical values for stability switches and Hopf bifurcations can be analytically determined. Subsequently, the effect of a fluctuating environment on the singular stochastic delayed predator-prey bioeconomic model obtained by introducing Gaussian white noise terms to the aforementioned deterministic model system is discussed. The fluctuation intensity of the population and harvest effort are calculated by Fourier transform method. Numerical simulation results are presented to verify the effectiveness of the conclusions.

Control of Closed Kinematic Chains Using A Singularly Perturbed Dynamics Model

2005 ◽  
Vol 128 (1) ◽  
pp. 142-151 ◽  
Author(s):  
Zhiyong Wang ◽  
Fathi H. Ghorbel
Keyword(s):  
Kinematic Chain ◽  
Perturbation Parameter ◽  
Parallel Robot ◽  
Differential Algebraic Equations ◽  
Singularly Perturbed ◽  
Algebraic Equations ◽  
Kinematic Chains ◽  
Novel Approach ◽  
Lyapunov Function Method ◽  
Perturbed Model

In this paper, we propose a novel approach to the control of closed kinematic chains (CKCs). This method is based on a recently developed singularly perturbed model for CKCs. Conventionally, the dynamics of CKCs are described by differential-algebraic equations (DAEs). Our approach transfers the control of the original DAE system to the control of an artificially created singularly perturbed system in which the slow dynamics corresponds to the original DAE when the perturbation parameter tends to zero. Compared to control schemes that rely on solving nonlinear algebraic constraint equations, the proposed method uses an ordinary differential equation (ODE) solver to obtain the dependent coordinates, hence, eliminates the need for Newton-type iterations and is amenable to real-time implementation. The composite Lyapunov function method is used to show that the closed-loop system, when controlled by typical open kinematic chain schemes, achieves asymptotic trajectory tracking. Simulations and experimental results on a parallel robot, the Rice planar Delta robot, are also presented to illustrate the efficacy of our method.

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