Modeling and Simulation of Stochastic Lorenz Systems by Polynomial Chaos Approach

2007 ◽  
Author(s):  
Lin Li ◽  
Corina Sandu
Keyword(s):  
Polynomial Chaos ◽  
Initial Conditions ◽  
Nonlinear Problems ◽  
System Response ◽  
Structural Vibrations ◽  
System Parameters ◽  
Highly Nonlinear ◽  
Body Dynamics ◽  
The Lorenz System ◽  
Multi Body

The Lorenz problem is one of the paradigms of the chaotic systems, which are sensitive to initial conditions and for which the performance is hard to predict. However, in many cases and dynamic systems, the initial conditions of a dynamic system and the system parameters can’t be measured accurately, and the response of the system must indeed be explored in advance. In this study, the polynomial chaos approach is used to handle uncertain initial conditions and system parameters of the Lorenz system. The method has been successfully applied by the authors and co-workers in multi-body dynamics and terrain profile and soil modeling. Other published studies illustrate the benefits of using the polynomial chaos, especially for problems involving large uncertainties and highly nonlinear problems in fluid mechanics, structural vibrations, and air quality studies. This study is an attempt to use the polynomial chaos approach to treat the Lorenz problem, and the results are compared with a classical Monte Carlo approach. Error bars are used to illustrate the standard deviation of the system response. Different meshing schemes are simulated, and the convergence of the method is analyzed.

Comparative Study of Evolutionary Algorithms for Parameter Identification of an Impact Oscillator

2014 ◽  
Author(s):  
Amit Banerjee ◽  
Issam Abu Mahfouz
Keyword(s):  
Evolutionary Algorithms ◽  
Differential Evolution ◽  
Parameter Identification ◽  
Optimization Techniques ◽  
System Response ◽  
Impact Oscillator ◽  
Swarm Optimization ◽  
System Parameters ◽  
Highly Nonlinear ◽  
The Impact

The use of non-classical evolutionary optimization techniques such as genetic algorithms, differential evolution, swarm optimization and genetic programming to solve the inverse problem of parameter identification of dynamical systems leading to chaotic states has been gaining popularity in recent years. In this paper, three popular evolutionary algorithms — differential evolution, particle swarm optimization and the firefly algorithm are used for parameter identification of a clearance-coupled-impact oscillator system. The behavior of impacting systems is highly nonlinear exhibiting a myriad of harmonic, low order and high order sub-harmonic resonances, as well as chaotic vibrations. The time-history simulations of the single-degree-of-freedom impact oscillator were obtained by the Neumark-β numerical integration algorithm. The results are illustrated by bifurcation graphs, state space portraits and Poincare’ maps which gives valuable insights on the dynamics of the impact system. The parameter identification problem relates to finding one set of system parameters given a chaotic or periodic system response as a set of Poincaré points and a different but known set of system parameters. The three evolutionary algorithms are compared over a set of parameter identification problems. The algorithms are compared based on solution quality to evaluate the efficacy of using one algorithm over another.

Robust multi-steps input command for liquid sloshing control

2021 ◽  
pp. 107754632110177
Author(s):  
Abdullah Alshaya ◽  
Adel Alshayji
Keyword(s):  
Liquid Sloshing ◽  
Flexible Body ◽  
Transient Vibration ◽  
Resonant Frequencies ◽  
System Parameters ◽  
Input Shaper ◽  
Body Dynamics ◽  
Time Optimal ◽  
Multi Body ◽  
Input Command

A robust input command based on multiple steps for eliminating the residual vibrations of a multimode linear system is proposed. Only the system resonant frequencies are needed to determine the step magnitudes in the shaped command. The command duration is selectable to help in designing an optimum command that compensates between the reduction in the transient vibration, the enhancement in the command robustness, and the increase in the total maneuver time. The induced transient and residual sloshing oscillations of a suspended water-filled container are suppressed using the proposed command. The dynamics of the sloshing is numerically simulated using finite element method that accommodates the interactions between the fluid, structural, and multi-body dynamics. A short move time penalty is incurred with the price of significant reduction in the liquid sloshing. The performance of the shaped command to the system parameters and the robustness to their uncertainty are investigated. An improved robust input command in the presence of uncertainties in the cable length and water depth is also introduced. The effectiveness and excellence of the proposed command is demonstrated through a comparison with multimode zero-vibration input shaper and time-optimal flexible-body control.

Explicit Poincaré equations of motion for general constrained systems. Part II. Applications to multi-body dynamics and nonlinear control

2007 ◽  
Vol 463 (2082) ◽  
pp. 1435-1446 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Control ◽  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Constrained Systems ◽  
The Third ◽  
Highly Nonlinear ◽  
Body Dynamics ◽  
Multi Body

The power of the new equations of motion developed in part I of this paper is illustrated using three examples from multi-body dynamics. The first two examples deal with the problem of accurately controlling the orientation of a rigid body, while the third example deals with the synchronization of two rigid bodies so that their relative orientations are ‘locked’ through prescribed dynamical relationships. The ease, simplicity and accuracy with which control of such highly nonlinear systems is achieved are demonstrated.

The Analysis of Mechanical-Hydraulic Governor of Diesel Based on the Multi-Body Dynamics with Rigid-Flexible Integration

2013 ◽  
Vol 823 ◽  
pp. 39-42
Author(s):  
Wei Dong Zhang ◽  
Hong Dong Wang ◽  
Zhen Ming Liu ◽  
Guang Yao Ouyang
Keyword(s):  
Hydraulic System ◽  
Initial Conditions ◽  
Working Process ◽  
Computational Dynamics ◽  
Analysis Process ◽  
Governing System ◽  
Body Dynamics ◽  
Flexible Integration ◽  
Set Up ◽  
Multi Body

The multi-body dynamics model with rigid-flexible integration of the mechanicalhydraulic governor of diesel was set up based on the theory of computational dynamics of multi-body system. The initial conditions and boundary conditions were analyzed and added correctly to the components of the model. Then the working process of the mechanicalhydraulic governor was simulated, and the dynamic characteristics were analyzed. The results show that the model and the solving process are correct and credible. These pave the way for the further study about structure and capability optimum of the speed governing system. The whole analysis process provides a simple and feasible approach which can be used in other mechanical and hydraulic system.

Uncertainty Propagation of System Parameters to the Dynamic Response: An Application to a Benchtop Pendulum

2017 ◽  
Author(s):  
David Petrushenko ◽  
Firas A. Khasawneh
Keyword(s):  
Dynamical Systems ◽  
Error Bounds ◽  
Random Quantity ◽  
Initial Conditions ◽  
Uncertainty Propagation ◽  
System Response ◽  
Pendulum System ◽  
System Parameters ◽  
Sources Of Uncertainty ◽  
Different Sources

Dynamical systems often involve uncertainties either in their parameters or in their initial conditions. Therefore, in order to reliably compute the system response, it is important to understand and quantify the different sources of uncertainty. This necessitates carefully measuring the system parameters and propagating the associated uncertainty to the system response. In this paper we demonstrate an approach for propagating uncertainties from the system parameters to the output response using a benchtop pendulum system. Since the input parameters are treated as random variables, the output is also reported as a random quantity with appropriate error bounds.

A Study on the Characteristics of a Nonlinear Oscillatory System With Dry Friction

2003 ◽  
Author(s):  
Liming Dai ◽  
Liang Xu
Keyword(s):  
Nonlinear System ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Approximate Solutions ◽  
Weakly Nonlinear ◽  
System Parameters ◽  
Weakly Nonlinear System ◽  
Highly Nonlinear ◽  
Nonlinear Oscillatory System

Nonlinear oscillatory system involved with friction is very common in nonlinear dynamics of engineering fields. This paper is to investigate the motions a nonlinear oscillatory system with involvement of dry friction. The cases of weakly and highly nonlinearity of the system are considered. Approximate and numerical solutions for the system are developed via the author’s newly developed P-T method. As demonstrated in the present work, the properties of the weakly and highly nonlinear systems exhibit great differences, though the governing equations of the two systems employ identical system parameters. The approximate solutions developed for the system are continuous everywhere on the time range considered. Under the conditions of weakly nonlinearity, the approximate solutions developed can therefore be conveniently implemented for the purpose of an analytical studying the properties of the system with numerous system parameters and various initial conditions. Taking this advantage, the behavior of motion of the weakly nonlinear system is analyzed and compared with the corresponding solutions developed with Van der Pol’s method. It is found in the present work, the system may undergo a self-excited oscillation under certain conditions. The highly nonlinear system is a physically much involved one. Its behavior is thus much complex in comparing with that of the weakly nonlinear system. Based on the approximate solutions developed for the highly nonlinear system, recurrence relations are generated for numerical calculations. For the sake of comparison with the oscillation of the weakly nonlinear system, numerical simulations for the highly nonlinear system are performed under the same initial conditions and identical system parameters. The conditions of convergence and divergence of the weakly nonlinear system are also established for application. Behavior of the oscillatory motion of the highly nonlinear system is investigated on the basis of the corresponding numerical solutions developed.

A Comprehensive Investigation of Ensembles of Gaussian-Based and Gradient-Based Transformed Reliability Analyses: When and How to Use Them

2016 ◽  
Author(s):  
Po Ting Lin ◽  
Wei-Hao Lu ◽  
Shu-Ping Lin
Keyword(s):  
Design Optimization ◽  
Design Space ◽  
Optimization Problems ◽  
Nonlinear Problems ◽  
Kernel Functions ◽  
Sensitivity Analyses ◽  
Normal Space ◽  
Highly Nonlinear ◽  
Standard Normal ◽  
Gradient Based

In the past few years, researchers have begun to investigate the existence of arbitrary uncertainties in the design optimization problems. Most traditional reliability-based design optimization (RBDO) methods transform the design space to the standard normal space for reliability analysis but may not work well when the random variables are arbitrarily distributed. It is because that the transformation to the standard normal space cannot be determined or the distribution type is unknown. The methods of Ensemble of Gaussian-based Reliability Analyses (EoGRA) and Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) have been developed to estimate the joint probability density function using the ensemble of kernel functions. EoGRA performs a series of Gaussian-based kernel reliability analyses and merged them together to compute the reliability of the design point. EGTRA transforms the design space to the single-variate design space toward the constraint gradient, where the kernel reliability analyses become much less costly. In this paper, a series of comprehensive investigations were performed to study the similarities and differences between EoGRA and EGTRA. The results showed that EGTRA performs accurate and effective reliability analyses for both linear and nonlinear problems. When the constraints are highly nonlinear, EGTRA may have little problem but still can be effective in terms of starting from deterministic optimal points. On the other hands, the sensitivity analyses of EoGRA may be ineffective when the random distribution is completely inside the feasible space or infeasible space. However, EoGRA can find acceptable design points when starting from deterministic optimal points. Moreover, EoGRA is capable of delivering estimated failure probability of each constraint during the optimization processes, which may be convenient for some applications.

scholarly journals Estimating maximum bite performance in Tyrannosaurus rex using multi-body dynamics

2012 ◽  
Vol 8 (4) ◽  
pp. 660-664 ◽  
Author(s):  
K. T. Bates ◽  
P. L. Falkingham
Keyword(s):  
Body Size ◽  
Feeding Behaviour ◽  
Ontogenetic Change ◽  
Positive Allometry ◽  
Musculoskeletal Models ◽  
Terrestrial Animal ◽  
Body Dynamics ◽  
Mode Of Life ◽  
Scaling Analyses ◽  
Multi Body

Bite mechanics and feeding behaviour in Tyrannosaurus rex are controversial. Some contend that a modest bite mechanically limited T. rex to scavenging, while others argue that high bite forces facilitated a predatory mode of life. We use dynamic musculoskeletal models to simulate maximal biting in T. rex . Models predict that adult T. rex generated sustained bite forces of 35 000–57 000 N at a single posterior tooth, by far the highest bite forces estimated for any terrestrial animal. Scaling analyses suggest that adult T. rex had a strong bite for its body size, and that bite performance increased allometrically during ontogeny. Positive allometry in bite performance during growth may have facilitated an ontogenetic change in feeding behaviour in T. rex , associated with an expansion of prey range in adults to include the largest contemporaneous animals.

Sir Robert Stawell Ball and methodologies of modern screw theory

2002 ◽  
Vol 216 (1) ◽  
pp. 1-11 ◽  
Author(s):  
H Lipkin ◽  
J Duffy
Keyword(s):  
Computational Geometry ◽  
Rigid Body ◽  
Lie Algebra ◽  
Screw Theory ◽  
Mechanical Design ◽  
Ball Screw ◽  
Dual Numbers ◽  
Body Dynamics ◽  
Rigid Body Mechanics ◽  
Multi Body

The theory of screws was largely developed by Sir Robert Stawell Ball over 100 years ago to investigate general problems in rigid body mechanics. Nowadays, screw theory is applied in many different but related forms including dual numbers, Plilcker coordinates and Lie algebra. An overview of these methodologies is presented along with a perspective on Ball. Screw theory has re-emerged after a hiatus to become an important tool in robot mechanics, mechanical design, computational geometry and multi-body dynamics.

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