Stability Analysis of Complex Multibody Systems

2005 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Jielong Wang
Keyword(s):  
Stability Analysis ◽  
Floquet Theory ◽  
Numerical Models ◽  
Characteristic Exponent ◽  
Fitting Method ◽  
Prony's Method ◽  
Linearized Stability ◽  
The Stability ◽  
Curve Fitting Method ◽  
Prony’S Method

The linearized stability analysis of dynamical systems modeled using finite element based multibody formulations is addressed in this paper. The use of classical methods for stability analysis of these system, such as the characteristic exponent method or Floquet theory, results in computationally prohibitive costs. Since comprehensive multibody models are “virtual prototypes” of actual systems, the applicability to numerical models of the stability analysis tools that are used in experimental settings is investigated in this work. Various experimental tools for stability analysis are reviewed. It is proved that Prony’s method, generally regarded as a curve fitting method, is equivalent, and sometimes identical, to Floquet theory and to the partial Floquet method. This observation gives Prony’s method a sound theoretical, footing, and considerably improves the robustness of its predictions when applied to comprehensive models of complex multibody system. Numerical applications are presented to demonstrate the efficiency of the proposed procedure.

Stability Analysis of Complex Multibody Systems

2005 ◽  
Vol 1 (1) ◽  
pp. 71-80 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Jielong Wang
Keyword(s):  
Stability Analysis ◽  
Multibody Systems ◽  
Floquet Theory ◽  
Numerical Models ◽  
Characteristic Exponent ◽  
Prony's Method ◽  
Linearized Stability ◽  
The Stability ◽  
Curve Fitting Method ◽  
Prony’S Method

The linearized stability analysis of dynamical systems modeled using finite element-based multibody formulations is addressed in this paper. The use of classical methods for stability analysis of these systems, such as the characteristic exponent method or Floquet theory, results in computationally prohibitive costs. Since comprehensive multibody models are “virtual prototypes” of actual systems, the applicability to numerical models of the stability analysis tools that are used in experimental settings is investigated in this work. Various experimental tools for stability analysis are reviewed. It is proved that Prony’s method, generally regarded as a curve-fitting method, is equivalent, and sometimes identical, to Floquet theory and to the partial Floquet method. This observation gives Prony’s method a sound theoretical footing, and considerably improves the robustness of its predictions when applied to comprehensive models of complex multibody systems. Numerical and experimental applications are presented to demonstrate the efficiency of the proposed procedure.

Time Domain Approaches to the Stability Analysis of Flexible Dynamical Systems

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Jielong Wang ◽  
Xiaowen Shan ◽  
Bin Wu ◽  
Olivier A. Bauchau
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Time Domain ◽  
Floquet Theory ◽  
Characteristic Exponent ◽  
Periodic Coefficient ◽  
Theoretical Background ◽  
System Stability ◽  
Lyapunov Characteristic Exponent ◽  
The Stability

This paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition (POD) method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition (SVD). Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent (LCE) theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the LCE closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show that the proposed approaches are robust enough to deal with the stability analysis of flexible dynamical systems and the predictions of the two approaches are found to be in close agreement.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

A Nonlinear Stability Analysis for Difference Equations Using Semi-Implicit Mobility

1977 ◽  
Vol 17 (01) ◽  
pp. 79-91 ◽  
Author(s):  
D.W. Peaceman
Keyword(s):  
Porous Media ◽  
Stability Analysis ◽  
Finite Difference ◽  
Nonlinear Stability ◽  
Difference Equations ◽  
Nonlinear Stability Analysis ◽  
Time Step ◽  
Two Phase ◽  
Linearized Stability ◽  
The Stability

Abstract The usual linearized stability analysis of the finite-difference solution for two-phase flow in porous media is not delicate enough to distinguish porous media is not delicate enough to distinguish between the stability of equations using semi-implicit mobility and those using completely implicit mobility. A nonlinear stability analysis is developed and applied to finite-difference equations using an upstream mobility that is explicit, completely implicit, or semi-implicit. The nonlinear analysis yields a sufficient (though not necessary) condition for stability. The results for explicit and completely implicit mobilities agree with those obtained by the standard linearized analysis; in particular, use of completely implicit mobility particular, use of completely implicit mobility results in unconditional stability. For semi-implicit mobility, the analysis shows a mild restriction that generally will not be violated in practical reservoir simulations. Some numerical results that support the theoretical conclusions are presented. Introduction Early finite-difference, Multiphase reservoir simulators using explicit mobility were found to require exceedingly small time steps to solve certain types of problems, particularly coning and gas percolation. Both these problems are characterized percolation. Both these problems are characterized by regions of high flow velocity. Coats developed an ad hoc technique for dealing with gas percolation, but a more general and highly successful approach for dealing with high-velocity problems has been the use of implicit mobility. Blair and Weinaug developed a simulator using completely implicit mobility that greatly relaxed the time-step restriction. Their simulator involved iterative solution of nonlinear difference equations, which considerably increased the computational work per time step. Three more recent papers introduced the use of semi-implicit mobility, which proved to be greatly superior to the fully implicit method with respect to computational effort, ease of use, and maximum permissible time-step size. As a result, semi-implicit mobility has achieved wide use throughout the industry. However, this success has been pragmatic, with little or no theoretical work to justify its use. In this paper, we attempt to place the use of semi-implicit mobility on a sounder theoretical foundation by examining the stability of semi-implicit difference equations. The usual linearized stability analysis is not delicate enough to distinguish between the semi-implicit and completely implicit difference equation. A nonlinear stability analysis is developed that permits the detection of some differences between the stability of difference equations using implicit mobility and those using semi-implicit mobility. DIFFERENTIAL EQUATIONS The ideas to be developed may be adequately presented using the following simplified system: presented using the following simplified system: horizontal, one-dimensional, two-phase, incompressible flow in homogeneous porous media, with zero capillary pressure. A variable cross-section is included so that a variable flow velocity may be considered. The basic differential equations are (1) (2) The total volumetric flow rate is given by (3) Addition of Eqs. 1 and 2 yields =O SPEJ P. 79

Steinkamp's Toy Can Hop 100 Times But Can't Stand Up

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
Gregg Stiesberg ◽  
Tim van Oijen ◽  
Andy Ruina
Keyword(s):  
Stability Analysis ◽  
Experimental Observation ◽  
Limit Cycle ◽  
Periodic Motion ◽  
Largest Eigenvalue ◽  
Unstable Limit Cycle ◽  
Stable Attractor ◽  
Linearized Stability ◽  
The Stability ◽  
Basic Motion

We have experimented with and simulated Steinkamp's passive-dynamic hopper. This hopper cannot stand up (it is statically unstable), yet it can hop the length of a 5 m 0.079 rad sloped ramp, with n≈100 hops. Because, for an unstable periodic motion, a perturbation Δx0 grows exponentially with the number of steps (Δxn≈Δx0×λn), where λ is the system eigenvalue with largest magnitude, one expects that if λ>1 that the amplification after 100 steps, λ100, would be large enough to cause robot failure. So, the experiments seem to indicate that the largest eigenvalue magnitude of the linearized return map is less than one, and the hopper is dynamically stable. However, two independent simulations show more subtlety. Both simulations correctly predict the period of the basic motion, the kinematic details, and the existence of the experimentally observed period ∼11 solutions. However, both simulations also predict that the hopper is slightly unstable (|λ|max>1). This theoretically predicted instability superficially contradicts the experimental observation of 100 hops. Nor do the simulations suggest a stable attractor near the periodic motion. Instead, the conflict between the linearized stability analysis and the experiments seems to be resolved by the details of the launch: a simulation of the hand-holding during launch suggests that experienced launchers use the stability of the loosely held hopper to find a motion that is almost on the barely unstable limit cycle of the free device.

Stability Analysis of On-Board Friction Modifier Systems at the Wheel–Rail Interface

2015 ◽  
Vol 137 (5) ◽  
Author(s):  
C. P. Sharma ◽  
A. Srikantha Phani
Keyword(s):  
Stability Analysis ◽  
Frictional Contact ◽  
Strong Dependence ◽  
Friction Modifier ◽  
Design Engineers ◽  
Design Changes ◽  
Linearized Stability ◽  
Stability Maps ◽  
And Performance ◽  
The Stability

Friction control at the wheel–rail interface, using on-board solid stick friction modifier systems can lead to enhanced track life, reduced wear, and increased fuel economy in railroads. Frictional contact between the solid stick and the railway wheel itself can potentially cause vibrations within the modifier systems, influencing their stability and performance. A frequency domain linearized stability analysis of the state of steady sliding at the frictional contact between the solid stick and the wheel is performed. The proposed approach relies on individual frequency response functions (FRFs) of the wheel and the applicator–bracket subsystems of the on-board friction modifier. Stability characteristics of three representative bracket designs are qualitatively compared, using the FRFs generated by their respective finite element (FE) models. The FE models are validated by comparing the predicted natural frequencies with corresponding experimentally measured values on a full wheel test rig (FWTR) facility. The validated FE models are then used to compute stability maps which delineate stable and unstable regions of operation in the design parameter space, defined by train speed, angle of applicator, friction coefficient, and bracket design. Strong dependence of stability upon the bracket designs is observed. The methodology developed here can be used by design engineers to assess the effectiveness of design changes on the stability of the applicator–bracket assembly in a virtual environment—thus avoiding costly retrofitting and prototyping. Directions for further model refinement and testing are provided.

Stability Analysis of a Pilot Operated Counterbalance Valve for a Big Flow Rate

2014 ◽  
Author(s):  
Yeming Yao ◽  
Hua Zhou ◽  
Yinglong Chen ◽  
Huayong Yang
Keyword(s):  
Stability Analysis ◽  
Flow Rate ◽  
Numerical Optimization ◽  
Dynamic Performance ◽  
Optimization Method ◽  
System Stability ◽  
Normal Range ◽  
Alternative Approach ◽  
Linearized Stability ◽  
The Stability

Counterbalance valves are widely used in hydraulic deck machinery to balance the overrunning loads. However, as is well known, counterbalance circuit designed with poor choice of counterbalance valve tends to introduce instability to the system. This paper investigates the dynamic behavior of a pilot operated counterbalance valve which can operate at a flow rate about 2000L/min. A linearized stability analysis of such a hydraulic circuit which consists of a slip in cartridge, a pilot counterbalance valve and a hydraulic winch is presented. Pole-zero plots are employed to reveal the effect of the volume of control cavity, the hydraulic resistance on pilot line and counterbalance valve pilot area ratio on the stability of the system. The analysis results indicate that such a system will be unstable within the normal range of each parameter. An alternative approach that guarantees system stability by adding an accumulator on the pilot line is put forward. The approach stabilizes the pilot pressure by reducing the hydro-stiffness of pilot control cavity, thus the system can reach its stability condition. Finally, a numerical optimization method is putted forward, with the optimized parameters, the dynamic performance of considered system become better.

Efficient and Robust Approaches to the Stability Analysis of Large Multibody Systems

2007 ◽  
Vol 3 (1) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Jielong Wang
Keyword(s):  
Stability Analysis ◽  
Large Scale ◽  
Equations Of Motion ◽  
Classical Stability ◽  
Linearized Stability ◽  
Signal Synthesis ◽  
Common Framework ◽  
Synthesis Procedure ◽  
The Stability ◽  
Proper Orthogonal

Linearized stability analysis methodologies that are applicable to large scale, multiphysics problems are presented in this paper. Two classes of closely related algorithms based on a partial Floquet and on an autoregressive approach, respectively, are presented in common framework that underlines their similarity and their relationship to other methods. The robustness of the proposed approach is improved by using optimized signals that are derived from the proper orthogonal modes of the system. Finally, a signal synthesis procedure based on the identified frequencies and damping rates is shown to be an important tool for assessing the accuracy of the identified parameters; furthermore, it provides a means of resolving the frequency indeterminacy associated with the eigenvalues of the transition matrix for periodic systems. The proposed approaches are computationally inexpensive and consist of purely post processing steps that can be used with any multiphysics computational tool or with experimental data. Unlike classical stability analysis methodologies, it does not require the linearization of the equations of motion of the system.

Stability Analysis of the Earth Embankments Subjected to Natural Cyclic Processes

2017 ◽  
Vol 21 ◽  
pp. 389-396 ◽  
Author(s):  
Smita Tung ◽  
Kaustuv Bhattacharya ◽  
Gupinath Bhandari ◽  
Sibapriya Mukherjee ◽  
Ancuţa Rotaru ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Numerical Models ◽  
High Tide ◽  
Phreatic Surface ◽  
Deltaic Plain ◽  
Freeze Thaw ◽  
The Earth ◽  
Cyclic Processes ◽  
The Stability ◽  
Permeable Soil

The paper brings out the stability analysis of the earth embankments subjected to specific natural cyclic processes such as tides in India and freeze-thaw in Romania. The Sundarban, along the Bay of Bengal, is a low tide-dominated deltaic plain of the Ganga-Brahmaputra-Meghna basin (GBM) spreading over India and Bangladesh around 25,500 sq.km. The 3520km of riverside embankments had been erected to protect the deltaic plain from tidal flooding. The tidal cycle in this region is twice high tide and low tide daily. Presently, the Indian Sundarban Delta is under the threat to embankment failure. The study analyses the stability of an 8 m height embankment in the Gosaba region of Sundarban considering a 2m wide earth filled berm at 6m from the bottom of varying clay core thickness under the transient groundwater flow. Pore pressure, Flow net and Phreatic surface have been obtained by numerical models using FLAC 2D software, thus the factor of safety being analysed. In Romania, built on the of the highway A1 Orăştie-Sibiu section crossing the clayey hills from Aciliu and Apoldu known for their instability, the embankment around the abutment pier from Apoldu broke up at the entry to the Aciliu Viaduct. The layers of permeable soil soaked with water become heavy instead the waterproof layers keep the water above them creating sliding surfaces. The water is channelled to dozens of springs that appear or disappear depending on each micro-slip or fallen water amount. By freezing the water expands on the ground determining additional efforts, the cycle iterating with each rain and freeze-thaw.

Assessment of the Prony's method for BWR stability analysis

2011 ◽  
Vol 241 (5) ◽  
pp. 1469-1477 ◽  
Author(s):  
Javier Ortiz-Villafuerte ◽  
Rogelio Castillo-Durán ◽  
Javier C. Palacios-Hernández
Keyword(s):  
Stability Analysis ◽  
Prony's Method ◽  
Prony’S Method
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