Preliminary Diagnostics of Dynamic Systems From Time Series

2014 ◽  
Author(s):  
C. A. Kitio Kwuimy ◽  
M. Samadani ◽  
C. Nataraj
Keyword(s):  
Time Series ◽  
State Space ◽  
Dynamic Systems ◽  
Short Description ◽  
Structural Defects ◽  
Reconstruction Technique ◽  
Coupling Mechanism ◽  
Single Degree Of Freedom ◽  
State Space Reconstruction ◽  
Single Degree

The state space reconstruction technique was recognized by Edward N. Lorenz as “one of the most surprising developments in nonlinear dynamics” [1]. Nowadays, the technique is applied in various scientific areas for prediction, analysis and diagnostics. This paper aims to discuss the possibility of using the embedding dimension of a reconstructed state space of time series as a tool for preliminary diagnostics. After a short description and illustration of the method, the paper considers two case studies: a single degree of freedom (DOF) and a 2 DOF system. The results of the analysis help detect a class of structural defects, including defects connected to a coupling mechanism. There is clearly a huge potential of such an approach for diagnostics of complex machinery.

State Space Reconstruction of Nonstationary Time-Series

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Hong-Guang Ma ◽  
Chun-Liang Zhang ◽  
Fu Li
Keyword(s):  
Nonlinear Dynamics ◽  
Time Series ◽  
State Space ◽  
Phase Angle ◽  
Analytical Form ◽  
Nonstationary Time Series ◽  
Original Time Series ◽  
State Space Reconstruction ◽  
Original Time ◽  
The Hilbert Transform

In this paper, a new method of state space reconstruction is proposed for the nonstationary time-series. The nonstationary time-series is first converted into its analytical form via the Hilbert transform, which retains both the nonstationarity and the nonlinear dynamics of the original time-series. The instantaneous phase angle θ is then extracted from the time-series. The first- and second-order derivatives θ˙, θ¨ of phase angle θ are calculated. It is mathematically proved that the vector field [θ,θ˙,θ¨] is the state space of the original time-series. The proposed method does not rely on the stationarity of the time-series, and it is available for both the stationary and nonstationary time-series. The simulation tests have been conducted on the stationary and nonstationary chaotic time-series, and a powerful tool, i.e., the scale-dependent Lyapunov exponent (SDLE), is introduced for the identification of nonstationarity and chaotic motion embedded in the time-series. The effectiveness of the proposed method is validated.

scholarly journals On the Investigation of State Space Reconstruction of Nonlinear Aeroelastic Response Time Series

2006 ◽  
Vol 13 (4-5) ◽  
pp. 393-407 ◽  
Author(s):  
Flávio D. Marques ◽  
Eduardo M. Belo ◽  
Vilma A. Oliveira ◽  
José R. Rosolen ◽  
Andréia R. Simoni
Keyword(s):  
Experimental Data ◽  
Time Series ◽  
Response Time ◽  
State Space ◽  
The State ◽  
Aeroelastic Response ◽  
State Space Reconstruction ◽  
Wing Model ◽  
Linear Behavior ◽  
Non Linear

Stall-induced aeroelastic motion may present severe non-linear behavior. Mathematical models for predicting such phenomena are still not available for practical applications and they are not enough reliable to capture physical effects. Experimental data can provide suitable information to help the understanding of typical non-linear aeroelastic phenomena. Dynamic systems techniques based on time series analysis can be adequately applied to non-linear aeroelasticity. When experimental data are available, the methods of state space reconstruction have been widely considered. This paper presents the state space reconstruction approach for the characterization of the stall-induced aeroelastic non-linear behavior. A wind tunnel scaled wing model has been tested. The wing model is subjected to different airspeeds and dynamic incidence angle variations. The method of delays is used to identify an embedded attractor in the state space from experimentally acquired aeroelastic response time series. To obtain an estimate of the time delay used in the state space reconstruction from time series, the autocorrelation function analyis is used. For the calculation of the embedding dimension the correlation integral approach is considered. The reconstructed attractors can reveal typical non-linear structures associated, for instance, to chaos or limit cycles.

Parametric Identification of Nonlinear Systems Using Chaotic Excitation

2007 ◽  
Vol 2 (3) ◽  
pp. 225-231 ◽  
Author(s):  
M. D. Narayanan ◽  
S. Narayanan ◽  
Chandramouli Padmanabhan
Keyword(s):  
Time Series ◽  
Nonlinear Systems ◽  
Periodic Orbits ◽  
Parametric Identification ◽  
System Response ◽  
Unstable Periodic Orbits ◽  
Single Degree Of Freedom ◽  
System Parameters ◽  
Single Degree ◽  
Quadratic Damping

The use of a time series, which is the chaotic response of a nonlinear system, as an excitation for the parametric identification of single-degree-of-freedom nonlinear systems is explored in this paper. It is assumed that the system response consists of several unstable periodic orbits, similar to the input, and hence a Fourier series based technique is used to extract these nearly periodic orbits. Criteria to extract these orbits are developed and a least-squares problem for the identification of system parameters is formulated and solved. The effectiveness of this method is illustrated on a system with quadratic damping and a system with Duffing nonlinearity.

scholarly journals Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Anna Krakovská ◽  
Kristína Mezeiová ◽  
Hana Budáčová
Keyword(s):  
Time Series ◽  
State Space ◽  
Time Window ◽  
Nearest Neighbour ◽  
Embedding Dimension ◽  
State Space Reconstruction ◽  
Dimensional Phase Space ◽  
Nearest Neighbours ◽  
Takens Theorem ◽  
The One

If data are generated by a system with a d-dimensional attractor, then Takens’ theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in 2d+1-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems.

Nonuniform State Space Reconstruction for Multivariate Chaotic Time Series

2019 ◽  
Vol 49 (5) ◽  
pp. 1885-1895 ◽  
Author(s):  
Min Han ◽  
Weijie Ren ◽  
Meiling Xu ◽  
Tie Qiu
Keyword(s):  
Time Series ◽  
State Space ◽  
Chaotic Time Series ◽  
State Space Reconstruction

Computational Approaches to the Min-Max Response of Dynamic Systems With Incompletely Prescribed Input Functions

1967 ◽  
Vol 34 (1) ◽  
pp. 87-90 ◽  
Author(s):  
Eugene Sevin ◽  
Walter Pilkey
Keyword(s):  
Dynamic Systems ◽  
Initial Conditions ◽  
Restoring Force ◽  
Maximum Displacement ◽  
Single Degree Of Freedom ◽  
Load Duration ◽  
Analytical Technique ◽  
Single Degree ◽  
Forcing Function ◽  
Total Impulse

Linear, nonlinear, and dynamic programming formulations are developed for the solution of the min-max response of a single-degree-of-freedom dynamic system with incompletely prescribed input functions. The problem is: Given an oscillator whose equation of motion is mx¨ + g(x, x˙) = f(t), subject to stated initial conditions, and acted upon by a forcing function, f(t), which is nonnegative, and of specified finite duration and total impulse, find the particular forces which produce the least possible maximum displacement of the oscillator, and find this bounding value. Previously, Sevin developed an analytical technique for the solution which is inherently dependent upon a linear undamped form for the restoring force g(x, x˙). In the current work, an alternate statement of the problem is presented which lends itself to tractable computational formulations involving less stringent restrictions on g(x, x˙). Results obtained by dynamic and linear programming for specified forms of g(x, x˙) are given as functions of load duration.

scholarly journals Complex-Damped Dynamic Systems in the Time and Frequency Domains

2004 ◽  
Vol 11 (3-4) ◽  
pp. 209-225 ◽  
Author(s):  
Elvio Bonisoli ◽  
John E. Mottershead
Keyword(s):  
Frequency Response ◽  
Structural Dynamics ◽  
Dynamic Systems ◽  
Dynamic Behaviour ◽  
Root Locus ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Frequency Domains ◽  
Spring System ◽  
Mass Spring

The fact that a complex-damped model may represent the dynamic behaviour of elasto-mechanical systems when acted upon by a magnetic field was brought to the attention of the structural dynamics community very recently by Professor Bruno A. D. Piombo and his colleagues at the Politecnico di Torino. In this paper a thorough analysis of the single degree-of-freedom complex-damped mass-spring system is presented. The analysis includes the root locus, the (non-causal) impulse response, the frequency response and the transmissibility. Regions of different behaviour in the frequency response and transmissibility are described in detail. The stiffening behaviour observed in Prof. Piombo's experiments and known as the "phantom effect" is demonstrated by the complex-damped model.

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