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System identification of a linearized hysteretic system using covariance driven stochastic subspace identification

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Anela Bajric at University of Oxford
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Abstract

A single mass Bouc-Wen oscillator with linear static restoring force contribution is approximated by an equivalent linear system. The aim of the linearized model is to emulate the correct force-displacement response of the Bouc-Wen model with characteristic hysteretic behaviour. The linearized model has been evaluated by the root-mean-square error between simulated response and the response history from two sets of experimental test data. Hysteretic behaviour is encountered in engineering structures exposed to severe cyclic environmental loads, as well as in vibration mitigation systems such as magneto-rheological dampers.
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Workshop on Nonlinear System Identification Benchmarks, Brussels, Belgium, April 25-27, 2016
System identification of a linearized hysteretic system using covariance
driven stochastic subspace identification
Anela Bajri´c
Department of Mechanical Engineering
Technical University of Denmark
Kongens Lyngby, Denmark
A single mass Bouc-Wen oscillator with linear static restoring force contribution is approximated by an equivalent
linear system. The aim of the linearized model is to emulate the correct force-displacement response of the Bouc-Wen
model with characteristic hysteretic behaviour. The linearized model has been evaluated by the root-mean-square error
between simulated response and the response history from two sets of experimental test data.
Hysteretic behaviour is encountered in engineering structures exposed to severe cyclic environmental loads, as well as
in vibration mitigation systems such as magneto-rheological dampers. The mathematical representation of a hysteretic
force can be obtained using the non-linear first-order differential equation
˙z(y, ˙y) = α˙y−βγ|˙y|z|z|ν−1+δ˙y|z|ν(1)
known as the Bouc-Wen model [1, 2]. The shape and smoothness of the hysteresis loop is controlled by model parameters:
α,β,γ,δand ν.
The non-linear hysteretic force governed by the Bouc-Wen model in (1) is approximated by a linear model of the form
z(y, ˙y) = λ y(t) + κ˙y(t) (2)
The individual coefficients in (2) are determined by assuming sinusoidal motion constant over one period with amplitude
A, applying harmonic averaging and integration over the full vibration period with angular frequency ω,
λ=9π2
32 A2µ+ 9 π2, κ =12 A µ π
ω(32 A2µ2+ 9 π2)where =α−β δ , µ =β γ (3)
These approximations are exact for vanishing non-linearity in z, associated with the power coefficient νapproaching
unity.
The input-output test data has been provided in [3], wherein the individual data sets are described. The random
phase multi-sine data set contains the steady-state response, while the sine-sweep data set is excited such that the
response is not in steady state. The natural frequency and damping ratio of the test systems have been estimated from
the displacement time history, using a covariance driven stochastic subspace identification algorithm (COV-SSI) [4],
implemented in MATLAB. The number of block-rows in the Toeplizt matrix and the model order have been selected
based on a minimization of the squared residual between the estimated natural frequency and damping ratio, and the
corresponding equivalent parameters obtained from the decay of the unbiased covariance estimate.
The response has been simulated using a fixed time-step fourth-order Runge-Kutta time integration scheme, with the
sample interval of 1/20 of the period and with zero initial conditions. The root-mean-square errors erms , relative to
the model and the test output time series are 7.9·10−5m and 1.1·10−4m for the multi-sine and sweep-sine data sets,
respectively. The error measure erms has been scaled by the number of points in the time series, in this case 8192 and
153000 for the multi- and sweep-sine data sets, respectively.
The present approach is suitable for simulations of steady state response of a single-degree-of-freedom system, where
the main advantage lies in its simplicity. For simulations of unsteady response the assumption of sinusoidal motion
is obviously violated. Furthermore, the assumptions of the estimated covariance function will also be violated and in
particular the estimates of the damping ratio by the COV-SSI will be erroneous.
References
[1] Bouc R. Forced vibrations of a mechanical system with hysteresis. In Proceedings of the 4th Conference on Nonlinear
Oscillations, Prague, Czechoslovakia, 1967.
[2] Wen Y. Method for random vibration of hysteretic systems. ASCE Journal of the Engineering Mechanics Division,
102(2):249 263, 1976.
[3] No¨el J.P. and Schoukens M. Hysteretic benchmark with a dynamic nonlinearity, Brussels, Belgium, 2016.
[4] Peeters B., and De Roeck G. Reference-based stochastic subspace identification for output-only modal analy-
sis.Mechanical systems and signal processing,13(6): 855-878, 1999.
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Hysteretic benchmark with a dynamic nonlinearity
Conference Paper
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  • Apr 2016
Hysteresis is a phenomenology commonly encountered in very diverse engineering and science disciplines, ranging from solid mechanics, electro-magnetism and aerodynamics to biology, ecology and psychology. The defining property of a hysteretic system is the persistence of an input-output loop as the input frequency approaches zero. Hysteretic systems are inherently nonlinear, as the quasi-static existence of a loop requires an input-output phase shift different from 0 and 180 degrees, which are the only two options offered by linear theory. The root cause of hysteresis is multistability. A hysteretic system possesses multiple stable equilibria, attracting the output depending on the input history. In this sense, it is appropriate to refer hysteresis as system nonlinear memory. This document describes the synthesis of noisy data exhibiting hystere-sis behaviour carried out by combining the Bouc-Wen differential equations and the Newmark integration rules.
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Reference-Based Stochastic Subspace Identification for Output-Only Modal Analysis
Article
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When performing vibration tests on civil engineering structures, it is often unpractical and expensive to use artificial excitation (shakers, drop weights). Ambient excitation on the contrary is freely available (traffic, wind), but it causes other challenges. The ambient input remains unknown and the system identification algorithms have to deal with output-only measurements. For instance, realisation algorithms can be used: originally formulated for impulse responses they were easily extended to output covariances. More recently, data-driven stochastic subspace algorithms which avoid the computation of the output covariances were developed. The key element of these algorithms is the projection of the row space of the future outputs into the row space of the past outputs. Also typical for ambient testing of large structures is that not all degrees of freedom can be measured at once but that they are divided into several set-ups with overlapping reference sensors. These reference sensors are needed to obtain global mode shapes. In this paper, a novel approach of stochastic subspace identification is presented that incorporates the idea of the reference sensors already in the identification step: the row space of future outputs is projected into the row space of past reference outputs. The algorithm is validated with real vibration data from a steel mast excited by wind load. The price paid for the important gain concerning computational efficiency in the new approach is that the prediction errors for the non-reference channels are higher. The estimates of the eigenfrequencies and damping ratios do not suffer from this fact.
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