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Technical Brief

# Identification of a Subsystem Located in The Complex Dynamical Systems Subjected to Random Loads

[+] Author and Article Information
Krzysztof Jamroziak

Faculty of Management,
Land Forces,
Czajkowskiego 109,
Wroclaw 51-150, Poland
e-mail: krzysztof.jamroziak@wso.wroc.pl

Miroslaw Bocian

Department of Mechanics, Materials Science and Engineering,
Wroclaw University of Technology,
Smoluchowskiego 25,
Wroclaw 50-370, Poland
e-mail: miroslaw.bocian@pwr.edu.pl

Maciej Kulisiewicz

Faculty of Technology and Engineering,
Wroclaw University of Technology,
Smoluchowskiego 25,
Wroclaw 50-370, Poland
e-mail: maciej.kulisiewicz@pwr.edu.pl

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 16, 2016; final manuscript received July 12, 2016; published online September 16, 2016. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 12(1), 014501 (Sep 16, 2016) (5 pages) Paper No: CND-16-1141; doi: 10.1115/1.4034274 History: Received March 16, 2016; Revised July 12, 2016

## Abstract

The paper presents a new way to determine some dynamical properties of materials modeled by the so-called degenerate system. The system is an element (subsystem) of any complex multidegree-of-freedom system. This subsystem follows from assumption of standard rheological model of stress–strain law of the materials. It is assumed that on the complex system act a set of random excitation forces. For this coincidence, a so-called energy balance equation was developed and was used to create a suitable identification method. The equations were derived for any differentiable function of elasticity. The stationary random process of the system response was assumed in the whole algorithm. As it was proved, in this case, instead of calculating appropriate fields of the hysteresis loop of suitable signals, an application of average values of the input and output signals and their proper combinations can be used. It is assumed that the elastic damping interaction force in the complex dynamical subsystem is described by the function $F(x,x˙)$, in which x is a deformation of the identified degenerated element and denotes a relative displacement between some appropriate neighboring masses of the system. Some numerical examples of the application are shown.

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## Figures

Fig. 1

The model (b) of the tested element (a) assumed in this paper

Fig. 2

The diagram of considered dynamic system

Fig. 3

The diagram and data of the system subjected to testing: (a) for the elasticity force of the form (14)m = 1 kg, mA = 2 kg, c0 = 1.0 N·m−1, c1 = 0, c2 = 10.0 N·m−1, c3 = 10 × 108 N·m−1, k = 6 kg s−1, and k1 = 12 kg s−1, and (b) for the elasticity force of the form (20)m = 1 kg, mA = 2 kg, c0 = 2.0 N·m−1, c2 = 5.0 N·m−1, k = 42 kg s−1, k1 = 50 kg s−1, β = 100, and C = 30

Fig. 4

An example of excitation p(t) (a), of response xm (b), and of response xA (c) of the system tested

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