Experimental dynamic system identification of damaged beam volkova V icon

Experimental dynamic system identification of damaged beam volkova V

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Volkova V.

Dnepropetrovsk National Univesity of the Railway Transport

UA 49005, Ukraine, Dnepropetrovsk, Televizionnaya 12 ap.48

The construction of mathematical model satisfactorily describing or predicting operation of object, process or system is an integral part of any problem of prediction of dynamic behaviour of mechanical systems. There are many types of damage that can cause an initially linear structural system to respond to its operational and environmental loads in a non-linear manner. One of the most common types of damage is cracks that subsequently open and close under loading. This type of damage may include fatigue cracks and cracks that result from excessive deformation. Vibration-based structural health monitoring consists in detecting damages in structures from changes in vibration features obtained from periodically spaced measurements.

Dynamic behaviour of mechanical systems is usually presented as oscillating processes in various graphic forms such as time processes, the Lissajous patterns and hodograph. Such patterns of presentations enable to determine the type of a process and to perform numerical estimations of its characteristics, but do not disclose any properties of the governing system. Unlike them classic phase trajectories have the row of advantages.

A phase space in classic mechanics is represented as a multidimensional space. The number of measured values for a phase space is equal to the doubled number of degrees of freedom of the system being investigated. The state of the system is presented as a point in the phase space, and any change in the system state in time is depicted as the displacement of the point along a line called a phase trajectory. The image on phase plane (, ) is a more vivid presentation because it depicts inharmonious oscillations particularly well. Each phase trajectory represents only one definite clearly defined motion. A disadvantage of phase trajectories (, ) consists in the fact that they do not provide for the immediate presentation of oscillating process in time. However, this drawback is compensated by a significant advantage. The geometric presentation of a single phase trajectory or a set of trajectories allows coming to important conclusions about the oscillation characteristics. It is, foremost, true with the oscillations, which are described with nonlinear differential equations.

As is has been shown by the investigations of several authors, the expansion of a phase space by taking into account the phase planes (, ) and (, ) substantially promotes the efficiency in analyzing a dynamic system behaviour. Hereby, we pass on to a three-

dimensional phase space confined with three co-ordinate axes, i.e. displacement, velocity and acceleration. An interest taken into accelerations in dynamic systems is conditioned by the fact that these accelerations are more sensitive to high-frequency components in oscillating processes. Phase plane (, ) is of particular interest in the analysis of dynamic system behaviour, because it allows a more evident interpretation of power relations in the dynamic system under investigation. Namely, the area confined by curve () and axis (0) is equal to work, and the anticlockwise motion around its contour corresponds to the energy spent by the system for one cycle of oscillating. Another important characteristic of phase trajectories on plane (, ) is the fact that dependence () for autonomous non-conservative systems is a mirror symmetric image in relation to axis (0) to the graph of changes in elastic force characteristic.

In the past two decades, the issues of construction of mathematical models and prediction of dynamic behaviour of structural elements proceeding from recorded experimental data have attracted considerable interest.

In spite of intensive investigations into the above mentioned matter, which have been undertaken in the scientific centres in different countries (supported by numerous publications on theoretical research and experiments, a number of specialized conferences, as well as the important results obtained, there is no, so far, the only universal effective approach, which would allow for correct determination, prediction and analysis of dynamic properties in construction elements. In most of the previous analyses, damages are characterized by changes in the modal parameters, for example, natural frequencies, modal damping ratios and mode shapes. The calculating process is done using data from the structure in some initial and usually assumed undamaged condition, and then is repeated at periodic intervals or after some potentially damaging event that triggers the assessment process. Structural parameters such as stiffness matrix constructed from identified modal parameters may also be used for damage detection and localization. These methods have prevented their use in most "real-world" applications. At first, it involves fitting a linear physics-based model to the measured data from both the healthy and potentially damaged structure. Often these models do not have the fidelity to accurately represent boundary conditions and structural component connection, which are prime locations for damage accumulation. Also, this process does not take advantage of changes in the system response that are caused by non-linear effects. There is also research using other damage-sensitive features without the need to identify the modal parameters, such as novelty analysis with auto-regressive models.

et us consider a mechanical system described by the following differential equation:

where m is mass; functions H(, ) and R(y) describe dissipative and elastic force, respectively.

In order to obtain information about the structure offerees H(, ) and R(y) , let us apply outer periodic excitation to the system (1). Thus, we investigate the system


The qualitative analysis of T - periodic oscillations of the system (1) is based on studying Poincare trajectories on phase plane (, ). It consists in studying distinctive trajectories (equilibrium conditions, limit cycles, separatrix) and their stability on plane (, ) or so-called the phase plane. Acceleration of system y at every moment of time t is uniquely determined by displacement y and velocity y according to equation (2).

The main concept of this paper is that, if a given type of damage converts a linear system into a non-linear system, then any observed manifestations of nonlinearity serve to indicate that damage is present. The simply supported beam was considered. In its undamaged state an assumption that the beam can be modelled as a linear system is quite adequate, but consider what happens when a crack is introduced, as shown in Fig. 1.

When the beam hogs, the effects of the crack are negligible because the two faces of the crack come together and the beam behaves as though the crack was not there. When the beam sags however, the presence of the damage must affect the beam because the crack opens and the effective cross-sectional area of the beam are reduced. Under these circumstances, an appropriate model of the beam has the general equation of motion (1).

The time processes of free vibrations had been selected from the general of experimental records. Time processes of free vibrations are damped process with the insignificant deposit of ultraharmonicss and subhormonics. Examples of the phase trajectories obtained from the experimental records are given in Fig 2. The system has settled onto a attractor. Length of the loops of phase trajectories and angle of slope change with time. Phase trajectories and their mappings are built in the extended phase space (Fig 3). The crosses on phase trajectories are caused the presence of subarmonis of order ?/3 in the records of time processes.

By averaging the experimental data, a polynomial trend was derived. As it seen from obtained results, cracked beam posses features of "soft" system.

The suggested method for analysis of the determinately chaotic processes provides fresh approach to data processing. The most significant feature of this method is that, in spite of its simple appearance, it enables to obtain maximum information about an investigated process or a phenomenon. The applicability of the suggested method is limited by noise levels, measuring errors or duration of a process under study. The engineering applications of the suggested method are very promising in identification of parameters in the determinated chaotic systems.


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