ANALYSIS OF DYNAMIC PROPERTIES OF ACCELEROMETER USING FRACTIONAL DERIVATIVES ANALYSIS OF DYNAMIC PROPERTIES OF ACCELEROMETER USING FRACTIONAL DERIVATIVES

The paper outlines the use of fractional derivatives for dynamic measurements while developing a method of improved description of dynamic properties of accelerometer, which the authors consider to be the original and unique achievement of their work. The main objective of the work is an implementation of a fractional calculus-based method which allows for a description of dynamic properties of signal processing of measuring transducers with fractional orders. A dynamics of transducer is identified by the ARX method (AutoRegressive with EXternal input identification method). Identification of accelerometer was accomplished with the use of the MATLAB&Simulink package.


Introduction
The problem of describing dynamic properties of objects by means of fractional calculus although well known since the times of Gottfried Wilhelm Leibniz [1], and [2], yet due to restrictions resulting from lack of appropriate calculation methods and possibilities of their verification has always been ignored.
At present technical and calculation possibilities cause that the problems related to these limitations have, to a large extent, been solved. There are more and more publications dealing with the topic of fractional order differential equations.
Majority of them, however, deal with theoretical aspects of the problem. There are no publications which put strong emphasis on the practical application of fractional calculus and combine theory with real applications. The paper outlines the use of fractional calculus for dynamic measurements while developing a method of improved description of dynamic properties of measuring transducers which the authors considers to be the original and unique achievement of this work. Proposed by the authors of this paper method of description of the dynamic properties of measuring transducer in terms of signal processing, based on fractional calculus, allows for a description of dynamic properties of broader class of measuring transducers, i.e. integerorder and fractional-order accelerometers. The aim of this paper is to investigate how models of accelerometers based on the fractional calculus ( [3], [4], [5], [6], [1], [7] and [2]) description convey their dynamic behaviour in comparison to models represented by differential equations of integer orders and in comparison to processing characteristics of their real counterparts.

Model of the second order accelerometer
Simulation and laboratory testing of the second order accelerometer is shown in Fig. 1. Fig. 1 Kinetic diagram of an accelerometer: m-vibrating mass, k s -spring constant, B t -damping coefficient, x -object motion relative to a fixed system of coordinates, y -motion of a vibrating mass relative to a fixed system of coordinates, w -motion of a vibrating mass relative to a vibrating object [4] and [7] where (6) is defined as the reverse difference of the discrete function and h is the increment of f (t) defined in the range [t 0 ,t]: Introducing a non-integral order to the measuring transducer's equation (1) converts it into: where v is order of a fractional derivative. The concept of this work is based on a comparison of different models of an accelerometer's dynamic behaviour (based on differential equations of integer and fractional orders) with the processing characteristics of a real accelerometer so as to obtain answer to the question about which method of modelling is more accurate and whether there are any criteria for which a certain model is better at reproducing the dynamic behaviour of the real accelerometer. The research results has included algorithms of determining models describing an accelerometer's dynamic behaviour based on differential equations of integer and fractional orders for the definitions by Grünwald-Letnikov (5). The results were presented in works [2], [3], [4], [7] and [8].
On comparing responses of the measuring transducer to the input sinusoid signal, it was described by means of three models using the Grünwald-Letnikov derivative (5): Classic model described with operator transmittance (9) -the same as formula (4). The operation of a transducer described with the equation was simulated for appropriately selected parameters: natural angular frequency and degree of damping: . G s s s 880 484 10 Classic discrete mode, derived from the operator transmittance model (9), described by means of discrete transmittance (10) Response of a continuous object to a discrete input depends not only on values of this signal at discrete moments of time but also on sampling time and the extrapolator used.
The quasi-fractional discrete model described by discrete transmittance (11): Discrete transmittance (11) is produced by implementation of the method of determining quasi-fractional expression of the measuring transducer in MATLAB&Simulink programme [3], [5], [7] and [9]. It can be noted that the model described by means of the discrete transmittance (11) correctly reproduces values of the input signal amplitude -the same as the model of where: (2b) Considering the motion of the vibrating mass relative to the vibrating object ( Fig. 1): Depending on the selection of k s , m and B t , a transducer can serve to measure acceleration as an accelerometer [7] assuming a high k s , a low m and B t . In practical vibration measurements, acceleration-measuring transducers, the so-called accelerometers, are employed. For purposes of simulation testing, a measuring transducer was assumed of the frequency f = 350 Hz, that is, circular frequency of free vibrations

Model of a laboratory system with accelerometer
An overview of the measurement system is shown in Fig.  3. In order to determine measuring transducer's operator transmittance, we built a system with two accelerometers A1 and A2. Accelerometer A1-DeltaTron by Bruel&Kjaer type 4507, which sensitivity 10.18mV/ms -2 and the range of frequency measurements from 0.4 Hz to 6 kHz was tested. The operating ranges of conditioner was between 1Hz and 20 kHz. The transducer was mounted on an electrodynamic inductor. A model accelerometer A2 -by VEB Metra, type KB12, which sensitivity 317 mV/ms -2 was aligned with the tested transducer A1.
The operator transmittance (12) describing dynamics of the measurement system was determined by identification with an transmittance (10). It can be noted in Bode frequency diagrams [6] (Fig. 2) that the measuring transducer model determined by the derivative-integral method presents the dynamics of the classically determined model (the diagrams of the models coincide). This confirms that integral-order differential-integral calculus is a special case of differential-integral calculus of nonintegral orders.
Measuring transducer models (10) and (11) have only been subject to simulation testing and do not fully represent real models but the simulations indicate that the quasi-fractional model (11) exhibits the same dynamics as the classic model. The 'apparent' time of stabilization of time diagrams -the time after which description of a model is independent from time -for the quasi-fractional model is the same as for the classic model.  [9] Discrete transmittances of measuring transducer models for varying increment Table 1 The model of the real measurement system in the form of discrete transmittance and models expressed by means of a differential-integral equation were then compared. Both types of the models were based on the classic model derived by ARX identification method. Figure 4 shows logarithm frequency amplitude and phase diagrams of the measurement system models. It can be observed that for the adopted increment of h, the diagrams clearly diverge. This means that other increments h far lower than the sampling frequency, must be adopted. external ARX (AutoRegressive with EXternal input) [7]. The voltage signal from the end of the tested measurement track is the identified signal, and the signal from the model accelerometer in response to the sinusoidal function of the generator of 100 Hz is the comparative signal.
The ARX identification method produced the operator transmittance G(s) describing the dynamics of the system: 1.
The discrete model (13) was produced by discretizing the classic model (20) by means of the 'Zero-Order-Hold' method [7] with the sampling time T p = 10 -4 s.

Discrete transfer function of fractional models was determined
with a method implemented in MATLAB&Simulink. For varying increment of h, quasi-fractional transducer models become discrete transmittances which is presented in the The effect of h on fractional measuring amplitude of fractional measuring transducer and phase diagrams was also shown in this paper. In this case increments h, far lower than the sampling frequency, must be adopted [4].
Application of the quasi-fractional method of describing dynamic properties of measuring transducers discussed in this paper, based on fractional calculus, will help to undertake analyses of simulated dynamics of various objects and processes which, due to their complexity, must be described by means of differential equations of any orders.