positive integer in the case of differentiation, a negative integer for integration and a ... integral derivative than by a classical integer order notation [9]. This fact ...
Effect of Fractional Orders in Differential Equation Describing Damping in the Measuring Transducer Mirosław Luft, Elżbieta Szychta, Radosław Cioć, and Daniel Pietruszczak Kazimierz Pułaski Technical University of Radom, Faculty of Transport and Electrical Engineering, 26-600 Radom, Malczewskiego 29, Poland {m.luft,e.szychta,r.cioc,d.pietruszczak}@pr.radom.pl
Abstract. The paper presents the way to write differential equations of measuring transducers by means of fractional orders. The impact on the dynamics of the transducer with fractional orders for part of the equation responsible for the damping is presented. Examples of the frequency characteristics for different orders of the equation are also given. Keywords: fractional calculus, measuring transducer, dynamic systems, discrete transmittance.
1 Introduction In the classical approach to modelling, the performance of measuring transducers as well as their dynamics are described by means of differential equations [1], [2], [3] and [5]. Usually it is enough to use ordinary second order equations in the form of:
Ai
d (i ) y d (i −1) y d ( j ) f ( x) d ( j −1) f ( x) + A + ... + A y ( t ) = B + B + ... + B0 f ( x) m−1 j m−1 0 dt (i ) dt (i −1) dt ( j −1) dt ( j −1)
(1)
The development of the fractional order integral derivative created new possibilities of modeling measurement transducer dynamics and, generally speaking, of all systems whose dynamics can be expressed in a differential form, not only those of integer order differentials. In the case of the derivative of an integral the known operators of differentiation n
and integration are combined into a single D operator [6], [7] and [9], where n is a positive integer in the case of differentiation, a negative integer for integration and a neutral operator for n = 0 , which can be written down as: d n f (t ) dt n D n f (t ) = f (t ) t τ1 τ − n − 1 ... f (τ− n )dτ− n ...dτ2 dτ1 t 0 t 0 t 0 J. Mikulski (Ed.): TST 2011, CCIS 239, pp. 226–232, 2011. © Springer-Verlag Berlin Heidelberg 2011
for n > 0 for
n=0
for
n
