The effect of random vibrations of linear acceleration transducers
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Measurement Techniques, VoL 36, No. 11, 1993
THE EFFECT OF RANDOM V IBRAT IONS OF L INEAR
A. M. Makarenkov, A. I. Trofhnov, and N. D. Egupov
A method is given for taking account of the action of random vibrations on linear acceleration transducers. This permits the transducers to be described and analyzed in a class of stochastic systems with varying lumped parameters. The method combines a block diagram description of the stochastic systems with spectral methods of analyzing linear nonstationary systems, and is oriented toward computer implementation.
Linear acceleration transducers or navigational accelerometer devices are the most sensitive components of the systems controlling the motion of missiles or aircraft. They measure the parameters of motion in a trajectory, for generating correction signals.
During flight, the mountings of instruments in the instrument bay are subjected to random vibrations from the propulsion motors. In addition to these vibrations of the mountings, there is a random acoustic pressure field around the instruments. All of this, as noted in , causes random modulation of the parameters of the electromechanical circuits.
An analysis of the dynamic properties of the accelerometers under the action of vibration is especially important, because they are primary information sensors. Errors arising in them have a direct effect on the operation of the system.
There is a necessity to construct mathematical models of the transducers in a class of stochastic systems, and to develop appropriate computational methods. In this paper, one such method is proposed. It is for describing and analyzing linear acceleration transducers, which are considered as stochastic systems with variable lumped parameters.
Mathematical Models of Linear Acceleration Transducers in a Class of Stochastic Systems. Let us consider accelerometers without force correction, and made according to two different constructional schemes: with rotational (a type of pendulum), and translational motion of the inertial element.
The equation of motion of the sensing element of an accelerometer of the first type, when vibrational acceleration is applied to the casing of the accelerometer, may, as in , be written in the form
where J is the moment of inertia of the pendulum armature; B is the damping coefficient; m is the mass of the moving system; l is the distance from the center of gravity to the axis of suspension; ~o is the angle of deviation from the vertical R and ~ are the components of the vibration acceleration vector along these axes.
For small angular deviations of the pendulum, using the notation a 2 = B/J; a 1 = mgl/J; Al(t) = ~m//J; f(t) = ~nl/J, Eq. (1) can be written in the form
q~ +~,~q~+ [.~,, + =% t t) ]~p =/(t). (2)
It follows from (2) that the action of the vibration causes a random coefficient Al(t) to appear in the model of the transducer.
The equation of motion of the sensing element of an accelerometer of the second type takes an analogous form :
~Tzx+2h.~+ kl ! +~F(t) ]x=P (t), (3)
Translated from Izmeritel'naya Tekhnika, No. 11, pp. 54-55, November, 1993.
0543-1972/93/3611-1281512.50 9 Plenum Publishing Corporation 1281
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iTi U-
~::~' 7< ~"~i ~,-,-~. ,F-.--,.-T~ " . - . . t i - - - - - :S +-i ,"
The Spectral Form of the Description of the Stochastic System. The description of a stochastic system by integral equations, as shown above, permits us to use the apparatus of a matrix representation of integral bounded operators in orthogonal bases, i.e., to introduce the so-called spectral form of the description  of dynamic systems, with reference to an orthogonal basis ,I,(t). The crux of this form consists of constructing integral equations as an expansion in terms of the basis
(D,,'/)=[qo,,.(t); re=l,2,,..; t~[0 ,T ]}T ,
where r are elements of a system of orthonormalized functions, and T is the transposition sign. The spectral form of the description supposes a transition from the time characteristics of the systems and signals to their discrete spectral characteristics (SC), through an expansion in terms of the basis of the orthogonal functions.
In particular, for the system of equations (6) in , we can write the spectral analog, which is given in  by system (7).
The system of matrix equations (7) in  is a mathematical description, based on the spectral method, for the acceleration transducer under the action of random vibrations.
From (7) in , it is easy to obtain a matrix equation for determining the SC of the averaged output signal from the transducer
m - . " A.(C,.:.,_A. C ey~) c ."~,(/-.4"/!~'c '" )-~ , (6)
and also a recurrence equation for computing the SC of the autocorrelation function of the transducer output signal:
where i is the number of the approximation, and matrix B is defined by
n , R R ")W ~C ~ +~C ' ~ (C ~)TI~,
As a zeroth approximation for Eq. (7), it is convenient to take
C~ sr =A*C "~r
which is equivalent to "switching off" the parametric disturbance in the block diagram of Fig. 1, The Statistical Analysis Algorithm. The matrix equations obtained above can be easily solved on the commonest types
of computer, by successive approximations, using standard matrix algebra sub-programs. For the transition from signals and functions to their spectral characteristics, and the reverse process, it is convenient to use Fast Transform algorithms appropriate to the chosen basis. As an orthogonal basis, a choice can be made of trigonomatric functions, Walsh functions, etc.
It should be noted that algorithms implementing the spectral methods possess, in principle, computational stability; since the matrix operator equations used in them are equivalent to integral equations of the second kind, the problem of fmding a solution of which belongs to the class of correctly formulated problems. Convergence conditions for the spectral method, a priori and a posteriori estimates of the error are considered in [5, 6].
Thus, the proposed method for taking account of random vibrations on linear acceleration transducers makes it possible to analyze the working of these transducers under the action of nonstationary Gaussian random processes, cross-correlated with the input signal to the transducer. The transducer is considered as a stochastic system with variable lumped parameters. The method has the following merks:
1) Simplicity in obtaining a mathematical description of the transducer as a stochastic system; 2) The possibility of carrying out the analysis without simplifying assumptions about the stationarity of the random
signals themselves, and of the determinate coefficients of the transducer model; 3) The mathematical description based on discrete SC permits the direct and inverse problem to be analyzed equally
easily, and also provides a basis for solving the optimal filtering and synthesis problems;
4) The method is, from the outset, oriented towards computer use, and permits the statistical analysis problem to be reduced to solving systems of matrix equations.
E. A. Fedosov (ed.), The Design of Guidance Systems [in Russian], Mashinostroenie, Moscow (1975). E. A. Fedosov, and G. G. Sebryakov, Automatic Control and Computing Techniques, Frequency Methods, No. 8 [in Russian], Mashinostroenie (1968), p. 207. A. M. Makarenkov, A. I. Trofimov, and N. D. Egupov, Izmer. Tekh., No. 10, 35 (1993). A. N. Malakhov, A Cumulant Analysis of Random Non-Gaussian Processes and Their Transformation [in Russian], Sov. Radio, Moscow (1978). V. V. Solodovnikov, A. N. Dmitriev, and N. D. Egupov, Spectral Methods for the Computation and Design of Control Systems [in Russian], Mashinostroenie, Moscow (1986). A. N. Dmitriev et. al, Machine Methods for the Computation and Design of Electrical Communication and Control Systems [in Russian], Radio i Svyaz', Moscow (1990).