Measurement Techniques, VoL 36, No. 11, 1993
MECHANICAL MEASUREMENTS
T H E E F F E C T O F R A N D O M V I B R A T I O N S O F L I N E A R
A C C E L E R A T I O N T R A N S D U C E R S
A. M. Makarenkov, A. I. Trofhnov,
and N. D. Egupov
UDC 620.178.53
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 [1], 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 [1], 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 [1]:
~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 ,"<;u
'~ L . . . . . . . . . . . < ) l . . . . . d
I
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F ' ~ , - : - / ~ ~ ..'----l
,j<ejj~.-F ~ I
I c. ' ,r;~ L_ . . . . . . . . . . . k_ra_t7
Fig. 1
where m is the mass of the sensing element; h is the damping coefficient; k is the overall stiffness of the system; t~ is the
coefficient of relative change in stiffness; F(t) are forces acting perpendicular to the measurement axis; P(t) are forces acting
along the measurement axis. From (2) and (3), it follows that the equation of motion of the accelerometer's sensing element can be put in the form
of a generalized differential equation of the stochastic system:
k~(O','4d:z(t) ~C+kra(f)x-~J, (4)
where y is the transducer input signal; x is the output signal; kl(t ) and k2(t) are determinate coefficients, which may, in the
general case, be variables, and kra(0 is a random coefficient. As mentioned above, the method described may be applied to an analysis of stochastic systems, the mathematical
models of which can be represented by a block diagram [2]. In particular, Eq. (4) may be represented by the block diagram
of a stochastic system (see Fig. 1). In this schematic, the random force coefficients are separated from the section making up
the determinate part of the system. Each random force coefficient is shown as a determinate force coefficient with a parallel branch containing the element for multiplying by the random component.
The circuits comprising the determinate part of the system, are described in the general case by differential equations of the type
S ~ (t)x~== Z b (t)v t') (5) ~ 0 ,s v = = 0 I
where y(t) and x(t) are the input and output signals of the circuit; and av(t ) and by(t) are the determinate variable coefficients.
Let us put the model of the acceleration transducer in the form of a block diagram of the stochastic system (see Fig. 1), where E(t) is the nonstationary determinate linear element defined by Eq. (5).
Let us take the input signal y(t) as a nonstationary Gaussian process with mean my(t) and autocorrelation function
Ryy(t 1, t2). The signal or(t) is a nonstationary Gaussian process with zero mean and autocorrelation function Ra,~(t 1, t2). The statistical connection between signals y(t) and c~(t) is expressed by the cross-correlation function Ry~(t 1, t2). For signal x(t),
it is required to determine the mean rex(t) and autocorrelation function Rxx(t 1, t2). The functions my(t) and mx(0 are, respectively, the useful signals at the input and output of the system.
By applying to the integral equations determining the signals in the block diagram of Fig. 1, an averaging over an
ensemble of samples of the random processes, equations (given in [3]) can be written for determining all unknowns of system
(6), where 6xa0- ) is the joint dispersion, found from the equation for Rxa(t 1, t2), with the conditions t I = t 2 = t; KA*(t, ~r) is the kernel of the integral equation for element E(t), inside the negative feedback loop through k(t) in Fig. 1.
It should be noted that Eqs. (6) in [3] are valid only in satisfying the normalizability of random process e(t) by linear
operator E(t). This condition is satisfied if the spectrum of the random process at the input of the linear operator is much wider than its pass-band. For non-Gaussian signals, it is possible to obtain a cumulant description [4].
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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 [4] 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 [3], we can write the spectral analog, which is given in [3] by system
(7).
The system of matrix equations (7) in [3] is a mathematical description, based on the spectral method, for the
acceleration transducer under the action of random vibrations.
From (7) in [3], 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:
(7)
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.
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REFERENCES
.
2.
3. 4.
5.
6.
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).
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