RU2568956C1 - Method to calibrate angular acceleration sensor - Google Patents

Abstract

FIELD: measurement equipment.

SUBSTANCE: output parameters of the sensor are taken in two positions of its installation on a bench for calibration. The angular acceleration sensor is installed on an axis of pendulum suspension, where it is exposed only to angular acceleration, and its readings are registered. At the same time at the distance L from the axis of rotation on the pendulum there is a dynamic analogue of the sensor fixed. After measurements the sensor and the dynamic analog of the sensor switch places, and they again register sensor readings at the specified amplitude and frequency of oscillations.

EFFECT: knowing sensor readings under action of only angular acceleration and sensor readings under action of additionally linear acceleration, during measurements on a bench they introduce corrections, making it possible to produce existing values of angular acceleration with high accuracy.

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Description

The invention relates to the field of mechanical measurements, in particular to the measurement of angular acceleration, as applied to stands for determining the moments of inertia of bodies.

Currently, TsAGI specialists have developed stands for measuring mass, coordinates of the center of mass and moments of inertia of engineering products for various purposes (Bogdanov VV, Veselov NV, Panchenko IN, etc. A stand for measuring mass, center coordinates mass and inertia tensor of the product. // Sensors and Systems, 2010, No. 6, pp. 24-28), (Bogdanov V.V., Veselov N.V., Panchenko I.N. et al. // RF Patent 2358880 , A stand for measuring mass, coordinates of the center of mass and inertia tensor of a product), (Bogdanov V.V., Panchenko I.N., Nyakk V.A., Chumachenko E.K. // RF Patent 2434213, Stand for measuring rhenium mass, coordinates of the center of mass and the inertia tensor of the product), (Bogdanov V.V., Panchenko I.N. On the theory of measuring mass, coordinates of the center of mass and moments of inertia tel. 12-15), (Bogdanov V.V., Panchenko I.N., Chumachenko E.K. Digital signal processing and bench test results. // Sensors and Systems, 2010, No. 5, p. 29-33).

When measuring the moments of inertia, the product is given angular vibrations around the stationary axes of the bench. According to the parameters obtained from the moment sensors and angular acceleration sensors, the moments of inertia of the products are calculated. In practice, due to the arrangement of the product on the stand, it is not possible to position the angular acceleration sensors on the axes around which oscillations occur, and they are installed at a certain distance L from the rotation axis (see the appendix), then the sensor, in addition to angular acceleration, also experiences linear acceleration . An angular acceleration sensor (DUU) has been developed for measuring angular acceleration, which allows high-precision selection of angular acceleration signals from a mixture of linear and angular accelerations acting simultaneously on the sensor.

The DUU used at the stands are periodically calibrated.

, ГОСТ 8.289-78). В качестве образцовых средств измерений применяют образцовые меры, работающие на принципах: падающего в аэростатическом подвесе винта, блока с падающим грузом, управляемых электродвигателей. Образцовые средства измерений применяют для проверки рабочих угловых акселерометров методом прямых измерений. В качестве рабочих средств измерений применяют угловые акселерометры, допускаемые относительные погрешности которых доходят 15%.A known method of reproducing and transmitting to working measuring instruments (including sensors) the size of a unit of constant angular acceleration (see State primary standard and the all-Union calibration scheme for measuring instruments of constant angular acceleration in the range

GOST 8.289-78). As exemplary measuring instruments, exemplary measures are applied that operate on the principles of: a propeller falling in an aerostatic suspension, a unit with a falling load, and controlled electric motors. Exemplary measuring instruments are used to test working angle accelerometers by direct measurements. As working measuring instruments, angular accelerometers are used, the permissible relative errors of which reach 15%.

The disadvantage of these methods for calibrating the remote control is as follows. In the stand system for determining the moments of inertia of the remote control system, the variable angular acceleration is measured at a certain fixed frequency of oscillations ƒ ₀ . Calibration of the sensor at constant angular acceleration does not allow to take into account the dynamic error of the sensor that occurs when measuring variable acceleration. In addition, the question remains of the effect of linear acceleration on the sensor readings.

Closest to the proposed solution is a method for calibrating a variable angular acceleration sensor used in measuring dynamic moments in electric drives (see Sokolov M.M., Masandilov L.B. Measurement of dynamic moments in alternating current electric drives, pp. 154 ... 156 // Publisher "Energy", Moscow. 1975)

The angular acceleration sensor is installed on the motor shaft, while the DUU axis is combined with the axis of rotation of the motor shaft and a standard angular velocity sensor ω _{0 is} installed on the shaft. DUU is subjected to the action of known angular acceleration and by processing the output signal DUU find the conversion coefficient of angular acceleration into an electrical signal.

The output signal of the angular acceleration sensor:

where: k is the conversion coefficient of angular acceleration into an electrical signal;

- мгновенное значение углового ускорения.

- instantaneous value of angular acceleration.

The electric drive is accelerated, angular acceleration acts on the shaft and the angular velocity increases.

For a certain interval of the acceleration curve, write the integral from the left and right sides of formula (1):

where: t ₁ and t ₂ are the boundaries of the interval of the curve of angular acceleration;

ω ₁ and ω ₂ are the boundaries of the interval of the angular velocity curve.

From:

- площадь под кривой сигнала датчика ускорения на интервале времени с границами t₁ и t₂.Where:

- the area under the curve of the signal of the acceleration sensor in the time interval with boundaries t ₁ and t ₂ .

Δω is the range of angular velocities with boundaries ω ₁ and ω ₂ .

The main disadvantage of this method of calibrating the remote control is that it is not suitable for alternating accelerations. If the acceleration varies according to the periodic law, which takes place on the stands measuring the moments of inertia, the integrals (2) turn to zero, or they are too small, unsuitable for further processing. In addition, this method does not allow calibration of the sensor under the simultaneous action of angular and linear accelerations.

The objective of the invention is to propose a calibration method in which the sensor is initially exposed only to angular acceleration (without linear acceleration), and then to the same angular acceleration and linear acceleration at the same time.

The technical result of the proposed method is to increase the accuracy of calibrating the remote control in conditions as close as possible to the conditions of its operation on the bench for measuring moments of inertia, determining the conversion coefficient of angular acceleration into the output signal of the sensor and determining the effect of the linear acceleration vector on this coefficient.

, по дискретным отсчетам сигнала датчика угла рассчитывают коэффициенты цифрового режекторного фильтра, которые используют для определения собственной круговой частоты колебаний маятника:The technical result is achieved by the fact that in the method of calibrating the angular acceleration sensor (angular acceleration sensor) is mounted on the shaft, subjected to the action of the known angular acceleration, and by processing the output signal of the sensor, the conversion coefficient of the angular acceleration into an electrical signal is found, while the angular acceleration sensor is installed on the pendulum, its axis is combined with the axis of rotation of the pendulum, and the axis of the dynamic analogue of the sensor - with the point of the pendulum, which is separated from its axis of rotation at a given distance, ma the casing is deflected by an angle φ _{(t = 0)} ≤8 ° allowed by the isochronism condition and released, discrete readings of the signals of the angle sensor α _(n) and angular acceleration are performed

, from discrete samples of the angle sensor signal, the coefficients of a digital notch filter are calculated, which are used to determine the natural circular frequency of the oscillations of the pendulum:

,

,

where: ω ₁ is the natural circular frequency of the oscillations of the pendulum;

- угол поворота собственного вектора характеристического уравнения;

- the angle of rotation of the eigenvector of the characteristic equation;

and ₁ and a ₂ are the coefficients of the digital filter;

- оптимальное целое число шагов задержки сигнала в цифровом фильтре;

- the optimal integer number of delay steps of the signal in the digital filter;

- частота дискретизации сигнала;

- signal sampling rate;

- коэффициент затухания колебаний;

- damping coefficient of oscillations;

from the discrete samples of the signals of the angle and angular acceleration sensors, the conversion coefficient of the angular acceleration into the output electrical signal of the sensor is calculated:

,

,

;where: s - conversion coefficient, dimension

;

- собственная недемпфированная частота колебаний; при этом на величину коэффициента затухания колебаний маятника накладывают ограничение: β≤0,01, после чего датчик углового ускорения и динамический аналог датчика меняют местами и снова определяют коэффициент преобразования датчика в условиях действия на него углового ускорения и вектора линейного ускорения.

- intrinsic undamped oscillation frequency; at the same time, the limitation coefficient of the oscillation pendulum is limited by β≤0.01, after which the angular acceleration sensor and the dynamic analog of the sensor are interchanged and the conversion coefficient of the sensor is again determined under the conditions of the action of angular acceleration and the linear acceleration vector on it.

For a more detailed explanation of the invention, we consider the layout of the stand, its design and principle of operation.

In FIG. 1 shows a perspective view of a bench for calibrating a child restraint;

In FIG. 2 shows the acceleration vectors acting on the pendulum;

In FIG. 3 shows a block diagram of a notch (non-recursive) digital filter.

The stand (see Fig. 1) consists of a bed 1, on which the turning unit 2 is fixed, an angle sensor 3 is mounted on its shaft on one side, and a rod 4, to which the load 5 is fixed, is mounted on the other side. The rod and the load form a pendulum. On the rod 4, the DUU 6 and the dynamic analogue of the sensor (DBP) 7 are installed, having the same mass as the DUU, the moment of inertia and the angular stiffness. If DUU 6 is installed on the axis of rotation, then DBP 7 is fixed at a distance L from the axis of rotation, moreover, the axis of DUU 6 is combined with the axis of rotation of the pendulum, and the axis of DBA 7 with the point of the pendulum, which is a predetermined distance from the axis of rotation. When installing the ДУУ 6 at a distance L, a DBP is installed on the axis of rotation 7. By moving the load 5 along the rod 4, the necessary oscillation frequency of the pendulum is selected. DUU 6 is subjected to the action of known angular acceleration by vibrational movements of the pendulum. The pendulum is deflected by an angle φ _{(t = 0)} ≅8 ° (see Fig. 2) from the equilibrium position, which is permissible under the condition of isochronism and released.

, по дискретным отсчетам сигнала датчика угла рассчитывают коэффициенты цифрового режекторного фильтра, которые используют для определения собственной круговой частоты колебаний маятника:Discrete readings of the signals of the angle sensor α _(n) and angular acceleration

, from discrete samples of the angle sensor signal, the coefficients of a digital notch filter are calculated, which are used to determine the natural circular frequency of the oscillations of the pendulum:

,

,

where: ω ₁ is the natural circular frequency of the oscillations of the pendulum;

- угол поворота собственного вектора характеристического уравнения;

- the angle of rotation of the eigenvector of the characteristic equation;

and ₁ and a ₂ are the coefficients of the digital filter;

- оптимальное целое число шагов задержки сигнала в цифровом фильтре;

- the optimal integer number of delay steps of the signal in the digital filter;

- частота дискретизации сигнала;

- signal sampling rate;

- коэффициент затухания колебаний;

- damping coefficient of oscillations;

from the discrete samples of the signals of the angle and angular acceleration sensors, the conversion coefficient of the angular acceleration into the output electrical signal of the sensor is calculated:

,

,

;where: s - conversion coefficient, dimension

;

- собственная недемпфированная частота колебаний; при этом на величину коэффициента затухания колебаний маятника накладывают ограничение: β≤0,01, после чего датчик углового ускорения и динамический аналог датчика меняют местами и снова определяют коэффициент преобразования датчика в условиях действия на него углового ускорения и вектора линейного ускорения.

- intrinsic undamped oscillation frequency; at the same time, the limitation coefficient of the oscillation pendulum is limited by β≤0.01, after which the angular acceleration sensor and the dynamic analog of the sensor are interchanged and the conversion coefficient of the sensor is again determined under the conditions of the action of angular acceleration and the linear acceleration vector on it.

The differential equation of free oscillations of the pendulum around the axis of rotation (OB) parallel to the axis Ζ (Fig. 2).

where: Jz is the moment of inertia of the pendulum relative to the Z axis;

l is the distance between the center of mass (CM) of the pendulum and OM;

g is the acceleration of gravity;

φ is the angle of rotation of the pendulum.

Equation (3) allows you to find the natural frequency of the pendulum, which, due to the nonlinearity of the equation, depends on the amplitude of the angle of rotation.

For small rotation angles, the natural frequency of the pendulum is determined by the dependence (see Markeev A.P. Theoretical Mechanics. // M. "Science", 1990, p. 155):

- приведенная длина маятника:

;Where:

- reduced length of the pendulum:

;

φ _m = maximum angle of rotation of the pendulum.

Since the pendulum makes damped oscillations, in accordance with equation (4), as the angle of rotation decreases, the frequency of the pendulum will change (increase), for this reason, the oscillations of the pendulum become non-isochronous.

As a result, an error arises in determining the frequency due to the non-synchronization of oscillations.

So, for the maximum deviation angle φ _m = 8 °, the error is 0.12%, and for φ = 6 °, 0.068%, respectively.

The differential equation of the oscillations of the pendulum, taking into account the smallness of the deflection angle and the loss resistance, proportional to the speed:

where: h is the resistance coefficient.

Equation (5) by simple transformations is reduced to the form:

where: ω ₀ - intrinsic undamped circular frequency of the pendulum:

:

:

β is the damping coefficient of oscillations:

If we set the pendulum to a small initial deviation φ _m , from equation (6) we obtain the following law of change in the angle of rotation in time:

;where: ω ₁ - own damped circular oscillation frequency:

;

.Ψ - phase angle:

.

The law of change in time of angular acceleration:

As can be seen from equation (8), to reproduce angular acceleration, it is necessary to know the amplitude φ _m , the natural circular frequency ω ₀ and the damping coefficient of the oscillations β.

The central task is to determine the natural frequency of oscillations from discrete samples of the signal of the angle sensor α _(n) .

To solve it, the digital sequence of the rotation angle φ (n) is passed through a notch (non-recursive) digital filter (Fig. 3) with a transfer function:

where: z is the operator of the discrete Laplace transform;

m is the delay coefficient of the signal in the digital filter - an integer greater than one (introduced to improve the accuracy of measuring the frequency of oscillations);

K - scaling factor:

.

.

Ζ - conversion of the discrete digital sequence of the angle signal:

After the transition in equation (7) to the discrete time t = m · (n · Δt) and performing the necessary transformations, one obtains:

where: Δt is the time step of signal sampling;

R is the modulus of the roots of the characteristic equation, determined by the coefficients b ₁ , b _n .

Expression (11) is common for the class of signals described by equation (7).

To determine the unknown coefficients b ₁ and b _2, the filter coefficients a ₁ and a _{2 are} adjusted so that the following conditions are met:

when implemented, the filter is “locked”, as a result of which the output sequence of the filter y (n) becomes zero.

The selection of the coefficients a ₁ and a ₂ as follows.

In accordance with the transfer function (9), the output digital sequence of the filter:

Dispersion y (n) (see Marple M.L. and Marple S.L. Digital spectral analysis and its applications. // Transl. From English. M., Mir, 1990):

where: N is the total number of samples of the signal.

To determine the coefficients, we have two conditions:

,

,

which lead to a system of N-2m linear equations solved by the least squares method.

The result is (see Hudson K. Statistics for physicists. // M., Mir, 1970, p. 152):

where: a is the column vector of the desired coefficients:

;

;

φ, s are one-column and two-column matrices:

Τ is the matrix transpose operator.

The roots of the characteristic equation of the filter:

Where:

The filter locking frequency is equal to the oscillation frequency of the pendulum:

The angle α included in expression (16) is determined by the coefficients a ₁ and a _{2 of the} digital filter.

Since the coefficients contain measurement errors, the frequency ω ₁ will also be measured with an error.

RMS error of frequency measurement:

и

- среднеквадратические погрешности коэффициентов а₁ и а₂.Where:

and

- standard errors of the coefficients a ₁ and a ₂ .

Performing the differentiation of equations (18), after simple transformations get:

- погрешность достигает минимума:At

- the error reaches a minimum:

and

, получим:Substituting the optimal value in (16)

we get:

;where: ƒ _d - signal sampling frequency:

;

.ƒ ₁ - oscillation frequency of the pendulum:

.

Attenuation coefficient β

In some cases, it is necessary to know the attenuation coefficient β. This need arises when measuring the amplitude of the angle of rotation.

From equation (15):

substitute in equation (22):

,

,

and making logarithms, after simple transformations get:

Angular acceleration sensor conversion coefficient

The source of the output signal of the remote control is the strain gauge bridge. The output signal of the bridge, as usual, is normalized in relative units.

,In our case:

,

- относительный выходной сигнал датчика;Where:

- relative output signal of the sensor;

ΔU is the signal voltage;

U _n is the supply voltage of the bridge.

,There is a linear relationship between the signal voltage and angular acceleration, which is confirmed by the static calibration of the sensor:

,

where: s is the conversion coefficient of the sensor.

To determine the conversion coefficient, it is necessary to have a reference signal that would coincide in shape with an angular acceleration signal (8) (see Helstrom K. Statistical Theory of Signal Detection. // M., Publishing House of Foreign Literature, 1963) A signal is taken as the reference angle of rotation in equation (7).

Dispersion of the signal difference taking into account discrete time:

.

.

, находят:Assuming

find:

Regarding expression (24), the following remarks should be made.

, и сигналы с точностью до знака совпадают по форме.The fact is that the rotation angle and acceleration signals differ in phase. In the first case, the phase angle is + Ψ, and in the second -Ψ. As β → 0 tends, the phase angle in accordance with equation (7) tends to

, and the signals coincide up to the sign in shape.

и подсчитывают зависимость δ% от коэффициента затухания β.The error in the measurement of the coefficient s due to the mismatch of the waveform is denoted by

and calculate the dependence of δ% on the attenuation coefficient β.

The calculation results are shown in table No. 1.

From table No. 1 we find that when:

the error from the mismatch of the waveform can be neglected.

At the same time, condition (25) ensures the accuracy of the frequencies ω ₀ and ω ₁ with an accuracy of 0.005%, and, as a result, the frequency ω0 ceases to depend on the damping coefficient β, which, in turn, depends on the ambient temperature, pressure, and other factors.

Amplitude of rotation angle

Discrete angle signal:

Given the smallness of the damping coefficient of oscillations β, we can write:

Where:

Signal difference dispersion:

.

.

, находим:Assuming:

we find:

Pendulum reproduction of linear acceleration

When installing the remote control at a distance l _c from the axis of rotation of the pendulum on the remote control, in addition to angular acceleration, linear acceleration also acts. The sensitivity of the sensor to linear acceleration is explained by a certain residual imbalance of the inertial mass, made in the form of a rim mounted on an axisymmetric tensometric element. The pendulum allows, if necessary, to balance the remote control system and thereby reduce its sensitivity to linear acceleration to an acceptable minimum.

To this end, the axis of the remote control system is initially combined with the axis of the pendulum, where only angular acceleration acts and there is no linear acceleration.

After that, the DUU and DBP are interchanged, i.e. axis DUU will be at a distance l _c from the axis of rotation of the pendulum.

In the General case, the linear acceleration vector at a point offset by a distance l _c from the axis of rotation:

where: u is the velocity vector of the point;

τ and ν are unit mutually perpendicular vectors.

The first term in equation (27) represents centripetal acceleration. When the sensor oscillates with a circular frequency ω ₀ , centripetal acceleration has a frequency of 2ω ₀ and is not taken into account when processing the sensor signal.

The second term is tangential acceleration with amplitude:

The pendulum most fully simulates the operation of the remote control in the bench system when measuring moments of inertia, while the calibration accuracy increases if the frequency and amplitude of the vibrations at which it is carried out is close to or coincides with the expected frequency of vibrations of the corresponding frames of the stand when measuring the moments of inertia of the product.

At existing stands, the maximum distance of the angular acceleration sensor from the axis of rotation of the product is l≅0.5 m.

When φ _m = 8 °; ƒ ₀ = 0.8 Hz, a _τ = 1.76 m / s ² or a _τ = 0.18 g.

Knowing the sensor readings under the action of only angular acceleration and the sensor readings under the action of additional linear acceleration, when making measurements at the bench, corrections are introduced to obtain the effective values of the angular acceleration with high accuracy.

Claims (1)

The method of calibrating the angular acceleration sensor, namely, that the angular acceleration sensor is mounted on the shaft, subjected to the action of known angular acceleration, and by processing the output signal of the sensor find the conversion coefficient of the angular acceleration into an electrical signal, characterized in that the angular acceleration sensor is installed on the pendulum, its axis is combined with the axis of rotation of the pendulum, and the axis of the dynamic analogue of the sensor with the point of the pendulum, which is a predetermined distance from its axis of rotation, the pendulum deviates dissolved in acceptable condition for isochronous angle φ _{(t = 0)} ≤8 ° dispensers produce discrete samples pickoff signals α _(n) and the angular acceleration

, from discrete samples of the angle sensor signal, the coefficients of a digital notch filter are calculated, which are used to determine the natural circular frequency of the oscillations of the pendulum:

,
where: ω ₁ is the natural circular frequency of the oscillations of the pendulum;

- the angle of rotation of the eigenvector of the characteristic equation;
and ₁ and a ₂ are the coefficients of the digital filter;

- the optimal integer number of delay steps of the signal in the digital filter;

- signal sampling rate;

- damping coefficient of oscillations;
the discrete samples of the signals of the angle and angular acceleration sensors calculate the conversion coefficient of the angular acceleration into the output electrical signal of the sensor:

,
where: s - conversion coefficient, dimension

;

- intrinsic undamped oscillation frequency;
at the same time, the limitation coefficient of the oscillation pendulum is limited by β≤0.01, after which the angular acceleration sensor and the dynamic analog of the sensor are interchanged and the conversion coefficient of the sensor is again determined under the conditions of the action of angular acceleration and the linear acceleration vector on it.

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