RU2658125C1 - Method for determining parameters of natural tones of structure vibrations in resonant tests - Google Patents

Abstract

FIELD: metrology.

SUBSTANCE: invention relates to metrology, in particular to resonant tests of mechanical structures. Method for determining the parameters of the natural tones of structural vibrations consists in isolating one's own tones by the phase resonance method by using a multichannel system of excitation and measurement of oscillations, determining the frequencies of phase resonances, representation of the oscillations of the structure for each of its own tones by a linear oscillator, characteristics of which are the generalized excitation force of oscillations, generalized mass, generalized damping and generalized rigidity of the corresponding tone, the definition of generalized masses, generalized damping and generalized stiffnesses of tones by the vibrational response of the structure. By the relationship between the forced monophasic and intrinsic oscillations, the dissipative properties of the structures are revealed, the definition of the generalized parameters of each characteristic vibration tone is made by the amplitude-frequency characteristic of the construction in the vicinity of the phase resonance frequency corresponding to this tone from the condition that the generalized excitation forces of the structure and a linear oscillator that realize the same oscillation amplitude of the structure and oscillator.

EFFECT: increase the accuracy of determining the generalized masses and the damping characteristics of natural vibration tones.

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Description

The invention relates to the field of testing equipment, in particular to resonance testing of mechanical structures.

In resonance tests, such characteristics of eigentones of structural vibrations as eigenfrequencies, shapes, generalized decrement of vibration, and generalized mass are determined. These dynamic characteristics are used to verify the design models of objects in mechanical engineering, aviation, and the space industry. The tests are carried out by the method of multipoint excitation of oscillations, which is based on the concept of phase resonance. The essence of the method: with the help of a controlled action, a large number of forces alternately distinguish structural vibrations for each own tone with the subsequent determination of the generalized characteristics of these tones.

The known energy method for determining the generalized mass and the generalized decrement of oscillations, which consists in equating the work of the forces of inelastic resistance to the work of the excitation forces during the period of oscillation. In this case, the work of damping forces is recorded for a specific model of energy dissipation and is expressed in terms of the decrement of vibrations and the generalized mass. Determining the decrement by known methods, we can calculate the generalized mass. And vice versa, if the generalized mass is known, then the energy method is used to determine the decrement of vibrations (Clerc, D. Metode de recherche des modes propres par calcul de l 'excitation harmonique optimum d' apres les res les resultats bruts d 'essais de vibrations / D . Clerc // Note technique: ONERA. - 1967. - No. 119. - 57 p.).

The disadvantages of the method are the high sensitivity to errors in the experimental data and a priori adoption of the energy dissipation model.

A known method of determining the generalized masses by the method of mechanical loading. The method involves evaluating the generalized tone mass by changing its natural frequency as a result of installing additional masses at the facility (Mikishev G.N. Dynamics of thin-walled structures with compartments containing liquid / G.N. Mikishev, B.I. Rabinovich. - M .: Mechanical Engineering , 1971. - 564 p.).

The disadvantage of this method is the high complexity, significant sensitivity to measurement errors in the test results and the assumption that the Basil hypothesis that the inertia, stiffness and damping matrices are reduced to a diagonal form by a single coordinate transformation is feasible.

A known method for determining the generalized characteristics of the natural tones of the oscillation method by selecting an approximating circle. It is based on the fact that the frequency response of a system with one degree of freedom describes a circle in the Nyquist diagram. To implement this method, a set of points of the frequency characteristic is allocated near the frequency of the phase resonance through which the approximating circle is drawn. After that, the natural frequency of the tone is found as a point with the maximum rate of change of the angle between the experimental values of the frequency response. The circle parameters are also the initial data for determining the damping coefficient (Heilen Ward. Modal analysis: theory and testing / Ward Heilen, Stefan Lammens, Paul Sas. - M .: Novatest LLC, 2010. - 319 p.).

The disadvantages of the method are:

- the absence of a criterion for substantiating the choice of the frequency range of forced oscillations in the vicinity of the resonant frequency in which the approximating circle is built;

- a priori use of the Basil hypothesis;

- a sufficiently high sensitivity to measurement errors of forced oscillation parameters. This is explained by the fact that in-phase and quadrature components of displacements are used to construct the approximating circle. But in the vicinity of the phase resonance, the in-phase component is small, therefore, it can be measured with a significant error.

Known methods and stands for determining the natural frequencies and generalized masses of oscillating structures, which differ from those described above in that they allow you to determine the mass of the test object without errors made by test equipment (Patents RU 2499239, RU 2489696, RU 2485468). But at the same time, the shortcomings of the method of determining generalized masses inherent in the energy method are not eliminated.

A known method for determining the generalized masses and decrements of oscillations by introducing the quadrature component of the excitation. To implement the method, vibration sensors (displacements, speeds, accelerations) are installed on the structure under study. Excitation of structural vibrations is carried out using several independent sources of vibration. Own structural tones are distinguished in a given frequency range by the phase resonance method by using a multi-channel excitation system and measuring vibrations. The eigenfrequencies of tone oscillations are determined from the frequencies of the phase resonances. The generalized characteristics of tones are calculated by changing the frequency of the phase resonance (the implementation of the modes of “fictitious” phase resonance) after the introduction of the quadrature component of the excitation (Vasiliev, KI. Experimental study of the elastic vibrations of aircraft using multichannel equipment AVDI-1N / K.I. Vasiliev, V.I.Smyslov, V.I. Ulyanov // Trucks of TsAGI named after N.E. Zhukovsky. - 1975. - Issue 1634. - S. 1-36.).

This method of determining the parameters of the intrinsic vibrations of structures in modal tests is selected as a prototype.

The disadvantages of the method are:

- the absence of a criterion for justifying the choice of the frequency range of forced oscillations in the vicinity of the resonant frequency, in which the modes of "fictitious" phase resonance are realized. Since the generalized characteristics are calculated from the difference in the frequencies of the phase resonance, each of which is determined with a certain error, with a small difference the error in the generalized characteristics can be significant, and with the removal of the frequency of the “fictitious” phase resonance, neighboring tones can influence the calculation results;

- a priori use of the Basil hypothesis;

- the mathematical expectation of the generalized mass determined by this method is a biased estimate, and this bias increases as the frequency of the “fictitious” phase resonance approaches the frequency of the phase resonance.

An object of the invention is to increase the accuracy of determining the generalized masses and damping characteristics of intrinsic vibrations of structures.

To achieve the technical result of the invention, in a method for determining the parameters of eigentones of oscillations, which consists in extracting eigen tones by the phase resonance method by using a multi-channel excitation and measuring system, determining the frequencies of the phase resonances, representing structural oscillations for each eigen tone by a linear oscillator, the characteristics of which are generalized force excitation of oscillations, generalized mass, generalized damping and generalized stiffness with corresponding tone, determination of generalized masses, generalized damping and generalized stiffnesses of tones according to the vibrational response of the structure, the relationship between forced monophasic and natural vibrations reveals the dissipative properties of the structures, the generalized parameters of each natural tone of vibration are determined by the amplitude-frequency characteristic of the structure in the vicinity of the corresponding tone of the phase resonance frequency from the condition of minimum differences in the generalized excitation forces design and a linear oscillator, realizing the same amplitudes of the oscillations of the structure and the oscillator.

In FIG. 1 shows the results of evaluating the relative error in determining the generalized mass ε _a for a different number of M points, amplitude-frequency characteristics (AFC);

in FIG. 2 - dependence of the parameter λ on the oscillation frequency;

in FIG. 3 - experimental (1) and calculated (2) amplitude-frequency characteristics of the frequency response.

The method for determining the parameters of the eigentones of vibration of structures in resonance tests is as follows. Sensors of vibrations (displacements, speeds, accelerations) are installed on the studied structure. The number of sensors depends on the size and degree of complexity of the design. The places of installation of the sensors are assigned so that it is possible to describe with the required detail the eigenmodes of vibration of the structure in a given frequency range. To do this, use the results of the calculated modal analysis performed during the design design.

Excitation of structural vibrations is carried out using several independent sources of vibration. The same requirements apply to the points of attachment of vibration sources to the structure as to the places where vibration sensors are installed. Sources of vibration create an exciting force that varies in harmonic law.

To describe the methodology for selecting the excitation forces in order to isolate the oscillations for each eigen tone, to determine the eigenfrequencies and shapes, we use the differential equations of the structural oscillation during the tests:

и

матрицы инерции и жесткости; R(N) - вектор сил демпфирования, к которым отнесены все силы, изменяющиеся в фазе со скоростью перемещений конструкции; E(N) и F(N) - векторы синфазной и квадратурной составляющих сил возбуждения, N - число собственных тонов колебаний конструкции в заданном частотном диапазоне.Here Z (N) is the displacement vector of construction points; ω is the frequency of the exciting force;

and

matrices of inertia and rigidity; R (N) is the vector of damping forces, to which are attributed all the forces that change in phase with the speed of movement of the structure; E (N) and F (N) are the vectors of the in-phase and quadrature components of the excitation forces, N is the number of intrinsic tones of the vibrations of the structure in a given frequency range.

The steady forced vibrations of the structure in the tests are of the form

where U (N) and V (N) are the in-phase and quadrature components of the structural displacements.

The concept of “forced monophasic oscillations” is used that satisfy the condition

where λ is a real number equal to the cotangent of the phase shift between the displacements of the structure and the real component of the excitation. For single-phase oscillations (3), differential equations (1), taking into account (2), correspond to systems of algebraic equations

- матрица демпфирования.Here

- damping matrix.

,

, i, j=1, 2,…, N, i≠j, следует, что если при этом монофазные колебания совпадают с собственными колебаниями на частотах, отличающихся от частот фазовых резонансов, то матрица демпфирования в обобщенной системе координат диагональная:

, i, j=1, 2,…, N, i≠j. Параметрами таких собственных тонов колебаний являются обобщенная масса, обобщенное демпфирование и обобщенная жесткость (собственная частота колебаний).From (4) it follows that with monophasic excitation (F = 0) and λ = 0 (phase resonance), monophasic oscillations coincide with the natural oscillations: V = W _i , i = 1,2, ..., N; W _i - i-th eigenvector (waveform). In addition, from (4) and (5), taking into account the conditions of orthogonality of the eigenvectors

,

, i, j = 1, 2, ..., N, i ≠ j, it follows that if in this case the monophasic oscillations coincide with the natural oscillations at frequencies different from the frequencies of the phase resonances, then the damping matrix in the generalized coordinate system is diagonal:

, i, j = 1, 2, ..., N, i ≠ j. The parameters of such natural vibration tones are generalized mass, generalized damping and generalized stiffness (natural vibration frequency).

The force vector E of monophasic excitation, which implements monophasic oscillations at a frequency ω, is determined by the formula

if there are real solutions to the eigenvalue problem

- матрица, столбцами которой являются векторы линейно независимых сил в N предварительных испытаниях; столбцы матриц

и

- векторы синфазных и квадратурных составляющих перемещений, зафиксированные в этих испытаниях.Here

- a matrix whose columns are linearly independent force vectors in N preliminary tests; matrix columns

and

- vectors of in-phase and quadrature components of displacements recorded in these tests.

If the number of excitation forces is L <N, then monophasic oscillations reproduce approximately from the condition

and the vector ξ in (6) is defined as an eigenvector in the problem

,

,

corresponding to the smallest eigenvalue α.

,

.Here

,

.

Condition (7) defines the parameter λ as

.

.

In tests, changing the frequency of oscillations ω in a given range, determine the frequency of the phase resonances of the structure according to the criteria

или

,

or

,

which follow from condition (7). The frequencies of the phase resonances are equated to the eigenfrequencies of the oscillation tones p _i , and the vectors V at these frequencies are taken for the eigenvectors W _i , i = 1, 2, ..., N.

So, natural frequencies and forms are found. The next task is to represent the oscillations of each tone by a linear oscillator and to determine the generalized masses and damping characteristics.

With constant forces and a step change in the frequency of oscillations in the vicinity of the natural frequencies, in-phase U and quadrature V components of the displacements at the control points are measured. Based on the measurement results, the frequency dependences of the parameter λ are built. If these dependences are the same at all control points and the shape of the vibrations does not change, then the vibrations of the structure for each own tone are described by the equation (linear oscillator):

; y - амплитуда, а

- фаза колебаний; a, h и с - обобщенные масса, демпфирование и жесткость; с=р²а, Q=W^TE - обобщенная сила внешнего воздействия. В качестве обобщенных координат g принимают перемещения точек нормирования собственных векторов.Here

; y is the amplitude, and

- phase of oscillation; a , h and c - generalized mass, damping and stiffness; c = p ² a, Q = W ^T E is the generalized external force. As the generalized coordinates g take the displacement of the normalization points of the eigenvectors.

, определяющих амплитуды колебаний осциллятора, равные амплитудам колебаний конструкции на частотах ω_к, (к=1, 2,…, М, М≥3):To determine the generalized mass, stiffness, and tone damping, the condition of the minimum difference between the generalized forces Q _k realized in the experiment and the forces is used

that determine the oscillation amplitudes of the oscillator, equal to the amplitudes of the structural vibrations at frequencies ω _k , (k = 1, 2, ..., M, M≥3):

.

.

The necessary condition for the extremum of the functional leads to a system of third-order nonlinear equations with respect to generalized characteristics. This system of equations can be solved exactly:

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

A change in the method for describing the damping forces in equation (8) does not lead to another solution for the generalized tone mass.

The natural tone frequency, which is determined independently and with high accuracy in modal tests, is not involved in this method of estimating generalized parameters. Therefore, the natural frequency is used to control the reliability of generalized masses and generalized stiffnesses.

Methods for determining parameters from the test results are checked for sensitivity to measurement errors in the experimental data. The most sensitive to measurement errors in modal tests are the methods for determining the generalized masses. The influence of random measurement errors on the accuracy of calculating the generalized masses by the proposed method was estimated by the method of statistical modeling. We studied the influence of only measurement errors of vibration amplitudes, since modern equipment allows us to maintain the frequency of forced vibrations with high accuracy (up to hundredths of a percent), and deviations in the given values of the excitation forces can (in the case of linear systems under consideration) be taken into account in the accuracy of determining displacements.

Errors in measuring the amplitudes of oscillations were considered distributed according to the normal law with zero mathematical expectation. In FIG. Figure 1 shows the results of evaluating the relative error in determining the generalized mass ε _a for a different number M of points of the amplitude-frequency characteristic (AFC) involved in calculating the mass. It is seen that the generalized mass can be calculated with high accuracy even with large errors in measuring the vibration amplitudes ε _y , if we take into account a sufficient number of frequency response points in the calculation. So at ε _y = 10%, the value ε _a = 5% is achieved at M = 9, and at M = 20-ε _a = 2.5%. The low sensitivity of the proposed method is explained by the fact that the calculation of the parameters of the natural vibration tones is performed according to the vibration amplitudes of the structure, which are maximum near the phase resonance frequencies and can be measured with sufficient accuracy.

So, the generalized masses and the generalized coefficients of damping tones are determined if the oscillations in their own tones are described by linear oscillators.

If in the vicinity of the phase resonance frequency the monophasic oscillations do not coincide with the natural ones, then the tone damping cannot be described by a generalized coefficient, and nonmonophasic excitation is used to determine the generalized mass. The quadrature component of the excitation is introduced from the condition that monophasic oscillations coincide with their own in the vicinity of the phase resonance frequency:

,

.

,

.

и матрица

определены на частоте фазового резонанса. Обобщенную массу тона рассчитывают по формуле:Here is the vector

and matrix

determined at the phase resonance frequency. The generalized mass of the tone is calculated by the formula:

.

.

Regarding the damping properties, the existence of a connection between the matrices A, C, and H is noted:

.

.

,Besides,

,

матрица демпфирования симметрична.and in the case of matrix symmetry

the damping matrix is symmetrical.

A device for implementing this method of determining the generalized characteristics of the intrinsic vibrations of structures is a traditional multi-channel system of excitation and measurement of vibrations.

The reliability of the generalized characteristics of the natural vibration tones determined by the proposed method is confirmed by the results of modal tests of a dynamically similar model of the Tu-334 aircraft and full-scale products Su-30 and Yak-152.

In FIG. Figure 2 shows an example of the dependences of the parameter of monophasic oscillations λ for several points of the Su-30 product in the vicinity of the phase resonance frequencies. In this case, the forms of monophasic vibrations are preserved and coincide with their own forms of vibrations.

In FIG. Figure 3 shows an example of experimental frequency response and calculated frequency response, based on the generalized characteristics of the natural oscillation tones.

The test results are also presented in table 1, in the first column of which are given the conditional number of tones, and Δp determines the difference between the calculated and experimental natural frequencies of vibrations.

The obtained results showed that the proposed method allows to determine the generalized masses, generalized stiffness and damping characteristics of the natural vibration tones of the test objects with high accuracy.

Claims (1)

A method for determining the parameters of eigentones of structural vibrations, which consists in extracting eigentones by the phase resonance method by using a multichannel excitation system and measuring vibrations, determining the phase resonance frequencies, representing the structural vibrations of each eigentone by a linear oscillator, the characteristics of which are the generalized excitation force of oscillations, the generalized mass , generalized damping and generalized stiffness of the corresponding tone, definition of generalized masses, generalized damping and generalized stiffnesses of tones according to the vibrational response of the structure, characterized in that the ratio between the forced monophasic and natural vibrations reveals the dissipative properties of the structures, the generalized parameters of each natural tone of vibration are determined by the amplitude-frequency characteristic of the structure in the vicinity of the corresponding tone the frequency of the phase resonance from the condition of a minimum difference between the generalized excitation forces of structural vibrations and the line oscillator, realizing the same oscillation amplitudes of the structure and the oscillator.

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