RU2310178C1 - Device for determining value and phase of unbalance - Google Patents

Abstract

FIELD: testing engineering.

SUBSTANCE: device comprises vibration system and measuring system for determining three subsequent time intervals of the rotor rotation. The rotor is set on the rocking platform. The three subsequent time intervals for determining the value and phase of unbalance are measured during forced vibration of the rotor together with the platform.

EFFECT: simplified design and balancing procedure.

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Description

The invention relates to balancing equipment and can be used in the construction of balancing machines.

A known method for determining the magnitude and phase of the imbalance is to determine three consecutive time intervals, which are used to calculate the rotor rotation angles proportional to them, giving a total revolution, and the actual magnitude and phase of the unbalance are calculated (Patent No. 2237878. BI 28 from 10/10/2004).

The device provided in the method for its implementation has the disadvantage that its natural frequency directly depends on the mass of the balanced rotor, which complicates the balancing procedure, because dynamic characteristics of the balancing stand change.

The purpose of the invention is to simplify the design and balancing procedure.

The goal is achieved due to the fact that in the oscillation system of the stand, the elastic forces that restore the equilibrium position are replaced by the weight component of the weight of the balanced rotor, which in the proposed pendulum design is always directed towards equilibrium, and three consecutive time intervals are measured during the execution of the forced swinging pendulum swinging rotor fluctuations. The natural frequency of linear oscillations of such a physical pendulum is practically independent of the mass of the balanced rotor.

The analysis of the prior art by the applicant, including a search by patent and scientific and technical sources of information, made it possible to establish that the applicant has not found an analogue characterized by features identical to all the essential features of the claimed invention.

Therefore, the claimed invention meets the requirement of "novelty" under applicable law.

The mechanical design of the device operates on the principle of a pendulum that performs forced oscillations (figure 1). The platform 1 with the balanced body 2 is suspended on two identical rigid rocker arms 3 and performs forced oscillating oscillations with an amplitude φ _m proportional to the imbalance under the influence of the imbalance. The balanced body is untwisted up to a given angular velocity ω by a drive through a belt drive 4.

The measuring system of the device is formed by an optical switch of the final position 5, located motionless at a distance r from the axis of rotation of the balanced body and triggered by the interruption of the light flux, as well as two mutually perpendicular opaque lines X ₁ and Y ₁ on the transparent disk 6. The disk is mounted and fixed on one axis with a balanced body and rotates with it with an angular velocity ω. Axis Y ₁ has a positive dimension larger than the opposite. This larger size is such that it can cross the optical axis of another limit switch 7, the output signal of which is used to turn on the sensor 5, which starts the timer for measuring time intervals Δt _i (i = 1, 2, 3). The opposite short side of the axis Y ₁ and the axis X ₁ cannot cross the optical axis of the switch 7. Measure the time intervals between successive intersections of the optical axis 9 with the movable axes X ₁ and Y ₁ .

At an arbitrary point in time, the end switch 7 is turned on and only the elongated part of the coordinate axis Y ₁ crosses its optical axis, thereby including the end switch 5. Moving further, the Y ₁ axis intersects the optical axis 8 of the switch 5, the signal of which starts the timer for measuring the time interval Δt ₁ . With further movement of the movable system X ₁ Y _{1, the} optical axis 8 crosses the coordinate axis X ₁ and stops the time count. The first time interval Δt _{1 is} obtained. Simultaneously with the stop of the account, a new time count starts and so on. The result is three consecutive intervals Δt ₁ , Δt ₂ , Δt ₃ , which uniquely determine the magnitude and phase of the imbalance.

In total, the intervals give the time of a complete revolution of the rotor, which allows you to indirectly measure and control the rotation speed ω, without involving an additional sensor:

ω = 2π / (Δt ₁ + Δt ₂ + Δt ₃ ).

A further algorithm of actions is aimed at implementing the well-known method of maximum marks [1]. According to this method, it is necessary to know the radius vector of displacement from the initial imbalance, the radius vector of displacement from the combined effect of unbalance with the test load and the angle γ between these vectors.

Since the oscillatory system of the device is close to linear, the values of these radius vectors are proportional to the amplitude of the angular oscillations φ _{m of the} rocker arm 3 (Fig. 1), which perform forced harmonic oscillations according to the law:

φ (t) = φ _m cosωt.

Here, the beginning of the process of moving a platform with a balanced body from the extreme right position is conditionally relied on. This is the moment t = 0. The calculation of the value of φ _m and the angle γ at known intervals Δt _i is based on the transformation of the coordinates of the position of the optical axis 8 from a fixed coordinate system to a moving one. To do this, we introduce a fixed system OXY with the beginning at the point O of the middle position of the equilibrium of the platform (figure 2). The optical axis 8 (point D) of the switch 5 in this fixed system has coordinates: along the X axis - 0, along the Y axis - r.

If the origin of the moving coordinate system X ₁ Y ₁ in the fixed XY has coordinates (x ₀ , y ₀ ) and, in addition, the system X ₁ Y _{1 is} rotated by an angle α relative to XY, then the connection between the coordinates (x, y) of the point D in fixed and mobile (x ₁ , y ₁ ) systems is defined as follows (This section is for explanation only). With the translational movement of the moving system (without rotation through the angle α), the coordinates of the point D in the moving system are specified by the vector: (x-x ₀ , yy ₀ ) ^T. When the movable system rotates through an angle α, the coordinate transformation is specified by the matrix [2]

The total transformation of the coordinates of the point D from fixed to moving in matrix form has the form:

We define the values of the coordinates x, x ₀ , y, y ₀ , α in the matrix expression. If there were no friction forces in the oscillatory system, then the maximum displacement of the platform and the maximum value of the force that caused this displacement would coincide in phase. This means that in the extreme right position, when φ (t) = φ _m , the projection of the centrifugal force vector F _c = mρω ² rotating with frequency ω on the direction of the tangent L to the platform trajectory has a maximum value. It follows that the centrifugal force vector F _i to the rightmost position would be directed along the tangent at the point L N (2). However, friction forces are present and this leads to a lag in the phase of the displacement from the force. As a result, the force vector F _c outstrips the direction of the tangent L at the point N in phase. Since the moving coordinate system X ₁ Y _{1 is} rigidly connected with the vector F _c , it will occupy a certain angular position defined by the force F _c in its extreme initial position. Then the direction vector of the tangent at the point N will also occupy a certain angular position in the system X ₁ Y ₁ . Let it be the angle β. In this position, the movable system X ₁ Y _{1 is} rotated relatively stationary by an angle (φ _m −β).

Moving from the initial position, the movable system X ₁ Y ₁ after a certain time interval t ₀ intersects the axis Y _{1 of the} optical axis of the sensor 5 (point D). The deviation angle φ _{0 of} links 3 at this moment is equal to: φ ₀ = φ _m · cosωt ₀ , and the coordinates of the beginning of the moving system X ₁ Y ₁ in the fixed XY are:

x ₀ = Rsin (φ _m cosωt ₀ ); y ₀ = R (1-cos (φ _m cosωt ₀ )).

The rotation angle α of the system X ₁ Y ₁ relative to XY in this position is: α = (φ _m -β + ωt ₀ ). The coordinates (x, y) of the point D in the fixed system are (0, r), i.e.

In the moving system X ₁ Y ₁ point D has the coordinates:

Substituting this data into matrix equality, we obtain:

Необходимо отметить, что матричному уравнению соответствуют два алгебраических уравнения, каждое из которых выражает теорему синусов для ΔCDN₀. Поэтому уравнения взаимозависимы и достаточно рассмотреть какое-то одно из них. Берется наиболее простое с нулевой левой частью. В этом матричном уравнении величины β, φ_m, t₀ являются неизвестными параметрами и подлежат определению. Раскрывая матричное уравнение, получим:

It should be noted that the matrix equation corresponds to two algebraic equations, each of which expresses the sine theorem for ΔCDN ₀ . Therefore, the equations are interdependent and it is enough to consider one of them. The simplest with the zero left side is taken. In this matrix equation, β, φ _m , t ₀ are unknown parameters and must be determined. Opening the matrix equation, we obtain:

Rsin (φ _m -β + ωt ₀ -φ _m cosωt ₀ ) - (Rr) sin (φ _m -β + ωt ₀ ) = 0.

After the time interval Δt _{1 has} passed, the point D will intersect the axis X ₁ , and its coordinate along the axis Y ₁ will be 0. At that, the system X ₁ Y ₁ will rotate through the angle α ₁ = (φ _m -β + ω (t ₀ + Δt ₁ )). The deviation of the rocker arm 3 will make the angle φ ₁ = φ _m · cosω (t ₀ + Δt ₁ ). The matrix expression of the coordinate transformation will take the form:

Opening it, we obtain the second equation for β, φ _m , t ₀ :

Rcos (φ _m -β + ω (t ₀ + Δt ₁ )) - φ _m cosω (t ₀ + Δt ₁ )) = (Rr) cos (φ _m- β + ω (t ₀ + Δt ₁ )).

Further, after a time interval Δt _{2, the} axis Y ₁ crosses the point D with a short "negative" direction. This will be the coordinate -y ₁ ^D. The coordinate along the X ₁ axis is 0. The angle of deviation of the rocker arm 3 is φ ₂ = φ _m cosω (t ₀ + Δt ₁ + Δt ₂ ), and the rotation angle α _{2 of the} moving coordinate system with respect to the fixed one is:

α ₂ = (φ _m -β + ω (t ₀ + Δt ₁ + Δt ₂ )).

The matrix equation has the form:

Opening it, we obtain the third equation for the unknown β, φ _m , t ₀ :

Rsin (φ _m -β + ω (t ₀ + Δt ₁ + Δt ₂ ) -φ _m cosω (t ₀ + Δt ₁ + Δt ₂ )) = (Rr) sin (φ _m -β + ω (t ₀ + Δt ₁ + Δt ₂ )).

A system of three nonlinear equations is obtained:

Rsin (φ _m -β + ωt ₀ -φ _m cosωt ₀ ) - (Rr) sin (φ _m -β + ωt ₀ ) = 0;

Rcos (φ _m -β + ω (t ₀ + Δt ₁ ) -φ _m cosω (t ₀ + Δt ₁ )) = (Rr) cos (φ _m -β + ω (t ₀ + Δt ₁ ));

Rsin (φ _m -β + ω (t ₀ + Δt ₁ + Δ ₂ ) -φ _m cosω (t ₀ + Δt ₁ + Δt ₂ )) = (Rr) sin (φ _m -β + ω (t ₀ + Δt ₁ + Δt ₂ )).

The system of equations can be solved, for example, by the Newton method or any of its modified method [3].

From the solution we find t ₀ , φ _m and β. In this case, the sign of β can be both positive and negative. If β> 0, then the tangent is in the first or second quadrants, and if β <0, then in the third or fourth. In addition, the value of the interval Δt ₃ (the time of approximately half a revolution) can serve as a refinement for the position of the tangent in the coordinate system X ₁ Y ₁ . If Δt ₃ ≥T / 2 = (Δt ₁ + Δt ₂ + Δt ₃ ) / 2, where T is the time of one revolution of the rotor, then this is the first or fourth quadrant. If Δt ₃ ≤T / 2 = (Δt ₁ + Δt ₂ + Δt ₃ ) / 2, then the second or third. Thus, the actual values of β and Δt ₃ uniquely determine the position of the tangent in the coordinate system X ₁ Y ₁ .

A further algorithm for determining the coordinates of the imbalance repeats the method of maximum elevations [1].

The value of φ _{m is} proportional to the imbalance mρ, if the rotor speed is the same. In addition, the phase lag of the direction vector of the tangent L from the centrifugal force vector F _c = mρω ² in the initial, “rightmost” position will also be the same value. Therefore, the angle γ between the vectors of centrifugal forces from the initial imbalance F _c1 and the imbalance with the test load F _C2 is the angle between the tangents corresponding to these forces, i.e.:

γ = β ₁ -β ₂ .

If φ _m1 and φ _m2 are the oscillation amplitudes of the rocker arms corresponding to the forces F _c1 and F _c2 and treated in the same way as vectors, and γ is the angle between them, then by the cosine theorem we can find the amplitude φ _m3 that would correspond only to the test load :

By the sine theorem in the vector triangle φ _m1 , φ _m2 , φ _m3 we find the angle ψ, by which it is necessary to shift the test load in order to balance the imbalance:

,

,

ψ = arcsin {(φ _m2 / φ _m3 ) sinγ}.

In this method, the absolute error of measuring time intervals Δt _{i by the} photoelectric method is 10 ^-7 s [4], which leads to an error in determining ψ of the order of 5 arc seconds.

Information sources

1. Steinwolf L.I. Dynamic calculations of machines and mechanisms. M.: Mechanical Engineering, 1961, 340 p.

2. Efimov N.V. Quadratic forms and matrices. M .: Nauka, 1975, 159 p.

3. Demidovich B.P. Fundamentals of Computational Mathematics. M .: Maron I.A. 1963, 659 p.

4. Kiselev M.I. et al. Measurement of the period of rotation of the turbine shaft shafting by the photoelectric method. // Measuring technique. 1996. No. 12.

Claims (1)

A device for determining the magnitude and phase of the imbalance, consisting of oscillatory and measuring systems for determining three consecutive time intervals of rotation of the rotor, characterized in that the action of balancing forces is provided by installing the rotor on a swinging platform, and three consecutive time intervals for determining the magnitude and phase of the imbalance are measured in the process of forced oscillations of the rotor with the platform.

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