RU2448336C2 - Method of determining body mass and centre of mass coordinates in given plane - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: radius vector of displacement of a platform in a centrifugal force field is measured for three different positions of mass distribution on the platform: the analysed body, the carriage and the load balancing the carriage. The mass of the body and the coordinates of the centre of mass in the plane of the carriage are determined from the measured radius vector, mass and displacement of the carriage.

EFFECT: invention widens the range of measured objects and increases measurement accuracy.

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Description

The purpose of the invention is the expansion of the range of objects and improving the accuracy of measurement.

A known method of determining the mass, coordinates of the center of mass and the central moment of inertia of the body in a given plane from the values of the period of natural torsional vibrations, measured for different positions of the body on the platform. According to the values of the period, the center of mass, the coordinates of the center of mass and the central moment of inertia of the body are calculated (ed. St. No. 1046633, BI No. 37, 1983).

The disadvantage of this method is not to take into account the ability of the test sample mounted on the platform to have vibrational properties. The elasticity and inertness of the structural elements of the sample give it the properties of an independent torsional vibrational system attached to a platform that performs its own torsional vibrations. As a result of the interaction of oscillatory systems, the law of motion of the platform as a sensitive element of the measuring system is distorted.

In addition, the time interval to be measured is a value in which the frequency composition of the total torsional vibrations of the platform is hidden. It is impossible to separate (filter) the vibrations of the platform and the own torsional vibrations of the sample on the platform. Their mutual influence leads to dynamic measurement errors of time intervals. Samples possessing the indicated properties may include thin-walled hull parts, spatial truss structures, and objects of zoological nature. The inability to adjust the dynamic loads acting on the sample during the measurement can lead to deformation of the spatial forms of the sample itself. In general, these are objects for which the "solid" model is applicable with restrictions.

The closest in technical essence is a method for determining the magnitude and phase of the body imbalance in the field of centrifugal forces, which are of constant sign in direction and magnitude and easily adjustable in magnitude by changing the speed of rotation of the object (patent No. 2237878, BI No. 28, 2004).

However, this method does not answer the question of the magnitude of the mass and the position of the center of mass, but allows one to determine only the product of the mass of the sample by the radius-vector of the center of mass. Each of the factors individually remains unknown. The magnitude of the product is known, since the mass of the test load and the installation radius of the test load are known [1, 2].

The purpose of the invention is to determine the mass and radius vector of the center of mass of the body.

The goal is achieved due to the fact that three consecutive time intervals are measured for each of two carriage positions of a carriage of mass m with a body under study mounted on it, and the corresponding radius displacement vectors of the platform are determined from the intervals. Using the values of these radius vectors, the mass of the carriage, and the radius vector of the position of the carriage on the platform, the mass and coordinates of the center of mass of the body are calculated. Own mass m of the carriage here plays the role of a test load.

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 method is as follows. The rotation of the platform 1 (Figure 1) is communicated from the engine through the shaft 2 with a double universal joint 3, which does not introduce kinematic errors in the angle of rotation. The platform 1 together with the base 4, three uprights 5 and elastic joints 6 form an oscillating system such as an astatic pendulum. The detailed work of the stand, the procedure for measuring time intervals and the algorithms for calculating the radius vector are given in [2] and are completely similar to the prototype. The carriage 7 of known mass m can be moved and fixed in rectilinear guides passing through the axis of symmetry of the shaft 2, coinciding with the beginning of the moving coordinate system Oξ ₁ η ₁ associated with the platform 1. The orthogonal coordinate axes ξ ₁ and η ₁ are located on a transparent carrier 8. When the platform rotates, they intersect the optical axis 9 of the sensor 10 located at a distance L from the axis of the shaft 2. The offset of the carriage from the axis of the shaft 2 and the origin can always be determined by a device (not shown). Under the carriage, balancing load 11 with a mass m equal to the mass of the carriage moves along similar guides. The load is designed to balance the centrifugal force from the action of the carriage if it is offset from the origin Oξ ₁ η ₁ . The movement of the load 11 in the opposite direction at an equal distance from the carriage is carried out by a double rack and pinion transmission, the elements of which are the carriage 7, balancing the load 11 and two gears 12 located on the same axis and capable of being blocked and disconnected due to their mutual displacement along the axis of rotation, without disengaging from the rails. One wheel is always engaged with the rail on the carriage, the other with the rail on the balancing weight 11 (the mechanism is not shown in detail). Being in a central position, the centers of mass of the carriage S ₁ and the load - S ₂ are at the beginning of the coordinate system and are balanced. The imbalance of the entire mobile system is possible only due to the displacement of the center of mass of the investigated body 13. Let this unknown radius vector be equal to R ₁ in the coordinates Oξ ₁ η ₁ , and the unknown mass of the body is equal to M (Figure 2).

The rotation of the platform 1 with a frequency of Ω will cause the appearance of a centrifugal force F ₁ equal to F ₁ = Ω ² MR ₁ .

The action of the force F ₁ will cause the displacement of the platform 1 by the radius vector r ₁ . The phase angle γ of the lag of the vector r ₁ from R _{1 is} always constant at constant Ω and M. The value and position of the radius vector r ₁ with the polar angle β _{1 are} determined from three time intervals in the coordinate system Oξ ₁ η ₁ according to the well-known algorithm. The platform stops.

The carriage 7 with the body 13 fixed on it shifts along the guides to an arbitrary but known radius vector k, the direction of which is determined by the guides, and coincides with the coordinate axis η ₁ . The balancing load 11 is shifted in the opposite direction by a value of k (Figure 3). The center of mass of the body 13 has taken a new position, determined by the unknown radius vector R ₂ = R ₁ + k (Figure 2). The corresponding centrifugal force is F ₂ = Ω ² MR ₂ . It will cause the platform to shift by the radius vector r ₂ with a polar angle β ₂ . These coordinates are in measured time intervals. The platform stops.

The gears 12 are disconnected from each other and the balancing load 11 is returned to its original central position. The carriage 7 with the body 13 remained in the same position.

During rotation, the platform experiences the action of two centrifugal forces at the same time. From an unbalanced carriage, the force F ₃ = Ω ² mk. The direction and magnitude of this force are known, because m and k are known (FIG. 2). From the action of this force, the displacement of the platform is the vector r ₃ . The lag angle of the vector r ₃ from the vector k is the same, i.e. γ. The magnitude and direction of this vector are still unknown, since γ and the dynamic coefficient λ of the measuring oscillatory system are unknown.

The second force is the centrifugal force F ₂ , and the reaction from its impact on the dynamic system is the already known radius vector r ₂ . The resulting displacement of the platform from the action of two forces is equal to the vector r: r = r ₂ + r ₃ .

The vector r and its polar angle β in the coordinate system Oξ ₁ η _{1 are} found from measurements of three time intervals.

The known values of r, r ₁ , r ₂ , β, β ₁ , β ₃ , mass m and the magnitude of the vector k allow us to uniquely determine the body mass M and the coordinates of the center of mass in the plane of the carriage in the coordinate system Oξ ₁ η ₁ .

.Indeed, Δ OAB is similar to Δ oab. This follows from the fact that the vectors r ₁ and ₂ are behind the vectors R ₁ and R ₂ by the same angle γ. It follows that ΔOAB is rotated by an angle γ relative to Δoab. The side ab in Δ oab is found by the cosine theorem

.

.From the similarity of these triangles follows

.

.The phase lag angle γ is determined from the scalar product of the known vectors k and ab

.

.From the known vectors r ₁ and r and the angle (β ₁ + β) between them, we find the length of the side AC in Δоас

.

Then the segment bc is equal to: | bc | = | ac | - | ab |.

.But | bc | = | r ₃ | by construction as the opposite side of the parallelogram. Module | r ₃ | is determined only by the action of a known force F ₃ = Ω ² m | k |, and there is a proportional relationship between | r ₃ | and F ₃ : | r ₃ | = λΩ ² m | k |. In addition, the vector ab is the vector difference ab = r ₂ -r ₁ and, therefore, the modulus of the vector ab is determined only by the imbalance M | k |, i.e. | ab | = λΩ ² M | k |. Both equalities for | r ₃ | and | ab | define M

.

Thus, the mass M and the coordinates of the center of mass — R ₁ and β _{1 — were found} in the polar coordinate system in the plane of the carriage.

Information sources

1. Steinwolf L.I. Dynamic calculations of machines and mechanisms. - M.: Mechanical Engineering, 1961.340 s.

2. Aleshin A.K. Method for determining the magnitude and phase of rotor imbalance // Problems of mechanical engineering and machine reliability. 2006. No. 6.

Claims (1)

A method for determining the mass and coordinates of the center of mass of a body in a given plane, which consists in determining the radius vectors of displacement of a rotating platform with a measured body, characterized in that they measure three radius vectors: the first is caused by the action of only an imbalance of the body fixed in a carriage of known mass and capable of move and fix on the platform; the second - the effect of an imbalance of the body, displaced together with the carriage by an arbitrary known value; the third is due to the combined action of the same imbalance and the known imbalance of the carriage due to its displacement from the axis of rotation, and the mass of the body and the coordinates of the center of mass in the plane of the carriage are determined by the three radius vectors, mass and magnitude of the carriage displacement.

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