RU2348020C1 - Method of defining tensor of inertia and coordinates of center of mass of body and device to this effect - Google Patents

Abstract

FIELD: physics, measurements.

SUBSTANCE: invention relates to measuring techniques and can be used in mechanical engineering for definition of the axial moments of inertia, tensors of inertia and centers of mass of the bodies. The method proposes to impart to the body, in conditions of notable power dissipation, a programmed spherical reverse symmetric acceleration-braking motion with synchronously-proportional change of the angles of own revolution and precession, with two consistently applied factors of proportionality with the angle of turn around its own horizontal axis of rotation not less than 600°. The matrix of tensor of inertia and polar coordinates of the center of masses of the body are computed from the calculated results of the work of the active twisting moment during programmed motion. The device to implement the proposed method, contains biaxial gimbal with mutually perpendicular axes of rotation, with the inner frame that features a cylindrical casing to fasten the body therein, the controlled electric drive, coaxial with the said inner frame, and the transfer mechanism containing pair of cylindrical cogwheels and bushing with two pairs of tapered cogwheels.

EFFECT: higher efficiency and accuracy, expanded performances.

2 cl, 4 dwg

Description

The invention relates to mechanical engineering and can be used to determine axial moments of inertia, inertia tensors and centers of mass of bodies on suspensions, rotating according to a given program under conditions of unknown friction and aerodynamic drag with the possibility of synchronously proportional rotation of two suspension frames.

A known method of determining the moment and tensor of inertia of the body (RF Patent No. 2112227, IPC G01M 1/10, priority date 07/20/94 publ. 05/27/98, bull. No. 15), in which the platform with an axis of rotation arbitrarily directed in space, installed in a support with a body fixed on it, is informed by a moment of forces in the form of an accelerating-braking moment of any physical nature that does not take values close to zero, a unidirectional dynamically quasi-symmetric two-stage rotation relative to the equilibrium position, having an acceleration stage with a substantially non-zero angular acceleration, sharply transforming into the stage of braking almost symmetrical to it, measure a finite set of values of the angular velocity of the platform at certain angular positions symmetrical with respect to the position of the static equilibrium of the platform-body system, change the applied variable to a certain value, satisfying certain conditions, apply the moment and remeasure the instantaneous angular velocities, which analytically determine the axial moment of inertia of the body, and after six tests are relatively differently directed x axes of rotation analytically determine the inertia tensor of the body.

The disadvantages of the method are the low accuracy and performance associated with the fact that it is necessary to know or determine, under friction conditions, the position of the system's static equilibrium corresponding to the lowest position of the center of mass, reconfigure the device, and retest.

A known method of determining the inertia tensor of the body with simultaneous determination of the coordinates of the center of mass of the body (RF Patent No. 2262678, IPC G01M 1/10, priority date 07/16/02, publ. 10/20/2005, bull. No. 29), by which the body is installed on a platform with the axis of rotation, this system is informed by the program slow rotation in a limited randomly selected angular zone and the reverse symmetric accelerated rotation in the same zone, repeating in the reverse order slow rotation with an angular velocity of opposite sign, determine the applied active rotate current moment and angular velocity on the set of angular positions of the system, which analytically determine the axial moment of inertia and two coordinates of the center of mass, and six similar tests performed at different angular positions of the body on the platform determine the inertia tensor at the point of the body and the location of the center of mass of the body .

The disadvantage of this method is the low productivity and accuracy associated with the fact that it is necessary to carry out six tests, separated by transients of stops, moving the body on the platform to new angular positions and six times the body to program movements.

Closest to the invention is a method of determining the inertia tensor of the body (RF Patent No. 2200940, IPC G01M 1/10, priority date 07/19/00, publ. 03/20/2003, Bull. No. 8), in which the platform with an arbitrary direction of rotation established in a bearing support, under conditions of unknown friction and an unknown position of the center of mass of the body, an active moment of forces of any physical nature reports a reversibly symmetric movement containing a complete braking revolution with a gradually decreasing angular velocity and a full accelerating revolution back m direction, repeating in reverse order the braking revolution with an angular velocity of opposite sign, with an acceptable transient between two revolutions, measure the set of values of the active torque, which determine the moment of inertia of the body, and determine the moment of inertia of the six differently directed axes of the body inertia tensor and simultaneously analytically determine the center of mass of the body.

The disadvantages of the method are the low productivity and accuracy associated with the fact that the inertia tensor of the body is determined by six tests, separated by the procedures for establishing and fixing the body on the platform in six different angular positions.

A device is known for determining the tensor and moment of inertia of a product (RF Patent No. 2112227, IPC G01M 1/10, priority date 07/20/94, publ. 05/27/1998, bull. No. 15), containing a platform with a rotation axis arbitrarily located in space, installed in motionless bearings, a grip for securing a body in it, a power mechanism interacting by means of a coupling with a rod, made in the form of an elastic four-spring mechanism, one pair of springs of which are made in the form of springs pre-compressed by adjustable stops, contacting x with the coupling rod, another pair of springs works within two measuring intervals and is switched off during repeated testing.

The disadvantages of the method are the low productivity and accuracy due to the fact that friction is not accurately taken into account, the position of the static equilibrium of the body-platform system is determined inaccurately and slowly, from which the angle of rotation is subsequently counted, the system is manually or by an additional drive removed from the equilibrium position, reconfiguration of the spring mechanical drive and re-testing is required; the device is not intended to simultaneously determine the center of mass of the body.

Closest to the invention of the device is a device for determining the inertia tensor of the body (RF Patent No. 2200940, IPC G01M 1/10, priority date 07/19/00, publ. 03/20/2003, Bull. No. 8), containing a platform with a shaft mounted in the bearing a support, a shaft with a grip for securing the body mounted on the platform with the possibility of its rotation and fixing in certain angular positions, a sensor of the angle of rotation and angular velocity of the platform, an automated electric drive containing an electric motor and a rotation switch of the platform shaft to rotated e capture shaft, designed as a bevel gear and two friction of electro.

The disadvantages of the device are low productivity and accuracy, due to the fact that it is designed to implement a method with insufficient productivity, it contains a sequence of switching body rotation from one axis to another and vice versa, with transients during stops, transferring the body to the required angular position in switched mode and acceleration to the required initial angular velocity.

The problem of increasing productivity, accuracy, and expanding the field of application on software control systems that are capable of executing programmatic non-uniform symmetrical movements around a fixed point, with a constant angle of nutation of the angular velocity vector, an uneven change in the angle of precession and a synchronously proportional change in the angle of rotation of its own, is being solved.

The essence of the invention lies in the fact that a body mounted in a biaxial gimbal under conditions of unknown friction in the bearings, the resistance of the medium and the location of the center of mass in the body are informed by a reversed symmetric spherical precession movement around a fixed point (CPF movement) containing a slow rotation in angle precession around the vertical axis of the outer frame, passing (with an allowable transient run-out process) into accelerated rotation in the opposite direction, repeating in the reverse order slows down Nogo rotation. At the same time, the internal frame of the gimbal, equipped with a cylindrical casing with a body installed in it, is informed of a reversed-symmetric proper rotational motion synchronous with the precession motion, with a certain constant ratio of the precession angle to the angle of proper rotation with an observed change in the angle of proper rotation in the range of 0-600 ° counted from any selected angular position, then repeat the test with another constant value of the coefficient of proportionality. At such a programmed spherical motion of a body, many values of the active torque work are measured, or the work of this moment is measured, which analytically determines the inertia tensor of the body at the point of intersection of the rotation axes and the coordinates of the center of mass of the body.

The invention is illustrated in figures 1-4, where figure 1 shows a device for implementing the method in the form of a biaxial cardan suspension with mutually perpendicular axes and an electric drive performing testing biaxial synchronous spherical movement with two successive values of the proportionality coefficient, figure 2 shows one of recommended for use testing movements in the angle of proper rotation with which the precession rotation is synchronized, with two values of the proportionality coefficients, in figure 3 so far en range of values of these coefficients of proportionality of the two conditions specified values of the deflection angle of the angular velocity of the spherical movement of its axis of suspension frame, Figure 4 shows the six basis vectors of the virtual axes of the icosahedron, conventionally associated with the test body, passing through the point of intersection of the axes of the gimbals.

The device (figure 1) contains a biaxial gimbal suspension, consisting of an external frame 1 with a vertical axis of rotation Oz ₁ and an internal frame with a cylindrical casing 2 with a horizontal axis of rotation Oz, a transmission mechanism for rotation from the shaft of the outer frame to the shaft of the inner frame, made in the form a pair of cylindrical gears 3-4 and a bevel coupling 5, providing alternate coupling of bevel wheels with two fixed bevel wheels 6-7, a controllable electric motor 8, a rotation angle sensor of the outer frame 9, the base 10. Inner frame suspension 2 is provided with a cylindrical casing ensure the independence of the drag moment from the rotational gimbals.

The reversed-symmetric synchronous precession of the body, or RSP-movement, is the spherical motion of the body around a fixed point, at which the nutation angle remains constant, equal to 90 °, the angle of proper rotation varies non-uniformly, reversely-symmetrically within 600 °, and grows slowly in a positive direction and decreases symmetrically when moving in the opposite direction, and the angle of precession changes synchronously-proportionally with a change in the angle of proper rotation. In this case, the precession movement is made around the vertical axis, and its own rotation is around the moving horizontal axis. Two RSP movements with two values of the proportionality coefficient, combined by a transition process into one motion, will be applied.

The method uses the measured values of the active torque on the software CPD movement, or the values of the torque (useful energy consumption).

The essence of the method lies in the fact that a solid body placed in a biaxial gimbal with mutually perpendicular axes of rotation is informed sequentially by a software CPF - movement in conditions of unknown friction, counted from an arbitrarily selected angular position, containing a step of accelerated rotation in an angle of 600 ° along the angle of its own cp synchronous rotation proportional to the angle of rotation of the precession ψ = λ _1,2 φ and slow reverse symmetrical movement, repeating in reverse order the accelerated motion, with wo sponds scalar negative values of the angular velocity. The transition to the second value of the coefficient λ can be performed on the fly, at low angular speeds, or after stopping the device.

Programmed spherical reverse-symmetric movement of the body and directly proportional to it precession rotation

при ε=const>0, начинающееся с положительного значения угловой скорости ω₀, задаваемое уравнениемEqually slow-motion, uniformly accelerated motion with constant negative acceleration can be taken as a BSP motion along the angle of own rotation φ in the interval [0.600 °]

for ε = const> 0, starting with a positive value of the angular velocity ω ₀ given by the equation

Under these conditions, assign

Such a movement from the moment of time t _p = ω ₀ / ε automatically passes from the braking to accelerating in the opposite direction. It is advisable to consider the movement at time intervals that do not include the reverse moment t _p , i.e. take t ₁ <t _p .

You can also apply uniformly accelerated motion of the form

which, after switching and reversing, goes into a symmetrical, equally slow motion.

Another example of a CPR motion is half the oscillation performed according to the equation

It is also recommended to use movement in the form of a pair of symmetrical parts in the first half of the oscillation of the form

Simultaneously with the “proper” programmed rotation of the body around a certain selected Oz axis, the “precession” rotation of this body axis around the vertical axis Oz ₁ , which is directly proportional to the proper rotation, should be carried out. In general, the body is given a spherical movement around a fixed point, divided into two synchronous rotational CSP movements.

Дальнейшим продолжением движения является медленный переходный процесс на интервале

c переключением на ходу коэффициента связи λ угла прецессии ψ со значения λ₁ на λ₂ и с выходом на повторное движение с новым значением коэффициентом λ₂.Figure 2 shows the CPF movement, consisting of acceleration according to a parabolic law in the time interval [t ₁ , t ₆ ] at t ₁ = 0, φ (t ₁ ) = 0, φ (t ₆ ) = 600 °, smoothly transitioning into the reverse movement according to another parabolic law, and then into the inverse symmetric braking movement on the interval

A further continuation of the movement is a slow transition process in the interval

switching on the fly the coupling coefficient λ of the precession angle ψ from the value of λ ₁ to λ ₂ and with the exit to the repeated movement with a new value of the coefficient λ ₂ .

Consider a controlled electromechanical system (Figure 1), consisting of a platform 2 with a body fixed in it, performing programmatic movement along the angle φ around the movable horizontal axis Oz and the second system - an electric drive that rotates the Oz axis around the Oz axis with unit vector k ₁ in angle ψ. The change in the angles ψ and φ is assigned synchronously directly proportional, and the proportionality coefficient is first given the value λ ₁ , and then λ ₂

к оси собственного вращения тела (фиг.3), обеспечивающие последовательное совпадение ее положения с осями виртуального икосаэдра, имеющие следующие значения: β₁=0.6524 рад = 37.38°, β₂=1.3821 рад = 79.19°.Here β ₁ and β ₂ are the required angles of inclination of the instantaneous angular velocity

to the axis of proper rotation of the body (Fig. 3), ensuring a consistent coincidence of its position with the axes of the virtual icosahedron, having the following values: β ₁ = 0.6524 rad = 37.38 °, β ₂ = 1.3821 rad = 79.19 °.

The body-platform system makes a spherical movement around a vertical axis with a precession angle ψ, a proper rotation angle φ, and a constant nutation angle θ = 90 °. Angular velocity of this movement

и

- орты угловых скоростей

и

Where

and

- unit vectors of angular velocities

and

при

at

выполняет асимметричную прецессию вокруг вертикальной оси Oζ с постоянным углом нутации α₁=90°-β₁, (α₂=90°-β₂) согласно программному уравнению ψ=λ₁φ(t), (ψ=λ₂φ(t)). Геометрической интерпретацией такой симметрической прецессии служит неравномерное разгонно-тормозное асимметричное качение кругового конуса с осью Oz, углом при вершине L₁Oz=β₁, (L₂Oz=β₂) по неподвижному круглому конусу с вертикальной осью Oζ с углом при вершине L₁Oz=α₁, (L₁Oz=α₂). Будет использовано следующее свойство: на реверсивном симметричном сферическом движении при угле поворота φ=600° и обратно каждая из трех осей виртуального икосаэдра через полный оборот совпадает четыре раза с мгновенной осью вращения OL₁ при λ=λ₁, а три другие оси икосаэдра на втором движении совпадают четыре раза с OL₂.We call this motion the precession motion around the vertical axis; it is determined by one equation of the form φ = φ (t), i.e. software equation for asymmetric precession motion. Instantaneous axis of rotation of the body OL ₁ (OL ₂ ) together with the vector

performs asymmetric precession around the vertical axis Oζ with a constant nutation angle α ₁ = 90 ° -β ₁ , (α ₂ = 90 ° -β ₂ ) according to the program equation ψ = λ ₁ φ (t), (ψ = λ ₂ φ (t )). The geometric interpretation of such a symmetric precession is the non-uniform accelerated-brake asymmetric rolling of a circular cone with the Oz axis, the angle at the apex L ₁ Oz = β ₁ , (L ₂ Oz = β ₂ ) along a fixed circular cone with a vertical Oz axis with an angle at the apex L ₁ Oz = α ₁ , (L ₁ Oz = α ₂ ). The following property will be used: in a reversed symmetric spherical motion with an angle of rotation φ = 600 ° and vice versa, each of the three axes of the virtual icosahedron coincides four times with the instantaneous axis of rotation OL ₁ at λ = λ ₁ , and the other three axes of the icosahedron on the second movement coincide four times with OL ₂ .

The introduced concept of symmetric precession is a generalization of the concept of regular precession (Bukhgolts NN The main course of theoretical mechanics. 4.2. M: Nauka, 1966. pp. 161-162, Besekersky VA, Fabrikant EA Dynamic synthesis of gyroscopic systems stabilization. L .: Shipbuilding, 1968. p. 18-23, Lurie A.I. Analytical mechanics. M: FM, 1961. p. 134).

Стороны треугольника приняты равными единице, ось Oz направлена из центра икосаэдра на центр пересечения медиан одной из граней.Figure 4 shows four of the twenty triangular faces of a virtual icosahedron linked to a test body. Through the vertices of the faces A, B, C, E, G, F, all six axes of the icosahedron are drawn with the unit vectors

The sides of the triangle are taken equal to unity, the Oz axis is directed from the center of the icosahedron to the center of intersection of the medians of one of the faces.

Radii of the inscribed and described spheres

высота грани

The distances from the tangent point of the inscribed sphere to the vertices of the ABC face are 2ρ, where

edge height

We have

Numerical values of the main quantities:

α ₁ = 52.622632 ° = 0.9184382 rad, α ₂ = 10.812317 ° = 0.1887105 rad,

β ₁ = 37.377368 ° = 0.652358 rad, β ₂ = 79.187685 ° = 1.382086 rad,

γ = 63.434949 ° = 1.107149 rad, R = 0.951057, r = 0.755761,

ρ = 0.288675, b = OK = 0.934172, h = KE = 0.178411

Above the Oxy plane are six vertices of the icosahedron. The vertices ABC are located at level z ₁ = r on a circle of radius ρ ₁ = 2ρ = 0.577350 with equal angular distances of 120 °, and the three vertices E, G, F are located at level z ₂ = h on a circle of radius ρ ₂ = b. Six axes of the icosahedron pass through the named peaks and the center O. Figure 4 shows the unit vectors of these axes. The directing cosines of the unit vectors are equal to the ratios of the coordinates of the vertices to the distance R. The unit vectors

Six-element column vectors:

From the columns we compose a matrix of unit vectors of the axes of the icosahedron of size 6 × 6:

U = [V ₁ , V ₂ , V ₃ , V ₄ , V ₅ , V ₆ ]

The numerical value of the matrix U:

The determinant of the matrix is significantly different from zero and is det (U) = 2.2897.

Suppose that as a result of the experiment, six moments of inertia of a rigid body with respect to the axes of a virtual icosahedron rigidly linked to the body and having a center at some point О of the body are found. Then the row vector from the axial and centrifugal moments of inertia of the body is calculated by the following matrix formula:

Consider a system consisting of a biaxial cardan suspension and a body installed in it (Fig. 1). The axis of rotation of the outer frame Oz _{1 is} vertical, and the axis of rotation of the inner frame of the platform Oz moves in the horizontal plane. The test body is rigidly fixed on the inner frame. The suspension is driven by an electric motor, the shaft of which is rigidly connected to the outer frame, and the movement to the inner frame is transmitted through a gear mechanism. We assume that the inner frame-platform has a cylindrical casing, which ensures equality of aerodynamic drag in symmetrical movements. The rotation of the inner frame is determined by the angle φ, and the outer frame is determined by the angle ψ. The body-platform system makes a spherical movement around the point O of intersection of the suspension axes. We assume that the holonomic constraint of the form ψ = λ _{1 (2)} φ is satisfied. Then the mechanical holonomic system: cardan suspension - electric motor rotor - transmission mechanism - the body under test will have only one degree of freedom, its position is determined by one generalized coordinate - the rotation angle φ.

Let I denote the reduced moment of inertia in the angle φ of the system without the test body. Obviously I = const, since the arrangement of the mass of the device around the axis of the outer frame does not change when the angle ψ (or φ) changes (we assume that the inner frame and wheels have axial symmetry). The unknown reduced moment of inertia of a body J making a spherical motion depends on the angle φ: J = J (φ).

этап симметричного равнозамедленного параболического движения при

этап переходного процесса при

с переключением коэффициента λ со значения λ₁ на λ₂ и повторного движения на интервале

Движение по углу ψ осуществляется синхронно согласно кинематическому уравнениюAs the RSP movement along the angle φ we take a piecewise-parabolic movement (Fig. 2), containing the acceleration stage at t <t ₁ , the stage of uniformly accelerated movement at t∈ [t ₁ , t ₆ ] in the angular interval 0≤φ≤600 ° , stage of a smooth transition process according to the parabolic law with

stage of symmetric equally slow parabolic motion with

stage of the transition process when

with switching the coefficient λ from the value of λ ₁ to λ ₂ and repeated movement on the interval

The movement along the angle ψ is carried out synchronously according to the kinematic equation

The square of the total angular velocity of the spherical CPR motion is determined by formula (6). At the stage of acceleration-braking, we distinguish three full turns displaced one relative to the other by 120 °:

implemented over a time interval

On the reverse motion, the angular intervals (14) pass into the intervals

in between

- кинетическая энергия механизма при изменении угла ψ и условно зафиксированном угле φ. Если же угол φ изменяется с угловой скоростью

то внутренняя рамка совершает сферическое движение вокруг полюса О с переносной угловой скоростью

и относительной скоростью

причем эти скорости взаимно перпендикулярны, а оси системы Oxyz являются главными осями инерции внутренней цилиндрической рамки. При этих условиях полная кинетическая энергия устройства определяется двучленной формулойLet I ₁ be the moment of inertia of the entire device with respect to the z ₁ axis. We have

is the kinetic energy of the mechanism with a change in the angle ψ and a conditionally fixed angle φ. If the angle φ changes with the angular velocity

then the inner frame makes a spherical movement around the pole O with a portable angular velocity

and relative speed

moreover, these speeds are mutually perpendicular, and the axes of the Oxyz system are the main axes of inertia of the inner cylindrical frame. Under these conditions, the total kinetic energy of the device is determined by the two-term formula

where I _z is the moment of inertia of the inner frame relative to its own axis, in which we include the moments of inertia of the wheels of the transmission mechanism.

Kinetic energy of the test body on a spherical motion

where J _L is the moment of inertia of the body relative to the instantaneous axis of rotation OL.

Based on formulas (18), (19), we determine the kinetic energy of the mechanism-body system, including the rotor of the electric motor and the transmission mechanism, on two movements ψ = λ ₁ φ and ψ = λ ₂ φ:

и

- приведенные моменты инерции механизма по углу поворота φ на двух режимах движения, J_L1(φ) и JL₂(φ) - моменты инерции тела относительно мгновенных осей L₁ и L₂ сферического движения.Here

and

- reduced moments of inertia of the mechanism with respect to the angle of rotation φ in two modes of motion, J _L1 (φ) and JL ₂ (φ) - moments of inertia of the body relative to the instantaneous axes L ₁ and L _{2 of} spherical movement.

учитывая равенство энергий на концах интерваловWe apply the kinetic energy change theorem to a device-body system on a pair of intervals τ ₁ = [t ₁ , t ₄ ] and

given the equality of energies at the ends of the intervals

- работы активного крутящего момента, создаваемого электродвигателем, V₁₄ и

- работы моментов трения и аэродинамического сопротивления.Here A ₁₄ and

- operation of the active torque generated by the electric motor, V ₁₄ and

- work moments of friction and aerodynamic drag.

ввиду динамической симметрии пары движений, обеспеченной, в частности, наличием цилиндрического кожуха. Работы силы тяжести тела не входят в уравнения, они равны нулю, поскольку на рассматриваемых полных оборотах тела его центр масс возвращается на прежнюю горизонтальную эквипотенциальную поверхность независимо от угла поворота тела вокруг вертикальной оси Oz₁. Почленно вычитая уравнения (22) и учитывая выражение (20), получим уравнение, не содержащее работы сил тренияNote that

in view of the dynamic symmetry of the pair of motions, provided, in particular, by the presence of a cylindrical casing. The work of the body’s gravity is not included in the equations, they are zero, because at the considered full revolutions of the body its center of mass returns to the previous horizontal equipotential surface regardless of the angle of rotation of the body about the vertical axis Oz ₁ . Subtracting equations (22) term by term and taking into account expression (20), we obtain an equation that does not contain the work of friction forces

or

From here we find the calculation formula for the moment of inertia of the body relative to the first axis of the icosahedron, which does not contain dissipative forces.

Let us similarly consider the two remaining pairs of turns from (14) and (16), we obtain the calculation formulas for the other two axes of the icosahedron

Moments of inertia relative to the three axes of the icosahedron of the second row are determined by similar calculation formulas

The given moments of inertia of the device can be found by calculation or by means of a previously performed experiment with an unloaded mechanism in the absence of a body according to the formulas

Formulas (27) are used to identify the device itself.

Thus, the inertia tensor of the body at the point O of the intersection of the two axes of the cardan suspension is determined by formulas (12), (11), (24), (25), (26) with the possible involvement of formulas. In these formulas, the work of the active torque M (φ) are found through the calculation of certain integrals of the active moment in the rotation angle,

It is also possible to directly measure these works by technical means for consuming energy for the work of the mechanism, including work to overcome dissipative forces.

The calculation formulas were obtained under the assumption that the inner frame is a symmetric body of revolution with an axis of symmetry Oz. If this condition is not fulfilled, the axial moments of inertia of the body together with the inner frame can be determined using these formulas and then the axial moments of inertia of the inner frame can be subtracted at six different angular positions of the frame.

To determine the center of mass of the body, we divide the first revolution (Fig. 1), performed for the time interval τ ₁ = [t ₁ , t ₄ ] by a quarter, the revolutions made for the time intervals

и другой полуоборот за интервал

Предполагаем, что центр масс платформы расположен на ее собственной оси и m - известная масса системы тело-платформа. Введем в рассмотрение цилиндрические координаты (ρ_C, α, z_C) неизвестного положения центра масс этой системы, где ρ_C - полярный радиус, α≥0 или α<0 - полярный угол, отсчитываемый от отвесного направления в начале эксперимента. Работа силы тяжести этой системы на интервалах

и

и работы на асимметричных полуоборотах:Consider the half-turn for the interval

and another half turn per interval

We assume that the center of mass of the platform is located on its own axis and m is the known mass of the body-platform system. We introduce the cylindrical coordinates (ρ _C , α, z _C ) of the unknown position of the center of mass of this system, where ρ _C is the polar radius, α≥0 or α <0 is the polar angle measured from the vertical direction at the beginning of the experiment. The work of gravity of this system at intervals

and

and work on asymmetric half-turns:

и

By analogy with formula (23), we compose two energy equations for two pairs of symmetric half-turns of the platform-body system, in which the differences in the work of gravity should be additionally included

and

Get

Here T ₁ , T _1C , T _2C , T _3C are the kinetic energy of the system at the ends of half-turns. They are considered known because the programmed values of the angular velocities are known and the inertia tensor of the body-platform system has already been calculated, and the kinetic energies of the system in any of its positions have been determined by the formula of spherical motion

From equations (31) we find two equations:

at

From the system we find the calculation formulas that determine the two cylindrical coordinates of the center of mass of the body-platform system:

The formulas determine a straight line parallel to the axis of the platform Oz, on which the center of mass of the body-platform system is located, and the angle α is counted from the section of the body by a vertical plane drawn at the initial time t ₁ through the axis of proper rotation. The coordinate z _c determining the position of the center of mass on this line can be found by performing an additional test by placing the body on the inner platform in a rotated position, rotated 90 ° from the Oz axis in the horizontal plane (Fig. 1).

The device operates as follows: the inner frame 2 of the cardan suspension with the body fixed in it performs a spherical programmed motion around a fixed point O, consisting of precessional rotation around the Oz axis, created by the outer frame 1 together with the electric motor 8 and the frame’s own rotation around the Oz axis created by the electric drive while the sensor 9 is a direct measurement of the angle of rotation and the angular velocity of rotation of the rotor of the electric motor. Motion control is carried out by an automatic program control system. The electromount 5 at one stage fixes the left bevel gear on the shaft 4, releasing the right bevel gear, and at the other stage, on the contrary, fixes the right bevel gear and releases the left wheel, thereby achieving a change in the proportionality coefficient connecting the precessional and proper rotation of the platform 2. As a result, the device provides the body with the required programmed spherical motion.

Thus, the alleged invention allows to solve the problem of increasing productivity, accuracy, expanding the scope.

Claims (2)

1. The method of determining the inertia tensor, which consists in the fact that the platform with the body fixed on it under conditions of unknown friction, resistance of the medium and the position of the center of mass of the body is informed by the active moment of forces a symmetrical reversible acceleration-brake rotation, counted from an arbitrarily selected angular position, containing full braking revolution with a gradually decreasing value of the angular velocity and a full accelerating revolution in the opposite direction, repeating in the reverse order the braking revolution with the angular velocity of a positive sign with an allowable transient process between revolutions, the active torque and angular velocities of the platform are measured or the work of the active moment is measured on a set of angular positions, which analytically determine the axial moment of inertia of the body and the moments of inertia relative to six axes directed along the axes of the virtual icosahedron, conditionally connected with the body, the inertia tensor of the body and the location of the center of mass are determined, characterized in that the platform is informed of one reversed-symmetric spherical rotation fixed-point region, combining a system of rotations around fixed axes, consisting of reversed-symmetric precessional motion around a vertical axis and directly proportional to it synchronous proper rotation in the angular interval [0-600 °] for two proportionality coefficients, ensuring sequential coincidence of the instantaneous axis of rotation with six axes of the vertical icosahedron, adopted as the axis of rotation of the body. 2. A device for determining the inertia tensor of the body and the center of mass of the body, containing a platform in the form of an external frame with a shaft mounted in a fixed bearing support, an internal shaft with a grip for securing the body mounted on the platform with the possibility of rotation, a rotation angle and angular velocity sensor platforms, an automated electric drive in the form of an electric motor and a gear transmission mechanism, characterized in that the shaft of the outer frame is rigidly connected to the rotor of the engine, the inner shaft with a grip is equipped with a frame with ilindricheskim housing, a transmission mechanism with a motor shaft and the inner shaft is formed as a pair of cylindrical friction wheels and the electrofusion with two conical wheels of alternatively fixing them on the clutch shaft, the engagement of the fixed bevel gears.

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