The present invention relates to a process for determining the centre of gravity and the inertial tensor of a body using a torsional pendulum, with the period of the torsional pendulum being measured and the moment of inertia being calculated on the basis of the measured period, where appropriate with correction for the moment of inertia of the apparatus which contains the torsional pendulum, and to a device to be used together with a torsional pendulum for determining the centre of gravity and the inertial tensor of a body.

The need to determine the complete inertial tensor is unusual and there are no known methods for satisfactorily determining the inertial tensors of a body. By contrast, there is commonly the need to measure the moment of inertia in a particular axial direction and, where appropriate, to measure asymmetries around this axis, for example in order to balance rotors, turbine wheels or car tyres. A balancing machine is used for this purpose. The imbalance in the point of support of the rotating body is measured and the radial position of the centre of gravity and the deviation of an adjacent main axis of inertia from the axis of rotation can be calculated. The input data which are required for these calculations are the axial and radial moments of inertia and also the mass and the axial position of the centre of gravity. The moments of inertia can be measured using a torsional axle and the centre of gravity in some type of centre of gravity balance.

A method of measurement as described above suffers from a number of limitations. If the body which is to be measured has a shape, for example is winged, such that substantial aerodynamic forces are generated at the rotational speed which is required, it is then difficult to distinguish these forces from the sought-

after inertial forces. The attachment of the body in the balancing machine can also give rise to problems. In addition, the method demands that the axial position of the centre of gravity should be well known, resulting in the requirement to develop a measuring method/equipment for determining this position. Consequently, the method of measurement is not suitable for carrying out measure¬ ments on, for example, winged submunitions.

The object of the invention is to provide a process and a device for determining the centre of gravity and the inertial tensor of a body which do not suffer from the abovementioned limitations. The object of the invention is achieved by means of a process which is characterized in that a) the body is moved relative to the axis of rotation of the torsional pendulum between a number, preferably three, of known positions with parallel axes of rotation through the body in order to determine the moment of inertia in each respective position, b) the period of the torsional pendulum is measured in the known positions in accordance with point a) , c) the moments of inertia of the known positions are calculated on the basis of the measured periods and, where appropriate, with correction for the moment of inertia of the apparatus which contains the torsional pendulum, d) Steiner's theorem, which is known per se and which states that the moment of inertia of an axis of rotation is equal to the moment of inertia through the centre of gravity of the body plus the mass of the body times the square of the perpendicular distance between the axis through the centre of gravity and the axis of rotation, is applied to the moments of inertia which were calcu¬ lated in accordance with point c) above, e) differences between the moments of inertia of the axes of rotation are established and the position of the centre of gravity is extracted from these differences and calculated, as a result of which the distance between the centre of gravity and the axes of rotation can be

ascertained, f) the moment of inertia through the centre of gravity is calculated knowing the moment of inertia of an axis of rotation, the mass of the body and the distance between the axis of rotation and the centre of gravity, and g) points a) - f) above are repeated for other, preferably 5, selected directions, and also by means of a device which is characterized in that a rig is affixed to the torsional pendulum for receiving the body whose centre of gravity and inertial tensor are to be deter¬ mined, which rig is attached to one end of the axle of the torsional pendulum and comprises a first hoop which is displaceable in two perpendicular directions in a plane which is perpendicular to the longitudinal direc- tion of the torsional axle, a second, tiltable hoop which is carried by the shanks of the first hoop, and a disc which is carried by the second hoop, is rotatable rela¬ tive to the hoop, and carries the body whose centre of gravity and inertial tensor are to be determined, which rig permits three-dimensional positioning of the body in combination with a displacement in a plane which is perpendicular to the longitudinal direction of the torsional axle.

The invention makes use of the change in the moment of inertia when the body is moved relative to the axis of rotation. Knowing the positions of the different axes of rotation and making use of Steiner's theorem, the centre of gravity of the body and the moments of inertia through this centre of gravity can be calculated. The six independent inertial tensor elements can be determined by taking measurements in six different directions.

According to a preferred embodiment, the process is characterized in that an inertia product is determined by measuring the moment of inertia with respect to a direction in accordance with points b and c in Patent Claim 1 and correcting this moment of inertia with respect to the influence of moments of inertia which are included in the inertia product.

In this connection it can be pointed out that the

main moment of inertia of the body and the direction of the respective axis are obtained if the eigenvalues and eigenvectors of the inertial tensor are calculated.

In order to counteract uneven loading of the torsional axle, the device is provided, in accordance with an advantageous embodiment, with a dismountable balance arm. The embodiment is characterized in that the device comprises a dismountable balance arm having a balance mass whose moment of inertia is known and which is arranged to balance the displaced rig, with the balance arm without the balance mass balancing the unloaded rig and with the balance mass balancing out the moment which arises as a result of the mass of the body.

The rotatable disc which is included in the device is advantageously rotatable in fixed steps of 45°. The second hoop is preferably tiltable into fixed posi¬ tions of 0°, 45° and 90° relative to the first hoop. These setting options render it possible, for example, to identify the following six positions as position XZ position YT position ZZ angle of angle of angle of tilt = 0° tilt = 90° tilt = 90° angle of angle of angle of rotation = 0° rotation = 90° rotation ■ 0°

position XY position TZ position ZZ angle of angle of angle of tilt = 45° tilt - 90° tilt m 45° angle of angle of angle of rotation = 90° rotation = 45° rotation = 0'

The rig can be fixed to one end of the torsional axle by means of a plate which is provided with pins, which pins are arranged to interact with grooves which are disposed in two perpendicular directions in the central part of the first hoop. This arrangement makes it easy to effect movements in a plane perpendicular to the longitudinal direction of the torsional axle.

According to an advantageous embodiment, the body can be positioned by means of a wedge-shaped centring

ring and precision gauge blocks.

The invention will be described in more detail below with reference to an exemplary embodiment of a device for determining the centre of gravity and the inertial tensor, where Figure 1 shows an example of a body whose centre of gravity and inertial tensor are to be determined. Figure 2 shows a perspective view of a rig which is intended to receive a body whose centre of gravity and inertial tensor are to be determined, Figure 3 shows a perspective view of the rig according to Figure 2 which is receiving a body in the form of a winged submunition, and Figure 4 shows evaluation and presenta¬ tion equipment which calculates and presents the centre of gravity and the inertial tensor on the basis of detected oscillations.

The theory of a torsional pendulum will first be described below.

After having been turned axially out of the equilibrium position, a vertical axle oscillates with a period T which is a function of the torsional resistance k and the moment of inertia I of the axle. The following relationships can be established:

T = 2 ir V I/k

I = C • T ² C = k/4ιr ²

which is a solution of the differential equation:

φ : torsional angle

In this case, I denotes the total moment of inertia of the system. In that which follows, the parame¬ ters of the measured body are indicated by _body and those of the measuring apparatus by _pp . The values for the entire measuring system are indicated by _tot . The following relationship applies:

~ ^body ⁺ ^app

I _app includes the moment of inertia from the axle and the rigs which are used.

In order to obtain the moment of inertia of the measured body around the axis of rotation, two measure¬ ments must be carried out; one measurement for determin¬ ing the period of the apparatus alone and one which gives the period of the apparatus together with the measured body.

T ^■ " ^■ tot s c *- • T ^ ² tot

The following can be calculated:

T ^■ •-body - - T -' ^■ tot - T ^x app a -s c * ^» • ( ^■J T' ¹² tot - τ ^A2 app '.

The value of the system constant C is determined by measuring the oscillation time of a calibration body having a known moment of inertia I _eal .

c = i _Ml / (T _ctl ^! - τ _app ² )

An account of the principles involved in measuring the centre of gravity and the moment of inertia of a body is given below.

By means of measuring the moment of inertia for three parallel axes of rotation whose connecting lines form a right angle, the position of the centre of gravity relative to the axes of rotation and the moment of inertia through the centre of gravity for this axial direction are determined.

Figure 1 shows a body with a coordinate system which is placed with the origin and one axis, the Z axis, along the first axis of rotation. The moment of inertia is measured for this axial direction 1 and for two further axial directions, 2 and 3, which are parallel

with the first direction and displaced perpendicularly with respect to each other.

Values l , I _j , ~ are obtained for the three moments of inertia through points 0, A and B. Steiner's theorem is applied in the following calculations. The moment of inertia through the centre of gravity is designated iξ.

3) - 2) =>I^I =ra-(r> - r _G > =>

The moment of inertia through the centre of gravity in the Z-axis direction is then

It is to be noted that the moment of inertia through the centre of gravity can be determined without knowing where the axis of rotation is located in the body. It suffices for making the calculations to know the mass and the size of the perpendicular movement.

Of course, if the distance from the centre of gravity to the axis of rotation is known, the moment of inertia through the centre of gravity is determined by one single measurement (formula 1) or 2)) . The moments of inertia I^, 1^ and I _IX are obtained by straightforward measurements around the x, y and z axes, respectively. However, the inertia product is a theoretical product which is not directly measurable and which indicates the degree of asymmetry around selected coordinate axes. In order to calculate the inertia products, the moments of inertia can be measured around three axes in the xy, yz and xz planes, respec-

tively, halfway between the coordinate axes (45°) . Using the measured values T^, T _yx and T„, the inertia products are then calculated with the aid of known relationships for the anisotropy of the moment of inertia. For example, the moment of inertia around an axis of 45° from the x axis in the xy plane is:

T = 1 _1 1 T 2 ^{+ X} yy ~~ ⁺ ->*

Separate out the inertia product (the two other inertia products are obtained in the same manner)

I^T^ - ⁽ I _^ +I^ ⁾ I„=T„ - ⁽ I«+I _tx ⁾ I _y _=τy_ " (Iyy+I,.)

A device according to the invention for deter¬ mining the centre of gravity and the inertial tensor is shown in Figures 2-4. The device includes a rig 10 which is mounted on a torsional pendulum (not shown) of conven- tional type. The rig is fixed to one end of the torsional axle of the torsional pendulum. The oscillations of the torsional pendulum are started by the body, which is to be studied, being manually rotated to a stop position and then released. A sensor 13, see Figure 4, detects the oscillations of the torsional axle and the period is presented continuously on a digital display 11. A com¬ puter 12 is arranged to process the periods detected by the sensor 13. The computer is equipped with software for facilitating calculation of moments of inertia from detected periods. The software of the computer includes various routines for calibrating and taring the rig and measuring/calculating the moment of inertia of the body. The software is also used to check that the oscillation time is not varying more than is permitted before a measured value is accepted as input data for the calcula¬ tion. The result is stored in a memory 15 and can be

presented on a display 14.

The rig 10, see Figures 2-3, is fixed to the torsional axle (not shown) via a plate 16. A balance arm 17 is fixed, by one of its ends, to the plate 16 in a dismountable manner. At its other end, the balance arm 17 carries a replaceable balance mass 18 which has a known moment of inertia. On its upper side, the plate 16 is provided with three pins 19, 20 and 21. The plate carries a first hoop 22, whose central part 23 is provided with guiding grooves 24 in two perpendicular directions. A second hoop 25 is carried rotatably by the two shanks 26 and 27 of the first hoop 22. The second hoop can be tilted into fixed positions of 0°, 45° and 90° relative to the first hoop. A rotatable disc 30, to which the body which is to be measured is affixed, is connected between the two shanks 28, 29 of the second hoop 25. The disc can be rotated in fixed steps of 45° relative to the second hoop. Figure 3 shows a body in the form of a winged submunition 31 in the affixed position. Precision gauge blocks 32 and a wedge-shaped centring ring 33 are present for positioning the body 31. A locking arm 34 keeps the body 31 in place by pressing it against the disc 30.

The first hoop can be displaced into fixed positions in two directions which are at right angles to each other by means of the pin and guiding groove arrangement. The pin 20, which, in Figure 2, is located where the guiding grooves 24 intersect, is provided with a nut 35 which has a conical lower part for securing and positioning the first hoop 22 against the plate 16. The balance arm 17 is used to counteract uneven loading of the torsional axle when the rig 10 is dis¬ placed. The balance arm 17 without the mass 18 balances the displaced rig when it is not carrying the load (the body) , and the balance mass 18 balances out the moment arising due to the mass of the body or the submunition

31.

The rig provides the possibility of easily turning the body which is to be measured into the desired rotational axis directions. For example, six different

rotational axis directions, which is the number which can be required for unambiguously determining the inertial tensor, can be set as follows: position . position YT position ZZ angle of angle of angle of tilt - 0° tilt « 90° tilt = 90° angle of angle of angle of rotation ■ rotation = 90' rotation = (

position ΣY position YZ position ZZ angle of angle of angle of tilt - 45 ^p tilt = 90° tilt = 45° angle of angle of angle of rotation = 90° rotation = 45° rotation - 0 ^~

Values for the tensor element of the inertial tensor of the body, as well as information on the centre of gravity of the body, are obtained by carrying out measurements in the above directions and by performing calculations in accordance with the relationships shown above.