JP3147400B2 - Elliptical vibration detection method of vibration feeder and display device thereof - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for detecting an elliptical vibration used for measuring a rotational direction of an elliptical vibration of a trough of a vibrating feeder using a vibrating motor as a vibration source, and a display for the method.

FIG. 7 shows a resonance type vibration feeder 1 using a vibration motor M as a vibration source. As is well known, a trough 2 is suspended on a part 6 of a building by a pair of coil springs 4 and 5. Hanged. A pair of wing plates 3 are attached to the bottom wall of the trough 2, and a pair of attachment plates 7a and 7b are fixed to the ends thereof. , 8b are fixed. The other surfaces of the plate rubbers 8a and 8b are fixed to the casing surface of the vibration motor 9. The casing 9 has a built-in vibration motor M, and semicircular unbalance weights 11 are fixed to both ends of the rotating shaft 10. The vibration feeder 1 using a conventionally known resonance type vibration motor M as a vibration source is configured as described above. When the vibration motor M is driven, the unbalance weights fixed to both ends of the rotating shaft 10 are driven. The rotation of 11 generates centrifugal force, which is a circular excitation force, which is the plate rubber 8a, 8b and the mounting plate 7a,
7b, the total mass m of the movable part including the trough 2 and the wing plate 3 and the total spring constant K of the elastic means including the pair of plate rubbers 8a and 8b are calculated by ΔK. /
Since the vibration motor M is driven at a frequency close to the resonance frequency determined by √m, the trough 2 performs a resonance vibration. The locus of the trough 2 (shown exaggerated) becomes V due to such vibration force. Performs elliptical vibration. This elliptical vibration V
8, an elliptical vibration V having a major axis A on an axis XX substantially parallel to the shear direction of the plate rubbers 8a and 8b and a minor axis B in a direction perpendicular to the axis XX. The inclination angle of the amplitude A with respect to the horizontal line x-axis in the major axis direction, that is, the vibration angle θ is substantially determined by the mounting angle of the plate rubbers 8a and 8b. When it is desired to know the amplitude A and the amplitude B in the short axis, an amplitude nameplate 21 as shown in FIG. 11 is conventionally used, which is made of a rectangular paper 22 as shown in FIG. Is coated on the surface, and graduation lines 23 are printed on the surface at regular intervals.
Display lines 24a and 24b that are inclined on both sides of −c are printed. When the amplitude of the vibrating body 200 is measured with the amplitude nameplate 21 as described above, the vibration direction is adjusted so as to match the scale line 23, and the vibrating body 200 is attached to the surface of the vibrating body 200. Now, if the vibrating body 200 performs a linear vibration in the direction a as shown in FIG. 12, the pair of display lines 24a and 24b vibrate in this direction with the width of the amplitude, and therefore, the display line 24a and 24b P and Q are observed as an afterimage, these composite portion is exhibited rhombic shape, observing the scale 23 in the distal end a, whereby it is to measure Fuhaba of the vibrator 200, the vibration feeder linear If vibration is applied, accurate measurement can be made by sticking the amplitude nameplate 21 such that the scale line 23 matches the vibration direction.

However, an elliptical vibration V as shown in FIG.
When measuring the amplitude in the major axis direction of the trough 2 for performing the
Even if 1 is attached to the trough 2, vibration in the short axis direction is applied, so that the entire graduation line 23 also has a band as a residual image of a certain width, and these composite residual images become considerably complicated, and in the case of linear vibration, Cannot accurately determine the intersection A of the afterimage. The Rukoto align your Fuhaba nameplate in the axial direction of the elliptical vibration V even difficult. Conventionally, in spite of such difficulties, the amplitude in the long axis direction has been measured with such a nameplate based on the experience of the measurer.

However, even if the amplitude in the major axis direction is measured as described above, the same amplitude in the major axis direction and the amplitude in the direction perpendicular thereto, that is, the amplitude in the minor axis direction is the same. Even so, it has been clarified theoretically and experimentally that the material transfer speed differs depending on the rotation direction of the elliptical vibration. That is, for example, when the side surface of the trough 2 is viewed with the material transfer direction by the trough 2 on the right side in FIG.
Is rotated clockwise, that is, clockwise counterclockwise, that is, the transport speed is greater than the counterclockwise elliptical vibration.

This elliptical vibration (usually 1500 to 1800)
r. p. m. It is difficult to see the direction of rotation of the trough 2 at all with the naked eye alone. Conventionally, the flash timing of the strobe flash is set to a frequency almost close to the frequency of this trough 2 vibration, and the vibration of the trough 2 is By sequentially reading the trajectory at a certain point during light emission, one point of the trough 2 is now performing an elliptical vibration, but the transfer direction is set to a fixed direction (when the transfer direction is viewed in the opposite direction). Of course, the direction of the same elliptical vibration is the opposite direction), and it was made to recognize whether this elliptical vibration was clockwise or counterclockwise.

However, the strobe flash is not always capable of being installed at all times. For example, the direction of the elliptical vibration of the trough is set clockwise so that the transfer speed thereof is larger by designating the rotation direction of the vibration motor. I was However, as is clear from the elliptical vibration theory, the vibration system
Depending on the weight of the material loaded on the trough, whether it is a mass system or a 3-mass system, the deviation of the resonance frequency at that time may increase or decrease, and may be reversed. It is not always possible to measure the rotation direction of the elliptical vibration. Also, it is dangerous to expose the unbalance weight of the vibration motor to the outside.

[0007]

SUMMARY OF THE INVENTION The present invention has been made in view of the above-mentioned problems, and the rotation direction of the elliptical vibration of the movable member for transferring the material increases the transfer speed regardless of the type of driving unit used. An object of the present invention is to provide an elliptical vibration detection method capable of easily measuring whether or not a direction is present, and an indicator thereof.

[0008]

Further objects Means for Solving the problems] The first vibrating off the vibration detecting element for detecting the vibration of the first predetermined direction
A second vibration detecting element mounted on a surface parallel to the vibration direction of the feeder and detecting a vibration in a second predetermined direction perpendicular to the first predetermined direction on a surface parallel to the vibration direction of the vibration feeder ; Mounting the first vibration detecting element and the second vibration detecting element
From the phase difference φ of detection output of the vibration detecting element, the vibration off
An elliptical vibration detection method is characterized in that a vibration rotation direction of an elliptical vibration of a seeder is obtained.
Alternatively, at least a microcomputer, a battery, and a display unit for receiving an output of the display element and displaying at least a rotation direction of the elliptical vibration are provided on a front panel portion of a casing containing a display element connected to an output terminal of the microcomputer. Body and
The vibration detector includes a vibration detector that is detachable from a part of the vibration feeder , and a conductive cord that derives a detection force of the vibration detector and is connected to a connector portion of the main body.
A first vibration detecting element for detecting vibration in a predetermined direction, a second vibration detecting element for detecting vibration in a second predetermined direction perpendicular to the first predetermined direction, and the first and second vibration detecting elements. Mounting means for supporting the vibration detecting element, the mounting means being detachable from a part of the vibration feeder , and being always parallel to the detection direction of the first and second vibration detecting elements. wherein displaying the rotation direction of the elliptical vibration of the vibrating feeder to the display unit from the phase difference detection Todoroki attachment is freely said vibration detecting element attached to a portion of the vibration feeder φ as the operation result of the microcomputer This is achieved by a portable elliptical vibration detection display characterized by the above.

[0009]

The detection output of the first vibration detecting element and the second vibration detecting element is such that the elliptical vibration at this time is from the resonance point, that is,
The phase difference φ is determined by how much the driving frequency is shifted at this time. This phase difference is electrically detected by a phase difference detector, for example.
The direction of the elliptical vibration can be specified in accordance with the size of, so that the direction of the elliptical vibration can be detected automatically and without any need for a strobe flash or the like. Therefore, even if the amplitude of the long axis and the amplitude of the short axis are the same, the vibrator can be set in the rotation direction at which the transfer speed is higher.

[0010] Further, by mounting the mounting member to a part of the vibration feeder so that the first vibration detection element and the second vibration detection element are mounted on a surface parallel to the vibration direction of the vibration feeder , the vibration is automatically generated. The phase difference φ of the vibration in the X direction and the Y direction of the feeder can be measured by a microcomputer, so that this is displayed on the display unit of the front panel unit, for example, in the right direction or the left direction by characters, or by an arrow. Can be displayed by a circle.
Other displays are possible.

[0011]

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a block diagram showing an embodiment of a method for detecting an elliptical vibration according to the present invention;

First, the electric circuit will be described with reference to FIG. In FIG. 1, the vibration detector 40 has a shape as shown in FIG. 3, and as will be described later, it incorporates detection elements 40a and 40b, and these detection outputs are supplied to low-pass filters 41a and 41b, respectively. Here, high-cycle signal components such as noise are removed, and this output is supplied to amplifiers 42a and 42b. The amplified output is supplied to A / D converters 43a and 43b to convert an analog value into a digital value, and the digital value is supplied to a microcomputer or a computer 45. The input terminal side of the computer 45 further includes a phase difference detector 44 receiving the outputs of the low-pass filters 41a and 41b.
Is connected. The computer 45 receives the digital output from the A / D converters 43a and 43b and the output of the phase difference detector 44, performs a predetermined operation, and supplies the operation result to the liquid crystal display units 46, 47, 48 and 49. ing.

That is, the amplitude of the elliptical vibration V in the long axis direction is displayed on the long axis LCD display section 47 by a computer 45 using an arithmetic expression described later, and the amplitude of the elliptical vibration V in the short axis direction is displayed on the short axis LCD. It is displayed on the display unit 48. The vibration angle LCD display unit 49 is also configured to display a vibration angle with respect to a certain axis, that is, a tilt angle θ of the long axis with respect to the horizontal direction, which is calculated by the computer 45 in the same manner. Further, the rotation direction of the elliptical vibration is displayed on the liquid crystal display unit 46.

Next, the details of the vibration detector 40 will be described with reference to FIG. This is an almost cylindrical casing 5
The first and second force detecting elements 40 a and 40 b made of, for example, lead zirconate are held on the mounting plate 52. The flange portion of the casing 50 is fixed to the mounting plate 52 by, for example, screwing. The detection outputs of the force detection elements 40a and 40b are led out through conductors 53a and 53b. According to the present invention, the first and second force detecting elements 40a,
The force detection directions F ₁ and F ₂ of 40b are perpendicular to each other. It is also assumed that a flat magnet is mounted on the back surface of the mounting plate 52. Therefore, the whole vibration detector 40 can be easily attached to and detached from the vibrating body by the magnet. The conductors 53a and 53b are collectively referred to as a cord 55.

Further, according to the present invention, A /
From the outputs A and B of the D converters 43a and 43b and the output of the phase difference detector 44, the amplitude and the vibration angle in the major axis and minor axis directions and the rotation direction of the vibration are calculated by the methods described later. However, these are displayed on the liquid crystal display units 47, 48, and 49 as digital values as described above. The vibration rotation direction is graphically displayed on the vibration rotation direction display section 46.

FIG. 8 shows the elliptical vibration V of the trough 2 in an enlarged manner. From the outputs of the detecting elements 40a and 40b in the above-mentioned vibration detector 40, the elliptical vibration V of the elliptical vibration V is obtained by the method of the present invention. The amplitude A in the direction of rotation (the direction of material transfer to the right), the longitudinal axis XX, and the amplitude B in the direction perpendicular thereto are calculated. In FIG. 8, the elliptical vibration V in the x-axis direction of the elliptical vibration V corresponds to the horizontal x-axis direction and the vertical y-direction in FIG. If the amplitude a in and the amplitude in the y direction is b,
These are x = asinωt and y = b sin (ωt +
φ), which are obtained from the detection elements 40a , 40b . An elliptical vibration force is obtained by a combined force of the vibration force in the x-axis direction and the vibration force in the y-axis direction, whereby the trough 2 performs the elliptical vibration V. According to the present embodiment, the detection element 40a and the The phase difference φ of the detection output of 40 b is detected by the phase difference detector 44 and supplied to the computer 45. Where the vibration force b in the y-axis direction
sin (ωt + φ) and vibration force x = a in the x-axis direction
The relationship between the phase difference φ from sinωt and the mode of the elliptical vibration will be described. When φ = 0 degrees, linear vibration is performed as shown by A in FIG. This vibration angle corresponds to θ in FIG. 8, and is a vibration easily obtained by, for example, a conventionally known electromagnet driving unit. The mode changes depending on the phase difference of the force in the x-axis direction and the y-axis direction. When the phase difference φ is 0 ° <φ <90 °, the mode is an elliptical vibration as indicated by B. The trajectory is different in the plus displacement direction and the minus displacement direction of the vibration. In the plus displacement direction, the vibration oscillates in an upper portion of the trajectory as indicated by an arrow, and in the minus displacement direction, oscillates in a lower trajectory. That is, the vibration is performed while drawing a locus around the clockwise direction. As shown by C in FIG. 5, at φ = 90 °, the direction of the major axis of the ellipse is parallel to the x-axis direction.
The direction of the short axis perpendicular to this is parallel to the y axis. Further, when φ is 90 ° <φ <180 °, the direction of the major axis of the ellipse is inverted with respect to the x-axis, but the trajectory of the vibration is clockwise as indicated by the arrow. When φ = 180 ° as shown in FIG. 6A, linear vibration is performed, but the inclination angle of the linear vibration is the inclination angle in the opposite direction to the linear vibration of FIG. 5A. Furthermore, φ is 180 ° <
When φ <270 °, elliptical vibration is performed, but the direction is opposite to the direction shown in FIG. 5B, and the trajectory of the vibration is counterclockwise. Further, as shown by C in FIG. 6, at φ = 270 °, elliptical vibration is performed, but the vibration is such that the major axis is parallel to the x-axis direction, and the trajectory of the vibration is counterclockwise as indicated by the arrow.
Further, when φ is 270 ° <φ <360 °, the direction of the long axis is the same as B in FIG. 5 as shown by D in FIG. 6, but the trajectory of the vibration is counterclockwise.

The above way but the vibration mode by a phase difference of the vibration force in the x-axis and y-axis directions is to change, such φ is detected by the phase difference detector 44 in FIG. 1, the size 5 and FIG.
The direction of vibration rotation can be easily understood .

The outputs from the A / D converters 43a and 43b are calculated by the computer 45. If the calculated values are a and b, the following equations (1) and (2) are used.
Fuhaba A of the major axis in FIG. 8 by the arithmetic expressions such as 2, the minor axis Fuhaba B, vibration angle θ is calculated. These A, B, θ
The transfer speed changes depending on the size of the
In addition to the direction, it is also necessary to obtain these values.
Above, it is effective.

[0019]

(Equation 1)

From the above equation 1, the amplitude A in the major axis direction, that is,
The maximum value of the vibration displacement in the X-axis direction from the origin 0 and the amplitude B in the short-axis direction, that is, the maximum value of the vibration displacement in the Y-axis direction from the origin 0
The value and the vibration angle θ are represented by the following Expression 2.

[0021]

(Equation 2)

The above A, B and θ are calculated by digital values in the CPU 45.
In this embodiment, b is measured with higher accuracy than in the prior art as follows. That is, each output of the vibration detector 40 has a sinusoidal waveform as a whole, but includes noise, and large noise often exists for some reason. When calculating the amplitude, the amplitude is measured from the maximum value and the minimum value by double integration, so that when a large noise is generated, the amplitude is measured larger than the actual amplitude. Will be. If it is an ideal sine waveform without such noise, there is no problem and accurate measurement can be performed.In general, noise of various sizes is often included. According to this, this effect can be eliminated.

That is, the amplitude component in the F ₁ direction to be measured (F
Consider only _one direction, the same applies to the F ₂ direction. ) Is At, the displacement x at each time is given by Equation 3 below.

[0024]

(Equation 3)

In this embodiment, the acceleration mean square R.S. M.
S. (Root Mean Square), that is, calculate the effective value. That is, the acceleration at each moment is integrated over time, divided by the time during this time, and the average value is obtained to obtain the following Expression 4.

[0026]

(Equation 4)

Thus, the amplitude component is obtained from the above equation (4), as shown in the following equation (5), and this is used in the arithmetic expression as the above a.

[0028]

(Equation 5)

That is, the twice-differentiated value of x is measured at each instant, and the amplitude component At is calculated by taking the time-averaged value. Therefore, even if a large noise is generated in a short time by taking an average value at a certain time, this effect can be almost ignored.

The F ₂ direction is calculated in the same manner, and the amplitude component in this direction is obtained as shown in the following Expression 6, which is the above b.

[0031]

(Equation 6)

Next, a portable vibration detecting and displaying device according to an embodiment of the present invention will be described with reference to FIGS.

FIG. 9 is a front view of the portable vibration detecting and displaying device of the present embodiment. In the drawing, the portable vibration detecting and displaying device is indicated by 60 as a whole, and has a casing 61 having a substantially rectangular shape. As shown in FIG. 10, the liquid crystal display device 64 and the battery 67 are built therein.
7 is stably supported via a mounting portion 68. The microcomputer 45 and the above-described liquid crystal display device 64 shown in FIG. At a predetermined position. The power switch 6 is located on the left side of the front panel 62.
9 is provided, which is a publicly known double push switch. When the switch is pressed once, the battery 67 supplies power to each section, and when the switch is further pressed, the power is shut off. On the right side, a lamp 70 that is turned on when the power switch 69 is turned on is provided. When it is on, it indicates that the power of the built-in battery 67 is being supplied to each unit. A display section 71 according to the present invention is provided below these. In the embodiment, the display section 71 is arranged in the vertical direction, and the rotation direction display section 46, the long axis amplitude display section 47, and the short axis amplitude display section 4 are arranged.
8, a vibration angle display section 49, which is a liquid crystal section of the above-described liquid crystal display device 64, which calculates a detection output from the vibration detector 40 by the microcomputer 45 as described above, and outputs a digital value as a result. Alternatively, the liquid crystal display device 64 is driven by the driver output as the output of the microcomputer with the rotation direction of the elliptical vibration, and the major axis and minor axis amplitude vibration angles are digital values on the display units 47, 48, 49 as shown in the figure, for example. Will be displayed. The display unit 46 shows the rotation direction in a graphic display.

In FIG. 10, the A / D converter 43
The terminals of a trimmer resistance device including trimmer resistance adjusters 85 and 86 are connected to the terminals a and 43b.
The analog inputs in the converters 43a and 43b are adjusted. The microcomputer 45 receives the digital output of the A / D converters 43a and 43b in the gain control units ga and gb of the amplifiers 42a and 42b.
a, a gain control signal for changing the gain of 42b.

In FIG. 10, reference numeral 100 denotes a connector, into which a plug connected to the end of the cord 55 from the vibration detector 40 in FIG. 3 is inserted.

As described above, according to this embodiment, the amplitude of the major axis and the minor axis of the elliptical vibration can be accurately obtained even if the vibration detector 40 is attached to the vibrating body arbitrarily. The vibration angle θ of the long axis is with respect to the detection direction of the force detection element 40a. Therefore, if the mounting direction of the force detecting element 40a is parallel to the bottom surface of the trough 2, the angle of vibration becomes the vibration angle.

Although the embodiments of the present invention have been described above, the present invention is, of course, not limited thereto, and various modifications can be made based on the technical concept of the present invention.

For example, in the above-described embodiment, a figure (circle) is used to display the rotation direction of the elliptical vibration, but instead, a character such as "clockwise" or "counterclockwise" is displayed. You may.

In the above embodiment, not only the rotation direction but also the amplitude and the vibration angle of the long axis and the short axis are displayed.
These may be omitted.

In the above embodiment, the display section 71 is provided with the rotation direction display section 46, the long axis amplitude display section 47, the amplitude display section 48 in the short axis direction, and the vibration angle display section 49. The entire display may be displayed on one display unit, or by providing a function key on the front panel unit, this push operation sequentially changes the rotation direction, the amplitude of the long axis, the amplitude of the short axis, and the vibration angle. You may make it display.

[0041]

As described above, the vibration filter according to the present invention is used.
According to elliptical vibration detection method and the indicator over da, wherein
According to the first aspect of the present invention, the first and second vibration detecting elements which are orthogonal to each other are provided.
As in the conventional child in a part of the vibrating feeder, the mounting angle what
Only attached without al adjusting, also according to claim 3
According to the invention, the vibration detector is also part of the vibration feeder.
Mounting without adjusting the mounting angle as before
The direction of rotation of the elliptical vibration can be immediately recognized only by touching .

[Brief description of the drawings]

FIG. 1 is a circuit diagram for implementing the method of the present invention.

FIG. 2 is a side view of a resonance type vibration feeder in which a vibration detector embodying the present invention is mounted and a vibration motor is used as a vibration unit.

FIG. 3 is a perspective view of the vibration detector.

FIG. 4 is a plan view showing an arrangement of force detection elements in the vibration detector.

FIG. 5 is a diagram showing a phase difference between vibration forces in each axis direction in rectangular coordinates and a mode of elliptical vibration generated by the phase difference, wherein A denotes elliptical vibration when the phase difference φ is 0, that is, a straight line; Vibration graph. B is a graph of the elliptical vibration when φ is 0 to 90 degrees. C is a graph of the elliptical vibration when the phase difference φ is 90 degrees. D is a graph of the elliptical vibration from 90 degrees to 180 degrees.

6A is a graph of an elliptical vibration at φ = 180 degrees, FIG. 6B is a graph of an elliptical vibration mode at a phase difference φ in a range of 180 degrees to 270 degrees, and C is φ = 270 degrees.
11 is a graph of the mode of the elliptical vibration in degrees, and D is a graph of the mode of the elliptical vibration when the phase difference φ is in the range of 270 degrees to 360 degrees.

FIG. 7 is a side view showing vibration of a resonance type vibration feeder using a vibration motor as a vibration source.

FIG. 8 is a graph showing an enlarged elliptical vibration in the vibration feeder.

FIG. 9 is a front view of a display according to an embodiment of the present invention.

FIG. 10 is a side view of the same device.

FIG. 11 is a front view of an amplitude nameplate used for measuring an amplitude by a conventional method.

FIG. 12 is a front view showing the same operation.

[Explanation of symbols]

Reference Signs List 40 Vibration detector 40a Detection element 40b Detection element 44 Phase difference detector 45 Microcomputer 60 Portable vibration display 64 Liquid crystal display 67 Battery 71 Display

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Claims (4)

(57) [Claims] 1. A first vibration detecting element for detecting vibration in a first predetermined direction is mounted on a surface parallel to a vibration direction of a vibration feeder , and a second predetermined direction perpendicular to the first predetermined direction. A second vibration detecting element for detecting the vibration of
Mounted on a plane parallel to the vibration direction of Ida, from the phase difference φ of detection output of said first oscillation detecting element and the second vibration detection element so as to obtain a vibration direction of rotation of elliptical vibration of the vibrating feeder A method for detecting an elliptical vibration of a vibration feeder . 2. The amplitude of the elliptical vibration of the vibration feeder in the major axis direction and the amplitude in the minor axis direction are A and B, respectively, and the detection outputs of the first and second vibration detecting elements are a , B, the A and B and the vibration angle θ are calculated by the following equation. A ²
= (A cos θ) ² + (b sin θ) ² + abs
in2θ · cosφ, B ² = (a sin θ) ² + (b
cos θ) ² -ab sin2θ · cos φ, tan
2θ = 2ab cosφ / (a ² −b ² ). 3. A front panel portion of a casing containing at least a microcomputer, a battery, and a display element connected to an output terminal of the microcomputer, receiving an output of the display element,
A main body provided with a display unit for displaying at least the rotation direction of the elliptical vibration, a vibration detector detachable from a part of the vibration feeder , and a detection force of the vibration detector, which is connected to a connector of the main body. The vibration detector comprises a first vibration detecting element for detecting vibration in a first predetermined direction and a vibration in a second predetermined direction perpendicular to the first predetermined direction. A second vibration detecting element, and mounting means for supporting the first and second vibration detecting elements, the mounting means being detachable from a part of the vibration feeder , and always providing the first and second vibration detecting elements. A phase difference φ of a detection force of the vibration detecting element, which is attachable to the part in parallel with the detection direction of the second vibration detecting element and is mounted on a part of the vibration feeder.
A portable elliptical vibration detection and display device, wherein a rotation direction of the elliptical vibration of the vibration feeder is displayed on the display unit as a result of calculation by the microcomputer. 4. The amplitude of the elliptical vibration of the vibration feeder in the major axis direction and the amplitude in the minor axis direction are A and B, respectively, and the detection outputs of the first and second vibration detecting elements are a 4. The elliptical vibration detection display according to claim 3, wherein A, B and the vibration angle θ are calculated by the microcomputer from the following formulas, and these are digitally displayed on the display unit. A ² = (a cos θ) ² + (b sin θ) ² +
ab sin2θ · cosφ, B ² = (a sinθ)
^{2 + (b cosθ) 2 -absin2θ} · cosφ,
tan2θ = 2ab cosφ / (a ² −b ² ).