Method for determining a mass of an attached implement for a utility vehicle

Abstract

A method for determining a mass of an implement attached to a vehicle includes providing a powerlift having at least one upper link and one lower link, a support structure, and the implement. The method also includes defining an angle () between the upper link and a vehicle horizontal line, an angle () between the lower link and a vehicle horizontal line, an angle of inclination () of a vehicle horizontal line relative to a terrestrial horizontal line, a path (AK) that represents a connection along the lower link between the support structure and the implement, and a force (F.sub.E) impinging on a connection between the upper link and the implement and acting along the upper link. The mass is determined as a function of at least one of the angle (), the angle (), the angle of inclination (), the path (AK), and the force (F.sub.E).

Claims

1. A method of deploying a utility vehicle for work using a mass of an implement attached at a front of the utility vehicle for moving the implement into different positions at the front of the utility vehicle, the method comprising: providing a front powerlift having at least one upper link and at least one lower link, a support structure formed at the front of the utility vehicle, and the implement being articulatably coupled to the support structure, wherein each of the at least one upper link and the at least one lower link is pivotably mounted to each of the support structure and the implement; defining an angle () between the at least one upper link and a vehicle horizontal line, an angle () between the at least one lower link and the vehicle horizontal line, an angle of inclination () of the vehicle horizontal line relative to a terrestrial horizontal line, a path (AK) that represents a connection along the at least one lower link between the support structure and the implement, and a force (F.sub.E) impinging on a connection between the at least one upper link and the implement and acting along the at least one upper link; and determining the mass of the implement as a function of at least one of the angle (), the angle (), the angle of inclination (), the path (AK), and the force (F.sub.E).

2. The method of claim 1, further comprising: providing the front powerlift with an adjustable-length lifting arm; and determining the mass of the implement as a function of an angle () that is enclosed by a vehicle-vertical line and a connecting path (BD) between two operative ends of the lifting arm.

3. The method of claim 2, further comprising determining the mass of the implement as a function of a force (F.sub.Zyl) acting between the two operative ends of the lifting arm.

4. The method of claim 2, wherein one operative end of the lifting arm comprises an articulated connection to the support structure, and another operative end of the lifting arm comprises an articulated connection to a link-connecting point of the at least one lower link.

5. The method of claim 4, further comprising determining the mass of the implement as a function of a path (AD) defined by a connection along the at least one lower link between the support structure and the link-connecting point.

6. The method of claim 2, further comprising providing a lifting arm including a piston-cylinder unit.

7. The method of claim 1, further comprising determining a center of gravity of the implement as a function of the determined mass of the implement.

8. The method of claim 1, further comprising: defining a coordinate system having an x-axis oriented parallel to a longitudinal direction of the utility vehicle and a z-axis oriented parallel to a vertical direction of the utility vehicle, wherein the x-axis and the z-axis intersect at a zero point of the coordinate system; and determining at least one of the mass or a center of gravity of the implement based on the coordinate system.

9. The method of claim 8, further comprising arranging the zero point on a rear axle of the utility vehicle.

10. The method of claim 8, further comprising determining an x-coordinate of the center of gravity of the implement as a function of the mass of the implement.

11. The method of claim 10, wherein the determining the x-coordinate of the center of gravity step is determined as a function of at least one of the angle () between the at least one upper link and a vehicle horizontal line, the force (F.sub.E) impinging on the connection between the at least one upper link and the implement and acting along the at least one upper link, an x-coordinate of an articulation point of the at least one lower link on the implement, a difference between the x-coordinates of an articulation point of the at least one upper link on the implement and an articulation point of the at least one lower link on the implement, and a difference between the z-coordinates of an articulation point of the at least one upper link on the implement and an articulation point of the at least one lower link on the implement.

12. The method of claim 1, wherein determining the mass of the implement includes determining the mass of the implement as a function of the angle (), the angle (), the angle of inclination (), the path (AK), and the force (F.sub.E).

13. A front powerlift system for work deployment of a utility vehicle having a vehicle horizontal line defining an angle of inclination () relative to a terrestrial horizontal line, the system comprising: at least one upper link defining an angle () between the at least one upper link and the vehicle horizontal line of the utility vehicle; at least one lower link defining an angle () between the at least one lower link and the vehicle horizontal line of the utility vehicle; a support structure formed at the front of the utility vehicle and having the at least one upper link and the at least one lower link pivotably mounted thereto; and an implement articulatably coupled to the support structure, the at least one upper link and the at least one lower link pivotably mounted thereto, and a mass of the implement being configured as a function of at least one of the angle (), the angle (), the angle of inclination (), a path (AK) representing a connection along the at least one lower link between the support structure and the implement, and a force (FE) impinging on a connection between the at least one upper link and the implement and acting along the at least one upper link.

14. The system of claim 13, wherein the mass of the implement is configured as a function of the angle (), the angle (), the angle of inclination (), a path (AK) representing a connection along the at least one lower link between the support structure and the implement, and a force (F.sub.E) impinging on a connection between the at least one upper link and the implement and acting along the at least one upper link.

15. The system of claim 13, further comprising an adjustable-length lifting arm having two operative ends including a first operative end of the lifting arm comprising an articulated connection to the support structure and a second operative end of the lifting arm comprising an articulated connection to a link-connecting point of the at least one lower link, the mass of the implement being further configured as a function of at least one of a force (F.sub.Zyl) acting between the two operative ends of the lifting arm, a path (AD) defined by a connection along the at least one lower link between the support structure and the link-connecting point, and an angle () that is enclosed by a vehicle-vertical line and a connecting path (BD) between the two operative ends of the lifting arm.

16. The system of claim 13, wherein a center of gravity of the implement is configured as a function of the mass of the implement.

17. The system of claim 13, further comprising a coordinate system having: an x-axis oriented parallel to a longitudinal direction of the utility vehicle; and a z-axis oriented parallel to a vertical direction of the utility vehicle, wherein the x-axis and the z-axis intersect at a zero point of the coordinate system, and the coordinate system forming a basis of at least one of the mass and a center of gravity of the implement.

18. The system of claim 17, wherein the zero point is arranged on a rear axle of the utility vehicle.

19. The system of claim 17, further comprising an x-coordinate of the center of gravity of the implement configured as a function of at least one of an angle between the at least one upper link and the vehicle horizontal line, a force (F.sub.E) impinging on the connection between the at least one upper link and the implement and acting along the at least one upper link, an x-coordinate of an articulation point of the at least one lower link on the implement, a difference between the x-coordinates of an articulation point of the at least one upper link on the implement and an articulation point of the at least one lower link on the implement, and a difference between the z-coordinates of an articulation point of the at least one upper link on the implement and an articulation point of the at least one lower link on the implement.

20. A front powerlift system for work deployment of a utility vehicle having a vehicle horizontal line defining an angle of inclination () relative to a terrestrial horizontal line, the system comprising: at least one upper link defining an angle () between the at least one upper link and the vehicle horizontal line of the utility vehicle; at least one lower link defining an angle () between the at least one lower link and the vehicle horizontal line of the utility vehicle; a support structure formed at the front of the utility vehicle and having the at least one upper link and the at least one lower link pivotably mounted thereto; an adjustable-length lifting arm having two operative ends including one operative end of the lifting arm comprising an articulated connection to the support structure, and another operative end of the lifting arm comprising an articulated connection to a link-connecting point of the at least one lower link; and an implement articulatably coupled to the support structure, the at least one upper link and the at least one lower link pivotably mounted thereto, and a mass of the implement configured to be determined to precisely transfer the implement into a desired target position by a function of the angle (), the angle (), the angle of inclination (), a path (AK) representing a connection along the at least one lower link between the support structure and the implement, and a force (FE) impinging on a connection between the at least one upper link and the implement and acting along the at least one upper link, an angle () that is enclosed by a vehicle-vertical line and a connecting path (BD) between the two operative ends of the adjustable-length lifting arm, a force (FZyl) acting between the two operative ends of the lifting arm, and a path (AD) defined by a connection along the at least one lower link between the support structure and the link-connecting point.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The above-mentioned aspects of the present disclosure and the manner of obtaining them will become more apparent and the disclosure itself will be better understood by reference to the following description of the embodiments of the disclosure, taken in conjunction with the accompanying drawings, wherein:

(2) FIG. 1 is a schematic side view of an attached implement articulated to a front powerlift at the front end of a utility vehicle;

(3) FIG. 2 is a schematic representation of forces impinging on a lower link of the front powerlift according to FIG. 1; and

(4) FIG. 3 is a schematic representation of forces impinging on the attached implement from FIG. 1.

DETAILED DESCRIPTION

(5) In FIG. 1, an attached implement 10 is schematically shown articulated to the front end of a utility vehicle (such as a tractor). A front powerlift 12 is articulated onto a support structure 14 of the utility vehicle. The front powerlift 12, having an upper link 16 and two parallel lower links 18, forms a three-point hitch for receiving the attached implement 10. The upper link 16 has an articulated connection via an articulation point C to the support structure 14 and via an articulation point E to the attached implement 10. In a transverse direction of the utility vehicle, running perpendicular to the plane of FIG. 1, the upper link 16 is arranged centrally between two parallel lower links 18. Each lower link 18 has an articulated connection via an articulation point A to the support structure 14 and via an articulation point K to the attached implement 10. The articulation points E, K on the attached implement are designed in the usual manner, e.g., as catch hooks for the front powerlift 12.

(6) In the present embodiment, the articulation point C permits three different positions in the vertical direction 20 for articulating the upper link 16. The respective position is defined by a user and accordingly installed. An angle is enclosed between the upper link 16 and a vehicle horizontal line 22. An angle is enclosed between each lower link 18 and a vehicle horizontal line 22.

(7) The angles formed between the vehicle horizontal line 22 and a vehicle vertical line 32, particularly angles and , relate to a fixed vehicle coordinate system 33 having an x-axis and a z-axis. The x-axis runs parallel to the longitudinal direction 24 of the utility vehicle or parallel to the vehicle horizontal line 22. The z-axis runs parallel to the vertical direction 20 of the utility vehicle or parallel to the vehicle vertical line 32. The inclination of the support structure 14, and thus of the vehicle horizontal line 22 of the utility vehicle, relative to the terrestrial horizontal line 21, is represented by an angle of inclination . The angle of inclination takes on values greater than zero when the utility vehicle is oriented uphill in the forward direction. The angle of inclination analogously assumes values less than zero when the utility vehicle is oriented downhill in the forward direction.

(8) A lifting arm designed in a conventional manner as an adjustable-length lifting cylinder 26 (piston-cylinder unit) is a component of the front powerlift 12. One operative end 28 of the lifting cylinder 26 has an articulated connection to an articulation point B of the support structure 14, whereas the other operative end 30 of the lifting cylinder 26 is articulated to a link-connecting point D of the lower link 18. A connecting path between the two operative ends 28, 30 encloses an angle with the vehicle vertical line 32 running in the vertical direction 20. Both lower links 18 are connected to a lifting cylinder 26 in the manner described.

(9) For mathematical/geometrical determination of a mass m and a center of gravity S of the attached implement 10, a coordinate system 33 is defined, the x-axis of which is oriented parallel to the longitudinal direction 24 and the z-axis of which is oriented parallel to the vertical direction 20. In the present embodiment of FIG. 1, the zero point of this coordinate system 33 lies on a schematically indicated rear axle 34 of the utility vehicle.

(10) FIG. 2 presents different forces impinging on the force system between support structure 14 and attached implement 10. A force of which the x-component is designated F.sub.Ax and the z-component is designated F.sub.Az impinges on the articulation point A. A cylinder force F.sub.Zyl is active between the two operative ends 28, 30 of the lifting cylinder 26. A force designated F.sub.K impinges on the articulation point K between the attached implement 10 and the lower link 18, and is oriented at an angle relative to the vehicle vertical line 32.

(11) Additional forces that impinge on the attached implement 10 are illustrated in FIG. 3. A force impinging on the articulation point E and acting along the upper link 16 is designated F.sub.E. The weight force of the attached implement 10 at center of gravity S is marked m.Math.g, where g is the gravitational constant.

(12) Based on the torque and force equilibria in FIG. 2, the following relationships can be assumed at the front powerlift 12:

(13) $\begin{array}{cc}.Math.M_i^{\left(A\right)}=0.fwdarw.0=-\stackrel{\_}{\mathrm{AD}}.Math.{\stackrel{.fwdarw.}{F}}_{\mathrm{Zyl}}.Math.\sin .Math.\sin +\stackrel{\_}{\mathrm{AD}}.Math.{\stackrel{.fwdarw.}{F}}_{\mathrm{Zyl}}.Math.\cos .Math.\cos -\stackrel{\_}{\mathrm{AK}}.Math.{\stackrel{.fwdarw.}{F}}_K.Math.\sin .Math.\sin -\stackrel{\_}{\mathrm{AK}}.Math.{\stackrel{.fwdarw.}{F}}_K.Math.\cos .Math.\cos ,& \left(1\right)\\ .Math.{\stackrel{.fwdarw.}{F}}_{\mathrm{ix}}=0.fwdarw.0={\stackrel{.fwdarw.}{F}}_K.Math.\sin -{\stackrel{.fwdarw.}{F}}_{\mathrm{Zyl}}.Math.\sin +{\stackrel{.fwdarw.}{F}}_{\mathrm{Ax}},& \left(2\right)\\ .Math.{\stackrel{.fwdarw.}{F}}_{\mathrm{iz}}=0.fwdarw.0={\stackrel{.fwdarw.}{F}}_K.Math.\cos -{\stackrel{.fwdarw.}{F}}_{\mathrm{Zyl}}.Math.\cos +{\stackrel{.fwdarw.}{F}}_{\mathrm{Az}}.& \left(3\right)\end{array}$

(14) Due to the torque and force equilibria in FIG. 3, the following additional relationships at the front powerlift 12 can be assumed:

(15) $\begin{array}{cc}.Math.M_i^{\left(K\right)}=0.fwdarw.0=\left(E_x-K_x\right).Math.{\stackrel{.fwdarw.}{F}}_E.Math.\sin +\left(E_z-K_z\right).Math.{\stackrel{.fwdarw.}{F}}_E.Math.\cos -\left(S_x-K_x\right).Math.m.Math.g.Math.\cos +\left(S_z-K_z\right).Math.m.Math.g.Math.\sin ,& \left(4\right)\\ .Math.{\stackrel{.fwdarw.}{F}}_{\mathrm{ix}}=0.fwdarw.0={\stackrel{.fwdarw.}{F}}_K.Math.\sin -{\stackrel{.fwdarw.}{F}}_E.Math.\cos -m.Math.g.Math.\sin .fwdarw.{\stackrel{.fwdarw.}{F}}_K.Math.\sin ={\stackrel{.fwdarw.}{F}}_E.Math.\cos +m.Math.g.Math.\sin ,& \left(5\right)\\ .Math.{\stackrel{.fwdarw.}{F}}_{\mathrm{iz}}=0.fwdarw.0={\stackrel{.fwdarw.}{F}}_K.Math.\cos +{\stackrel{.fwdarw.}{F}}_E.Math.\sin -m.Math.g.Math.\cos .fwdarw.{\stackrel{.fwdarw.}{F}}_K.Math.\cos =-{\stackrel{.fwdarw.}{F}}_E.Math.\sin +m.Math.g.Math.\cos .& \left(6\right)\end{array}$

(16) By inserting equations (5) and (6) into equation (1), and by solving equation (1) for the mass m of the attached implement 10, it follows that for the mass m

(17) $\begin{array}{cc}m=\frac{\stackrel{\_}{\mathrm{AD}}.Math.{\stackrel{.fwdarw.}{F}}_{\mathrm{Cyl}}.Math.\cos \left(+\right)+\stackrel{\_}{\mathrm{AK}}.Math.{\stackrel{.fwdarw.}{F}}_E.Math.\sin \left(-\right)}{\stackrel{\_}{\mathrm{AK}}.Math.g.Math.\cos \left(-\right)}.& \left(7\right)\end{array}$
Thus, the mass m is determined as a function of the angle between the upper link 16 and the vehicle horizontal line 22, the angle between the lower link 18 and the vehicle horizontal line 22, the angle between the lifting cylinder 26 and the vehicle vertical line 32, the angle of inclination of the support structure 14 or the vehicle horizontal line 22 of the utility vehicle in relation to the terrestrial horizontal line 21, the path AK along the lower link 18 as the connection between the articulation points A and K, the path AD along the lower link 18 between the articulation point A and the link-connecting point D, the cylinder force F.sub.Zyl on the lifting cylinder 26, and the force F.sub.E along the upper link 16.

(18) The lengths of paths AD and AK are known design parameters of the front powerlift 12. The force F.sub.Zyl can be measured by means of a pressure sensor or differential pressure sensor in the lifting cylinder 26. The force F.sub.E can be measured by means of pressure sensors and a length sensor on the upper link 16, for example, or alternatively (in the case of a mechanical sensor 16) by means of a two-axial force sensor at the articulation point C. The angles , and in equation (7) can be derived as follows:

(19) $={\tan }^{-1}\left(\frac{A_z-D_z}{A_x-D_x}\right)$$={\tan }^{-1}\left(\frac{D_x-B_x}{D_z-B_z}\right)$$={\tan }^{-1}\left(\frac{E_z-C_z}{E_x-C_x}\right)={\tan }^{-1}\left(\frac{{\stackrel{.fwdarw.}{F}}_{\mathrm{Cz}}}{{\stackrel{.fwdarw.}{F}}_{\mathrm{Cx}}}\right)$

(20) The x-coordinates A.sub.x, B.sub.x and the z-coordinates A.sub.z, B.sub.z are known since the articulation points A and B thereof on the support structure 14 have fixed coordinates relative to the coordinate system 33. The x-coordinate C.sub.x and the z-coordinate C.sub.z of the articulation point C are also known, and are therefore either determined by sensors or specified by the operator.

(21) The above-mentioned angles can also be measured by means of suitable angle sensors. Alternatively, biaxial force measuring pins can be used, as illustrated with reference to angle and the two force components F.sub.Cx (along the x-axis of the coordinate system 33) and F.sub.Cz (along the z-axis of the coordinate system 33) in schematic form (see FIG. 3).

(22) The variable x-coordinates and z-coordinates D.sub.x and D.sub.z of the link-connecting point D and E.sub.x and E.sub.z of the articulation point E can be derived mathematically, as described below.

(23) To calculate the connecting point D, two circles are defined. The first circle has a radius corresponding to the path BD with a circle center B and the second circle has a radius corresponding to the path AD with a circle center A. The associated circle equations are
(D.sub.xA.sub.x).sup.2+(D.sub.zA.sub.z).sup.2=AD.sup.2 (D.sub.xB.sub.x).sup.2+(D.sub.zB.sub.z).sup.2=BD.sup.2

(24) The two circle equations are solved for the z component of the coordinate D. This yields a straight-line equation of a straight line running through the two circle centers, of the form

(25) $\begin{array}{c}{D}_z=-\underset{\underset{u}{}}{\left(\frac{B_x-A_x}{B_z-A_z}\right)}.Math.D_x+\underset{\underset{s}{}}{\left(\frac{{\stackrel{\_}{\mathrm{AD}}}^2-{\stackrel{\_}{\mathrm{BD}}}^2+B_x^2+B_z^2-A_x^2-A_z^2}{2.Math.\left(B_z-A_z\right)}\right)}\\ D_z=-u.Math.D_x+s\end{array}$
With respect to the link-connecting point D, the quadratic equation that follows by inserting the straight-line equation into one of the two circle equations.

(26) $D_x^2+\underset{\underset{p_2}{}}{\frac{2.Math.\left(-B_x-u.Math.w\right)}{1+u^2}}.Math.D_x+\underset{\underset{q_2}{}}{\frac{B_x^2+w^2-{\stackrel{\_}{\mathrm{BD}}}^2}{1+u^2}}=0$
as well as the auxiliary variables u, s and w

(27) $u=\left(\frac{B_x-A_x}{B_z-A_z}\right)$$s=\left(\frac{{\stackrel{\_}{\mathrm{AD}}}^2-{\stackrel{\_}{\mathrm{BD}}}^2+B_x^2+B_z^2-A_x^2-A_z^2}{2.Math.\left(B_z-A_z\right)}\right)$$w=s-B_z$
are defined. This yields the following equations for the x-coordinate D.sub.x and the z-coordinate D.sub.z:

(28) $\begin{array}{cc}{D}_x=-\frac{p_2}{2}+\sqrt{\frac{p_2^2}{4}-q_2}=\frac{\left(B_x+u.Math.w\right)}{1+u^2}+\sqrt{{\left(\frac{\left(B_x+u.Math.w\right)}{1+u^2}\right)}^2-\frac{B_x^2+w^2-{\stackrel{\_}{\mathrm{BD}}}^2}{1+u^2}}& \left(8\right)\\ D_z=-u.Math.D_x+s.& \left(9\right)\end{array}$
The path BD can be measured by means of a length sensor or a distance sensor, for example.

(29) Two additional circles are defined for calculating the articulation point E. The first circle has a radius corresponding to the path EK with a circle center K and the second circle has a radius corresponding to the path CE with a circle center C. The associated circle equations are
(E.sub.xC.sub.x).sup.2+(E.sub.zC.sub.z).sup.2=CE.sup.2 (E.sub.xK.sub.x).sup.2+(E.sub.zK.sub.z).sup.2=EK.sup.2

(30) The two circle equations are solved for the z component of the articulation point E. This yields a straight-line equation of a straight line running through the two circle centers, of the form

(31) $\begin{array}{c}{E}_z=-\underset{\underset{n}{}}{\left(\frac{C_x-K_x}{C_z-K_z}\right)}.Math.E_x+\underset{\underset{b}{}}{\left(\frac{{\stackrel{\_}{\mathrm{EK}}}^2-{\stackrel{\_}{\mathrm{CE}}}^2+C_x^2+C_z^2-K_x^2-K_z^2}{2.Math.\left(C_z-K_z\right)}\right)}\\ E_z=-n.Math.E_x+b\end{array}$
With respect to the articulation point E, the quadratic equation that follows by inserting the straight-line equation into one of the two circle equations

(32) 0$E_x^2+\underset{\underset{p}{}}{\frac{2.Math.\left(-C_x-n.Math.z\right)}{1+n^2}}.Math.E_x+\underset{\underset{q}{}}{\frac{C_x^2+z^2-{\stackrel{\_}{\mathrm{CE}}}^2}{1+n^2}}=0$
and the auxiliary variables n, b and z

(33) $n=\left(\frac{C_x-K_x}{C_z-K_z}\right)$$b=\left(\frac{{\stackrel{\_}{\mathrm{EK}}}^2-{\stackrel{\_}{\mathrm{CE}}}^2+C_x^2+C_z^2-K_x^2-K_z^2}{2.Math.\left(C_z-K_z\right)}\right)$$z=b-C_z$
are defined. This yields the following equations for the x-coordinate E.sub.x and the z-coordinate E.sub.z:

(34) $\begin{array}{cc}{E}_x=-\frac{p}{2}+\sqrt{\frac{p^2}{4}-q}=\frac{\left(C_x+n.Math.z\right)}{1+n^2}+\sqrt{{\left(\frac{\left(C_x+n.Math.z\right)}{1+n^2}\right)}^2-\frac{C_x^2+z^2-{\stackrel{\_}{\mathrm{CE}}}^2}{1+n^2}}& \left(10\right)\\ E_z=-n.Math.E_x+b& \left(11\right)\end{array}$

(35) The path CE can also be measured by means of a length sensor or a distance sensor on the upper link 16, for example, or this path CE is known for a constant length of the upper link 16. The path EK is a defined value corresponding to the dimensioning of the attached implement 10.

(36) The x-coordinate S.sub.x of the center of gravity S relative to the coordinate system 33 can be determined by solving equation (4) for S.sub.x. In this case, the angle of inclination is assumed to be 0:

(37) $\begin{array}{cc}{S}_x=\frac{1}{m.Math.g}\left[\left(E_x-K_x\right).Math.{\stackrel{.fwdarw.}{F}}_E.Math.\sin +\left(E_z-K_z\right).Math.{\stackrel{.fwdarw.}{F}}_E.Math.\cos +K_x.Math.m.Math.g\right]& \left(12\right)\end{array}$

(38) Thus, the x-coordinate of the center of gravity S is determined as a function of the determined mass m, the angle between the upper link 16 and the vehicle horizontal line 22, the force F.sub.E on the upper link 16, the x-coordinate K.sub.x of the articulation point K of the lower link 18 at the attached implement 10, the difference between the x-coordinates E.sub.x and K.sub.x of the articulation points E and K at the attached implement 10, and the difference between the z-coordinates E.sub.z and K.sub.z of the articulation points E and K at the attached implement 10.

(39) The x-coordinate K.sub.x and the z-coordinate K.sub.z of the articulation point K follow from, for example
K.sub.x=A.sub.x+AK.Math.cos and K.sub.z=A.sub.zAK.Math.sin ,
where the path AK is a known design parameter. The other components and values of equation (12) can be derived according to the above description.

(40) While embodiments incorporating the principles of the present disclosure have been described hereinabove, the present disclosure is not limited to the described embodiments. Instead, this application is intended to cover any variations, uses, or adaptations of the disclosure using its general principles. Further, this application is intended to cover such departures from the present disclosure as come within known or customary practice in the art to which this disclosure pertains and which fall within the limits of the appended claims.