CONVEYANCE SYSTEM FOR TENSIONING IN ORDER TO POST-TREAT A RAPIDLY-SOLIDIFIED METAL STRIP, AND POST-TREATMENT METHOD

Abstract

A conveyance system for tensioning to post-treat a rapidly-solidified metal strip, and a method for post-treating the metal strip with the conveyance system is provided. The conveyance system comprises a tension roller assembly and a tensioning assembly, between which the metal strip is conveyed to be continuously post-treated under a predetermined tensile stress. The tension roller assembly comprises a single drive roller and a freely-rotating pressing roller. The metal strip is conveyed over an angle of wrap on the drive roller, and, with respect to the drive roller, the pressing roller is arranged at a contact point of the metal strip that defines one end of an angle of wrap . The method can include bridging the distance between the tensioning assembly and the tension roller assembly amid the insertion of the metal strip into the tension roller assembly, after a post-treatment region in space and in time.

Claims

1. A method for post-treating a rapidly-solidified metal strip, comprising: conveying a metal strip through a conveyance system, the metal strip having a composition of Fe.sub.100-a-b-c-d-x-y-zCu.sub.aNb.sub.bM.sub.cT.sub.dSi.sub.xB.sub.yZ.sub.z and up to 1 atom % impurities, wherein M is one or more of the elements Mo, Ta, or Zr; T is one or more of the elements V, Mn, Cr, Co, or Ni; and Z is one or more of the elements C, P, or Ge; and 0 atom %a<1.5 atom %, 0 atom %b<2 atom %, 0 atom %(b+c)<2 atom %, 0 atom %d<5 atom %, 10 atom %<x<18 atom %, 5 atom %<y<11 atom %, and 0 atom %z<2 atom %; the conveyance system including a tension roller assembly and a tensioning assembly, wherein the tension roller assembly a single drive roller and a freely rotating pressing roller; and the conveying step includes conveying the metal strip between the tensioning assembly and the tension roller assembly to continuously post-treat the metal strip under tension.

2. The method of claim 1, wherein the metal strip is conveyed over an angle of wrap on the drive roller; and, in relation to the drive roller, the pressing roller is arranged at a contact point of the metal strip that defines an end of an angle of wrap .

3. The method according to claim 1, wherein in order to achieve a desired value of the permeability or of the anisotropy field, a maximum value of a remanence ratio and a maximum value of a ratio of a coercivity field intensity to an anisotropy field intensity, H.sub.c/H.sub.a, as well as an allowable deviation range of each of these values are predetermined, wherein magnetic properties of the metal strip are measured continuously at an exit from a continuous furnace, and when deviations from the allowable deviation ranges of the magnetic properties are found, the tensile stress on the metal strip is adjusted correspondingly in order to bring the measured values of the magnetic properties back within the allowable deviation ranges.

4. A method for post-treating a rapidly-solidified metal strip, comprising the steps of: inserting a metal strip into a tensioning assembly before a post-treatment region in space and time; bridging the distance between the tensioning assembly and the tension roller assembly amid the insertion of the metal strip into the tension roller assembly, after a post-treatment region in space and in time; conveying the metal strip between the tensioning assembly and the tension roller assembly, wherein the metal strip is conveyed over an angle of wrap on the drive roller and, in relation to the drive roller, the pressing roller is arranged at a contact point of the metal strip that defines an end of an angle of wrap ; and post-treating the metal strip under tensile stress between the tensioning assembly and the tension roller assembly with continuous conveyance of the metal strip through the tensioning assembly and the tension roller assembly.

5. The method according to claim 4, wherein an optimum range for an angle of wrap of the metal strip on the drive roller is determined by recording and assessing a frequency of breaks of the metal strip based on the length of the metal strip of more than or equal to 1 km.

6. The method according to claim 4, wherein the metal strip is formed of a Co-based or Fe-based amorphous alloy, and the metal strip is post-treated at room temperature.

7. The method according to claim 4, wherein a heat treatment of the metal strip under tensile stress is carried out in the post-treatment region, between the tensioning assembly and the tension roller assembly.

8. The method according to claim 7, wherein the following are predetermined: a desired value of the permeability or of the anisotropy field; a maximum value of a remanence ratio J.sub.r/J.sub.s, of less than 0.1; and a maximum value of a ratio of a coercivity field intensity to an anisotropy field intensity H.sub.c/H.sub.a of less than 10%; as well as an allowable deviation range of each of these values; magnetic properties of the metal strip are continuously measured at an exit from a continuous furnace; and when deviations from the allowable deviation ranges of the magnetic properties are found, the tensile stress on the metal strip is adjusted correspondingly in order to bring the measured values of the magnetic properties back within the allowable deviation ranges.

9. The method according to claim 4, wherein metal strip has the following composition and is post-treated: Fe.sub.100-a-b-c-d-x-y-zCu.sub.aNb.sub.bM.sub.cT.sub.dSi.sub.xB.sub.yZ.sub.z and up to 1 atom % of impurities, wherein: M is one or more of the elements Mo, Ta, or Zr; T is one or more of the elements V, Mn, Cr, Co, or Ni; and Z is one or more of the elements C, P, or Ge; and 0 atom %a<1.5 atom %, 0 atom %b<2 atom %, 0 atom %(b+c)<2 atom %, 0 atom %d<5 atom %, 10 atom %<x<18 atom %, 5 atom %<y<11 atom %, and 0 atom %z<2 atom %.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0037] The invention shall now be described in greater detail with reference to embodiments depicted in the drawings.

[0038] FIG. 1 illustrates a schematic diagram of an S-roller system, as a conveyance system;

[0039] FIG. 2 illustrates a schematic diagram of a conveyance system according to a first embodiment of the invention;

[0040] FIG. 3 illustrates a schematic diagram of a conveyance system according to a second embodiment of the invention;

[0041] FIG. 4 illustrates a schematic diagram of a conveyance system according to a third embodiment of the invention; and

[0042] FIG. 5 illustrates a schematic diagram of a conveyance system according to a fourth embodiment of the invention.

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS

[0043] Components of the conveyance systems in the following FIG. 1 to 5 that fulfill the same functions are designated with the same reference numerals and not discussed further.

[0044] The S-roller system for tensioning in order to post-treat a rapidly-solidified metal strip according to a comparative example shall be described in greater detail with reference to FIG. 1.

[0045] FIG. 1 illustrates a highly simplified schematic diagram of this S-roller system, as a conveyance system 5. In this simplified depiction, only the essential components of the conveyance system 5 are shown, in order to clarify the forces acting on the roller pairs 22 and 23 as well as on the metal strip 6 to be post-treated. Acting on the two roller pairs 22 and 23 before and after the post-treatment region 20 is a contact pressing force F.sub.S, which acts on the metal strip 6 between the respective rollers of the roller pairs 22 and 23 and produces a frictional connection between the rollers and the metal strip 6.

[0046] While the roller pair 22 of the tensioning assembly 12 at the beginning of the post-treatment region 20 carry out a braking function with a braking force F.sub.B on the metal strip 6 to be post-treated, the roller pair 23 of the tension roller assembly 11 at the end of the post-treatment region generate a driving force FD, which is greater in a start-up phase or acceleration phase at the beginning of conveyance than during the post-treatment phase, during which the metal strip 6 passes through the roller pairs 22 and 23 at a constant speed in the direction of passage A with a tensile force F.sub.z=F.sub.D=F.sub.B.

[0047] Thus, the known S-roller system provides a conveyance system 5 with which it is possible to continuously post-treat metal alloy strips 6 under tensile stresses in the post-treatment region 20 of up to 1500 MPa. The metal strip 6 is then transported at a constant speed V.sub.F in the direction A. In a region a between the two roller pairs 22 and 23, the metal strip 6 is subjected to an adjustable tensile stress along the strip axis. The tensile stress in the metal strip 6 then arises from the tensile force F.sub.z and the cross-section A.sub.Band of the transported metal strip 6, in the entire post-treatment region 20 between the roller pairs 22 and 23. In the region b outside of the S-roller pairs 22 and 23, almost no tensile stress (or a significantly low tensile stress) predominates in the metal strip 6.

[0048] Of the two rollers 13 or 19 of each S-roller pair 22 and 23 is driven by a motor having a gear. The two rollers 13 and 24 or 19 and 24 of each S-roller pair 22 and 23 are mounted so as to be displaceable relative to one another in the vertical direction, and pressed together with an adjustable force F.sub.S, so as to apply the adjustable force F.sub.S to the metal strip 6 needing to be post-treated and transported that is between the rollers 13 and 24 or 19 and 24. This force F.sub.S also acts on the axes of the rollers 24 or 24. Due to this frictional connection of the two rollers 13 and 24 or 19 and 24 via the force F.sub.S and the metal strip 6 present therebetween, the rollers 24 or 24 of each S-roller pair 22 and 23 also appear to be driven with this known conveyance system 5 through an angle of wrap of 180 per roller. This angle of wrap of two times 180 is a typical feature of the known conveyance system 5.

[0049] The schematic diagram of FIG. 1 only indicates that the tensile force F.sub.z in the metal strip 6 is enabled by a braking function of the roller pair 22.

[0050] The retraction force can be introduced by various methods, such as, for example, one of the methods disclosed in WO 2013/156010 A1. This braking function can be generated by differences in the torque of the drives of the rollers 13 and 19, or by mechanical braking acting adjustably on one of the rollers of the roller pair 22 of the tensioning assembly 12 before the post-treatment region 20.

[0051] Due to the double deflection of the metal strip 6 by 180, the use of S-roller systems leads to a problem in that the metal alloy strips 6, when used, have a higher frequency of breaks. Passage through the S-roller system leads to breaking in particular with the use of very thin, amorphous Fe-based alloys, which are to be transferred into the nanocrystalline state under tensile stress along the strip axis and at temperatures around 700 C., within a tempering or heat treatment region 30 that is confined from the post-treatment region. When rapidly-solidified metal alloy strips 6 are heat-treated under tensile stress, a thermal relaxation occurs in the nanocrystalline strip material in the tempering or heat-treatment region 30. This can lead to brittling of the material as a whole, or to inhomogeneities in the metal alloy material, with an increased brittleness of the metal strip 6.

[0052] The nanocrystalline state is thus more brittle in comparison to the amorphous state, such that the metal alloy strips cannot readily be further bent or cut without fragments occurring. On the other hand, the nanocrystalline strip material can be subjected to very high tensile stresses along the longitudinal strip axis.

[0053] FIG. 2 illustrates a schematic diagram of a conveyance system according to a first embodiment of the invention, which comprises a drive roller 13 corresponding in diameter and in width to the rollers of the S-roller system as illustrated in FIG. 1, and a smaller freely rotating pressing roller 14. The drive roller 13 and the smaller freely rotating pressing rollers 14 form a tension roller assembly 7 of the first embodiment of the invention, at the end of the post-treatment region 20, wherein an associated tensioning assembly 12 at the beginning of the post-treatment region 20 may have an identical roller assembly.

[0054] The freely rotating smaller pressing roller then defines the end of an angle of wrap of the metal strip 6 on the drive roller 13.

[0055] The smaller freely rotating pressing roller 14 is pressed with a force F.sub.1 onto the continuous metal strip 6, and thus onto the drive roller 13. The contact pressure force F.sub.1 acts constantly, and is not dependent on the position. The contact pressure force F.sub.1 is in the range of 15 to at most 150 N, and is adapted to the strip width of the continuous metal strip 6, so that a surface pressure of .sub.1 of 7 to 10 MPa occurs, where at =F.sub.1/A.sub.Abplattung with a flattening surface from A.sub.Abplattung=strip width [m] times a flattening width of 0.001 m, because the flattening of the drive roller 13 is by about one millimeter.

[0056] The drive roller 13 is produced at least in a peripheral region 18 thereof from a flexible plastic material such as FRIBOFLEX, with a high hardness (Shore 90A). The smaller pressing roller 14 is made from a comparatively inelastic material, such as stainless steel, at least in a peripheral region 17 thereof.

[0057] The width of the pressing roller 14 is selected so as to fulfill the condition b.sub.ra>b.sub.r1>b.sub.Band, where b.sub.ra is the width of the drive roller 13, b.sub.r1 is the width of the pressing roller 14, and b.sub.Band is the width of the metal strip 6.

[0058] FIG. 2 illustrates only the right-side outgoing part of the conveyance system 1, namely, the tension roller assembly 7 of the first embodiment of the invention. The left-side part of the conveyance system 1, which is optionally located before a heat treatment oven, is constructed analogously to the part of the conveyance system 1 illustrated here. The metal strip 6 passes through the system from left to right in the direction of the arrow A. In the region a, the metal strip 6 is under the tensile force F.sub.z. In contrast, in the region b, there is a significantly lower other tensile stress in the strip. The tensile force F.sub.z is applied by a system (not specified in FIG. 2). The direction of the tensile force F.sub.z is always directed opposite to the direction of rotation of the drive roller 13.

[0059] The tensile force F.sub.z is necessary, for example, for adjusting the magnetic properties. Through the contact pressure force F.sub.1, the material combination of the roller pairing of the drive roller 13 and the pressing roller 14, and the selection of the above-described width condition, the roller system according to the invention made of the rollers 13 and 14 builds a holding force F.sub.A.sup.0 that acts opposite to the tensile force F.sub.z and protect the metallic strip from slipping through opposite to the direction of travel of the strip. The tensile force F.sub.z can be increased in the strip through a system (not further specified here) up to an equilibrium state F.sub.z=F.sub.A.sup.0. In the case where F.sub.z>F.sub.A.sup.0, the metal strip 6 slips through the roller pair made of the drive roller 13 and pressing roller 14, and the tensile force F.sub.z in the region a can no longer be kept constant.

[0060] The conveyance system 1 illustrated in FIG. 2 has a strip angle of wrap =almost 0, because the contact point 15 at the beginning of the angle of wrap of the metal strip 6 is equal to the contact point 16 at the end of the angle of wrap , which is defined by the pressing roller 14. Forming a sufficiently high holding force F.sub.A.sup.0 with this low angle of wrap, which develops only in the region of the flattening of 1 mm of the peripheral region 18 of the drive roller 13, requires a corresponding contact pressure force F.sub.1, a corresponding material combination of the roller pairing of the drive roller 13 and the pressing roller 14, and the maintenance of the width condition b.sub.ra>b.sub.r1>b.sub.Band.

[0061] In the roller pairing according to the invention, made of the drive roller 13 and the pressing roller 14, the tensile force in the metal strip 6 is formed within a very short range of, for example, 1 mm, and corresponding an angle of wrap 0, i.e., without any process to bend or curve the metal strip 6. This results in a very low material strain, which is reflected in an improved frequency of breaks. Table 3 shows the results. However, the maximum holding force F.sub.A.sup.0 that can be achieved with =0 and the above-mentioned measures is too low for some applications. With an increase in the angle of wrap , the holding force F.sub.A can be increased, such that it is then possible to have higher strip tensions, as illustrated by the following drawings and depicted by table 3 in relation to the increase in the frequency of breaks.

[0062] FIG. 3 illustrates a schematic diagram of a conveyance system 2 according to a second embodiment of the invention. The depiction in FIG. 3 is thereby limited to a tension roller assembly 8 of the second embodiment of the invention. In FIG. 3, in order to increase the angle of wrap , a deflection roller 21 that corresponds in diameter and width to the drive roller 13 is used. The deflection roller 21, however, is made out of stainless steel and is freely rotating, like the pressing roller 14. In order to realize different angles of wrap of the metal strip on the drive roller 13, the position of the deflection roller 21 can be freely selected relative to the drive roller 13.

[0063] In FIG. 3, the position of the deflection roller 21 is selected such that the deflection roller 21 defines a first support point 15 of the metal strip 6 on the drive roller, at the start of the angle of wrap , while the end of the illustrated angle of wrap of =25 is defined by the pressing roller 14. Through the angle of wrap 25 defined by the pressing roller 14, a circumferential path on the drive roller 13 is set, the metal strip 6 being pressed thereon against the drive roller 13, so that an additional static holding force F.sub.R acting in the same direction as the holding force F.sub.A.sup.0 is formed from the initial contact point 15 to the contact point 16 at the end of the angle of wrap , which is defined by the pressing roller.

[0064] The now effectively acting holding force F.sub.A results from the sum:

F.sub.A=F.sub.A.sup.0+F.sub.R(1)

[0065] The static holding force F.sub.R can be calculated with the aid of the equation (2), as described above. It is thus possible to realize higher tensile forces in the metal strip 6 and, in the magnetic case, to also induce a higher anisotropy, or achieve a lower material permeability. As long as an angle of wrap that is not too high is selected, there will also be no significant increase in the frequency of breaks, as shown in table 3.

[0066] FIG. 4 illustrates a schematic diagram of a conveyance system 3 according to a third embodiment of the invention. The depiction in FIG. 4 is thereby limited to a tension roller assembly 9 of the third embodiment of the invention. As shown in FIG. 4, the metallic freely rotating deflection roller 21 in FIG. 4 is arranged such that the initial contact point 15 of the metal strip 6 on the drive roller 13 is arranged relative to the end of the angle of wrap defined by the pressing roller 14 such that now an angle of =90 is realized. Nevertheless, this embodiment of the invention does not entail a typical roller system, because the drive roller 13 and the deflection roller 21 do not touch.

[0067] In the conveyance system 3 of the third embodiment of the invention, unlike the typical S-roller systems, the range of the very different tension ratios is kept extremely low during the process of bending the metal strip when wrapping around the drive roller 13. Table 1 shows, for example, that the base value of F.sub.A.sup.0 can be doubled with an angle of wrap =90. The effective holding force F.sub.A is then about 166 N, thus coming very close to the targeted maximum tensile force value of F.sub.z=170N.

[0068] However, the preferred embodiment range, as already listed above, is at =0 to about 40, because the consequently achievable maximum tensile forces F.sub.z of 120N are sufficient for magnetic applications. For example, as shown in table 2, a minimum permeability of =60 is achieved for an Fe-based alloy (VP800, Fe.sub.Rest, Cu.sub.1, Nb.sub.3, Si.sub.15.6, B.sub.6.6, at %) at a width of 6 to 12 mm and thickness of 19 m, which is subjected to a continuous heat treatment in the aforementioned tension range. Table 2 below shows further details.

[0069] FIG. 5 illustrates a schematic diagram of a conveyance system 4 according to a fourth embodiment of the invention. The depiction in FIG. 5 is thereby limited to a tension roller assembly 10 of the fourth embodiment of the invention. In this embodiment, the freely rotating deflection roller 21 is arranged so as to produce an angle of wrap on the drive roller 14 of 180, wherein, again, the contact point 15 at the beginning of the angle of wrap is clearly determined by the arrangement of the deflection roller 21 and the contact point at the end of the angle of wrap =180 is clearly determined by the position of the pressing roller 14.

[0070] Nevertheless, this fourth embodiment also does not entail a typical S-roller system, because the drive roller 13 and the deflection roller 21 do not touch, and, unlike the typical S-roller systems, the range of the very different tension ratios here is kept extremely low during the process of bending when the metal strip 6 is wrapped around the drive roller 13. In addition, as shown in table 1, the angle of wrap of 180 results in a static holding force of 129N, such that an effective holding force F.sub.A of F.sub.A.sup.0=80N+129N=209N results, which is well above the desired tensile forces of F.sub.z=120N for magnetic applications.

[0071] It should also be noted that the mechanical post-treatment region 20 for the metal strip 6, in which a tensile stress is induced in the metal strip, as illustrated in FIG. 1 to 5 with the reference numeral 20, is larger than the heat treatment region 30, which is marked only in FIG. 1. The reason is that the tensioning assembly 12 and the tension roller assemblies 7 to 11 are arranged on the outside of a continuous oven or tempering oven.

[0072] The following tables 1 to 3 show results in relation to a possible increase of the effectively acting holding force, the achievable permeability, and the number of breaks averaged per 1000 m for different angles of wrap .

TABLE-US-00001 TABLE 1 [] (1 1/e.sup.[rad]) F.sub.R [N] F.sub.A = F.sub.A.sup.0 + F.sub.R [N] 0 0 0 80 25 0.13 30 110 45 0.22 51 131 90 0.40 86 166 180 0.63 129 209

[0073] Table 1 illustrates the possible increase of the effectively acting holding force F.sub.A with the increase in the angle of wrap . The base value F.sub.A.sup.0 results from the corresponding contact pressure force F.sub.1 exerted by the pressing roller 14 at the end of the angle of wrap on the material combination, namely, from the material combination of the roller pairings made of the drive roller 13 as well as the width condition b.sub.ra>b.sub.r1>b.sub.Band.

[0074] For F.sub.A.sup.0, a value of 80 N to 100 N is reached at maximum. 170 N is preferable as a maximally necessary tensile force in the strip. This corresponds, for a metal strip 7 that is 6 to 12 mm wide and 19 m thick, to a tensile stress of 1500 MPa (where =F/A; where is tensile stress, F is force, and A is cross-sectional area). F.sub.z=170N was also set as the tensile force for calculating the data in table 1. For the material pairing of the metal strip and the FRIBOFLEX of the drive roller 13, a coefficient of sliding friction of =0.32 was used.

TABLE-US-00002 TABLE 2 F.sub.z [N] F.sub.z [N] [MPa] Permeability 6 mm, 19 m 12 mm, 19 m 30 1700 3.5 7 50 1000 6 12 200 200 23 46 500 90 57 114 750 60 85 170

[0075] Table 2 shows the achievable permeability () after a continuous heat treatment under tensile stress () for the alloy VP800 (Fe.sub.Rest, Cu.sub.1, Nb.sub.3, Si.sub.15.6, B.sub.6.6, at %) at a width b.sub.Band of the metal strip 6 of 6 to 12 mm and a thickness of the metal strip 6 of 19 m, with a continuous oven or tempering oven temperature of 690 F. and an annealing or tempering time of 4 s. The required tensile force F.sub.z is also shown in the table, and depends on the strip width b.sub.Band of the metal strip 6. A tensile stress of 750 MPa already constitutes a borderline material strain in the heat treatment temperatures used. With tensile stresses above there, the material of the metal strip 6 breaks off after previous elastic deformation and constriction.

TABLE-US-00003 TABLE 3 average number of applicable to [] breaks per 1000 m [MPa] Notes 0 0.20 200 See FIG. 2 25 0.65 400 See FIG. 3 45 0.85 750 See FIG. 3 90 1.5 1500 See FIG. 4 180 4.0 1500 See FIG. 5 S-rollers 7 1500 Standard S-roller system (see FIG. 1)

[0076] Table 3 illustrates the probability of a strip break on production lengths of 1 km, on the basis of average numbers of break depending on the angle of wrap between a first contact point 15 on the drive roller 13 at the beginning of the angle of wrap and a second contact point 16 at the end of the angle of wrap, which is defined by a pressing roller 14. As shown in table 3, the averaged number of breaks per 1000 m clearly falls below 1 as the angle of wrap decreases, as long as the new conveyance system is used in the preferable angle-of-wrap range of []=0 to []=40.

LIST OF REFERENCE SIGNS

[0077] 1 Conveyance system (first embodiment) [0078] 2 Conveyance system (second embodiment) [0079] 3 Conveyance system (third embodiment) [0080] 4 Conveyance system (fourth embodiment) [0081] 5 Conveyance system (prior art) [0082] 6 Metal strip [0083] 7 Tension roller assembly (first embodiment) [0084] 8 Tension roller assembly (second embodiment) [0085] 9 Tension roller assembly (third embodiment) [0086] 10 Tension roller assembly (fourth embodiment) [0087] 11 Tension roller assembly (prior art) [0088] 12 Tensioning assembly [0089] 13 Drive roller [0090] 14 Pressing roller [0091] 15 Contact point at the beginning of the angle of wrap [0092] 16 Contact point at the end of the angle of wrap [0093] 17 Peripheral region of 14 [0094] 18 Peripheral region of 13 [0095] 19 Brake roller [0096] 20 Post-treatment region [0097] 21 Deflection roller [0098] 22 Roller pair [0099] 23 Roller pair [0100] 24 24 Second rollers of the roller pairs [0101] 30 Tempering or heat treatment region [0102] A Direction of passage of the metal strip [0103] b.sub.RA Width of the drive roller [0104] b.sub.R1 Width of the pressing roller [0105] b.sub.Band Width of the metal strip [0106] F.sub.A effective force acting on the metal strip [0107] F.sub.A.sup.0 holding force acting on the metal strip at =0 [0108] F.sub.D Driving force [0109] F.sub.B Braking force [0110] F.sub.1 Contact pressure force [0111] F.sub.R Frictional force (static friction) [0112] F.sub.S Force on an S-roller pair assembly [0113] F.sub.Z Tensile force on the metal strip [0114] Angle of wrap of the metal strip [0115] [] Angle of wrap, in degrees [0116] [rad] Angle of wrap, in radians