Sonotrode, ultrasonic vibration system, and method for operating a sonotrode

Abstract

The invention relates to a sonotrode (101) for forming workpieces (50, 51) comprising a first longitudinal section (2) with a coupling area (21). The first longitudinal section (2) has a first maximum cross-sectional dimension (22). A second longitudinal section (4) of the sonotrode (101) has a machining surface (41) for forming the workpieces, which is arranged at least partially perpendicular to the longitudinal axis (L) of the sonotrode (101). The second longitudinal section (4) has a second maximum cross-sectional dimension (42), which is smaller than a first maximum cross-sectional dimension (22). An intermediate longitudinal section (3) of the sonotrode (101) is arranged directly adjacent to and between the first and second longitudinal sections (2, 4). According to the invention, the second longitudinal section (4) has a taper (43), preferably a conical taper (43), which extends to the end of the second longitudinal section (4).

Description

The invention relates to a sonotrode, an ultrasonic vibration system comprising the sonotrode, and a method for operating the sonotrode according to the independent claims. Further embodiments are described in the dependent claims. Formed or welded workpieces, such as those produced by rivet dome welding, often have to meet high quality standards. Specifically, workpieces made of plastics, especially polyethylene, are frequently used in applications that must meet high demands both visually and functionally. Particularly with painted workpieces, it is important that the forming process does not damage them. It is known that workpieces, such as rivet bosses, can be heated and reshaped by introducing ultrasonic vibrations. Conventional ultrasonic welding processes for reshaping or welding workpieces, especially rivet boss welding, often cause qualitative defects in the workpiece joint. Firstly, it has been observed that these ultrasonic welding processes can cause optical defects in painted surfaces, foamed workpieces, or sensitive coverings such as leather. These defects could not be completely eliminated by specific welding geometries of sonotrodes, coupling geometries (e.g., of rivet bosses), or optimized recording techniques. Furthermore, it has been found that unwanted focusing of the sound within workpieces can occur, which, due to stray sound waves, can lead to qualitative damage and defects in the workpiece / workpiece joint. EP 2 246 178 discloses torsional ultrasonic welding of an attachment and a component. However, torsional ultrasonic welding is often associated with higher costs than conventional ultrasonic welding processes. Torsional ultrasonic welding, especially of workpieces on plastic, often requires large amplitudes, which can be a problem, especially with small workpieces. It is therefore the object of the present invention to overcome these and other disadvantages of the prior art. In particular, a sonotrode, an ultrasonic vibration system, and a method are to be provided which enable effective, simple, cost-efficient, and reliable forming of workpieces. Specifically, large amplitudes with optimized internal stress distribution are to be made possible for torsional sonotrodes. This is intended to improve the required forces and times for forming workpieces and / or avoid high impact on the workpieces, so that, in particular, plastic workpieces and / or coated workpieces can be formed. According to the invention, a sonotrode for forming workpieces comprises a first longitudinal section with a coupling area. The coupling area serves for connection to an ultrasonic source such as a converter or booster. The first longitudinal section has a first maximum cross-sectional dimension. A second longitudinal section of the sonotrode comprises a machining surface for forming at least one workpiece. The machining surface is, in particular, arranged at least partially perpendicular to the longitudinal axis of the sonotrode. The second longitudinal section has a second maximum cross-sectional dimension, which is smaller than the first maximum cross-sectional dimension. An intermediate longitudinal section of the sonotrode is adjacent to and arranged between the first and second longitudinal sections. According to the invention, the second longitudinal section has a taper, in particular a conical taper, which extends to the end of the second longitudinal section. The sonotrode is preferably a torsional sonotrode. This allows for gentler plasticizing of workpieces through tangential introduction of the sound, as well as a reduction in the plasticizing time. Furthermore, the "rubbing" effect of the torsional ultrasonic vibration, instead of the "hammering" effect of longitudinal ultrasonic vibrations, prevents sound transmission to underlying workpieces and thus avoids damage to them. The solution according to the invention enables, in particular, the gentle and reliable forming of workpieces, preferably plastic workpieces such as polyethylene, at high torsional amplitudes and small forming areas. Furthermore, by reducing the sonotrode diameter to achieve smaller processing areas and an increase in amplitude, the sonotrode according to the invention allows internal stresses of the sonotrode to be distributed more evenly along the longitudinal axis. The end of the second longitudinal section has the smallest cross-sectional area of ​​the second longitudinal section, in particular of the entire sonotrode. The sonotrode can be formed entirely or essentially entirely in one piece, with the exception of the machining surface and / or centering aids on the working surface. The tapering can extend over the entire second longitudinal section. The second longitudinal section can have a length that extends at least along one torsional wavelength λ. Additionally or alternatively, at typical ultrasonic frequencies of about 20–40 kHz, the length can be selected from a range of 22 mm to 55 mm, in particular 24 mm to 52 mm, preferably 28 mm to 50 mm. The torsional wavelength λ is obtained by dividing the material-dependent torsional wave velocity ct by the ultrasound frequency f. The tapering according to the invention enables an optimized amplification of the amplitude while reducing internal stresses, since the amplitude transformation is distributed over a larger longitudinal area of ​​the second longitudinal section. The torsional wavelength λ can result from an impact frequency between 10 kHz and 60 kHz, in particular between 15 kHz and 40 kHz, preferably 20 kHz and 30 kHz. The sonotrode can comprise or consist of aluminum, titanium, steel (especially tool steel or stainless steel), tungsten (especially tungsten alloys or tungsten carbide), and / or ceramics. For forming plastic workpieces, a sonotrode made of aluminum, steel, or titanium is suitable. The conical taper can extend over the entire length of the second longitudinal section. Alternatively or additionally, the taper can have at least an angle relative to a longitudinal axis of the sonotrode between 5° and 50°, in particular between 10° and 40°, preferably between 15° and 30°. A first end of the taper can have a large cross-sectional dimension between 15 mm and 45 mm, in particular between 20 mm and 40 mm, preferably between 25 mm and 35 mm. A second end of the taper can have a smaller cross-sectional dimension of 4 mm to 30 mm, in particular 5 mm to 20 mm, preferably 6 mm to 18 mm. This allows for the reshaping of small areas with high amplitudes. The first longitudinal section can comprise or consist of a cylindrical area on a side facing the coupling area. The cylindrical area optionally has a cross-sectional dimension of 30 mm to 60 mm, in particular 35 mm to 55 mm, preferably 40 mm to 50 mm. The cylindrical area optionally has a length extending over at least the torsional wavelength λ. Alternatively or additionally, the length can be 15 mm to 45 mm, preferably 20 mm to 40 mm, in particular 25 mm to 35 mm. The coupling area can have an essentially cylindrical structure. The coupling area can be connected to or connected with one or more ultrasonic converters or a booster, and in particular with a sound transmission unit. The cylindrical area can extend to the immediate vicinity of the intermediate longitudinal section, in particular immediately adjacent to the narrowing of the intermediate longitudinal section. The intermediate longitudinal section can include a taper, in particular a conical taper, arranged adjacent to the first longitudinal section. The taper preferably has at least a partial angle of 15° to 45°, in particular 20° to 40°, preferably 22.5° to 30°, relative to the longitudinal axis. The tapered section of the intermediate longitudinal segment can exhibit a curvature in cross-section along the longitudinal axis. Preferably, the tapered section is curved at least partially inwards, resulting in a partially concave shape. The intermediate longitudinal section can comprise a first cylindrical intermediate section on the side adjacent to the first longitudinal section and a second cylindrical intermediate section on the side adjacent to the second longitudinal section. The first cylindrical intermediate section can have a smaller cross-sectional dimension than the second cylindrical intermediate section, in particular in a ratio of the cross-sectional dimensions of the first cylindrical intermediate section to the second cylindrical intermediate section of 0.5 to 0.99, more specifically 0.65 to 0.95, preferably 0.8 to 0.9. Such different cross-sectional dimensions allow the upper and lower ends of the sonotrode to be adapted to the operating frequencies of a generator, making resonant operation easier to achieve. Alternatively, the first and second cylindrical intermediate sections can have the same cross-sectional dimensions, thus forming a cylindrical intermediate section with a cross-sectional dimension that is constant in the longitudinal direction. The first cylindrical intermediate section can have a cross-sectional dimension of 30 mm to 60 mm, in particular of 35 mm to 50 mm, preferably of 40 mm to 45 mm. The second cylindrical intermediate section can have a cross-sectional dimension of 30 mm to 60 mm, in particular of 35 mm to 50 mm, preferably of 40 mm to 45 mm. The intermediate longitudinal section can have a length extending over at least the torsional wavelength λ, and / or a length of 45 mm to 85 mm, in particular 50 mm to 80 mm, preferably 55 mm to 75 mm. The length of the sonotrode can extend over the torsional wavelength λ. The variable n is chosen from the set of natural numbers. In particular, n = 2. The length of the sonotrode can alternatively or additionally be 100 mm to 180 mm, in particular 105 mm to 170 mm, preferably 110 mm to 150 mm. Such a relatively short length is particularly suitable for applications in confined spaces. Especially at frequencies in the range of 10–40 kHz and / or with a sonotrode made of steel or titanium with a small cross-sectional area at the machining surfaces, a high amplitude ratio can be achieved for a short sonotrode. The machining surface can have a concave machining area for dome welding. The concave machining area can optionally have a cross-sectional dimension of 5 mm to 20 mm, in particular 5 mm to 18 mm, preferably 6 mm to 15 mm. The machining surface can have a multitude of longitudinally projecting protrusions, which are particularly wedge-shaped in the longitudinal direction. The protrusions also extend in a radial direction. This allows for a more even pressure distribution and reduces slippage during ultrasonic forming or ultrasonic welding. This, in turn, improves the quality of the joint. A recess for receiving a positioning pin can be arranged on the longitudinal axis in the center of the machining surface. The sonotrode can encompass the positioning pin, which is inserted into the recess. This allows for the simple production of contours on the machining surface without the positioning pin. The positioning pin enables the sonotrode to be used for hollow riveting, thus allowing the machining of workpieces with reduced angularity. The positioning pin may be pressed into the recess with an excess of material. The sonotrode can be configured to achieve, when properly subjected to ultrasound in the coupling area at an impingement frequency of 10 kHz to 60 kHz, in particular 15 kHz to 40 kHz, preferably 20 kHz to 30 kHz, a maximum amplitude at the outer edge of the processing area of ​​120 µm to 20 µm, in particular 100 µm to 30 µm, preferably 90 µm to 40 µm. This enables reliable and efficient welding of plastics, especially polyethylene. In particular, the sonotrode can be configured to allow an amplitude translation greater than a factor of three. Another aspect of the invention relates to an ultrasonic vibration system comprising a previously described sonotrode and an ultrasonic converter which can be coupled or is coupled to the coupling area. The ultrasonic vibration system can include a sound transmission unit configured to convert longitudinal ultrasonic vibrations into torsional ultrasonic vibrations, preferably damping longitudinal components. The ultrasonic vibration system may include an ultrasonic generator that is coupled or can be coupled to the ultrasonic converter to impart ultrasound to the sonotrode. The ultrasound generator can be configured to produce an impact frequency of 10 kHz to 60 kHz, in particular of 15 kHz to 40 kHz, preferably of 20 kHz to 30 kHz. Another aspect of the invention relates to a method for operating a previously described sonotrode, in particular a previously described ultrasonic vibration system. The method includes forming or welding a workpiece to join two workpieces, wherein in particular at least one of the workpieces is coated with a coating material. The coating material may include or consist of at least one of the following: a varnish and / or paint, in particular acrylic resin, polyurethane, polyester resin, fluoropolymer, hybrid and / or epoxy resin-based varnishes / paints, a primer coating, a metallic coating, a polymer coating, a surface finish, a leather coating, a silicone coating, or a polytetrafluoroethylene coating. The invention will now be described with reference to certain embodiments and figures, which show: Fig. 1: a perspective view of an ultrasonic vibration system according to the invention comprising a sonotrode, Fig. 2: a side view of the sonotrode according to the invention according to Fig. 1, Fig. 3A and Fig. 3B: a longitudinal section of a portion of the second longitudinal section of the sonotrode according to Fig. 2 and a perspective view of the treatment surface of the sonotrode, Fig. 3C and Fig. 3D: a cross-sectional view of the processing surface 41 of a sonotrode according to the invention during rivet dome welding, Fig. 4A and Fig. 4B: a side view of the sonotrode according to the invention and a sonotrode not according to the invention and the stress distribution of the sonotrodes under torsional ultrasonic stress, Fig. 5A and Fig. 5B: a side view of the sonotrode according to the invention. Sonotrode and a non-inventive sonotrode and the associated amplitude, and Fig.5C : a longitudinal section of the end of the sonotrode according to Fig. 5A and the associated amplitude. Fig. 1 shows a perspective view of an ultrasonic vibration system 201 according to the invention, comprising a sonotrode 101. The ultrasonic vibration system 201 also comprises a connection 6 to an ultrasonic generator, an ultrasonic converter 5, and a sound transmission unit 7. The ultrasonic generator is configured to produce electrical oscillations in the ultrasonic range and to excite the ultrasonic converter 5. The ultrasonic converter 5 converts the electrical oscillations into mechanical ultrasonic vibrations. The ultrasonic converter 5 is operatively connected to the sound transmission unit 7, which converts longitudinal ultrasonic vibrations and at least partially into torsional ultrasonic vibrations, with which the sonotrode 101 is subjected. The sound transmission unit can be designed, in particular, as described in WO 2021 / 094329 or WO 2022 / 194383. Fig. 2 shows a side view of the sonotrode 101 according to the invention, comprising a first longitudinal section 2, an intermediate longitudinal section 3, and a second longitudinal section 4. The sonotrode 101 has a total length LT of 136.5 mm. A diameter D1 at the first end of the sonotrode 101 is 45 mm and a diameter D3 at the opposite end is 14.6 mm, enabling an amplitude amplification of more than a factor of 3. The sonotrode 101 is formed in one piece to ensure optimal ultrasound transmission. Only a centering pin (see Fig. 3A) of the sonotrode 101 may be detachably connected to the sonotrode 101. Furthermore, the sonotrode 101 is made of steel and / or titanium. A coupling area 21 is arranged at the first end of the sonotrode 101. For better visibility of the coupling area 21, a circular section of the sonotrode 101 at the first end is shown in a cross-sectional view. The coupling area 21 contains a cylindrical recess around a longitudinal axis L of the sonotrode 101, which is arranged centrally at the first end. An external thread of the sound transmission unit can be inserted into the recess. The first longitudinal section 2 has a length L2 of 29.5 mm, which corresponds approximately to a torsional wavelength λ. The first longitudinal section 2 is completely cylindrical and is located immediately adjacent to the intermediate longitudinal section 3. The intermediate longitudinal section 3 has a taper 31 on one side facing the coupling area 21. The taper 31 is directly adjacent to the first longitudinal section 2 and comprises two sections 311 and 312. In the first section 311, the taper is conical and has an angle W2 of 30° relative to the longitudinal axis L. In the second section 312, the taper 31 is curved inwards and forms a concave shape extending along a curve with a radius R of 20 mm. Overall, the taper 31 has a length of 20 mm along the longitudinal axis L. Immediately following the taper 31 along the longitudinal axis L, the intermediate longitudinal section 3 has a first cylindrical intermediate section 32, which has a diameter Z1 of 28 mm and a length of approximately 16 mm. Adjacent to the first intermediate section 32, a second cylindrical intermediate section 33 is arranged, which has a diameter Z2 of 33 mm and a length of 26.7 mm. Thus, the ratio of the diameters Z1, Z2 of the cylindrical intermediate sections 32, 33 is 0.85. This widening between the cylindrical intermediate sections 33, 38 along the longitudinal axis L is located approximately at [distance missing] on the intermediate longitudinal section. The two cylindrical intermediate sections 32, 33 are connected by a connecting area 34, which forms a step and, in the example shown, includes a conical widening of the sonotrode 101 at an angle of 60° relative to the longitudinal axis L. Overall, the intermediate longitudinal section 3 has a length of 64.3 mm and thus extends over approximately The second longitudinal section 4 of the sonotrode 101 extends immediately following the end of the second cylindrical intermediate section 33. The second longitudinal section 4 extends over a length L4 of 40.9 mm, which is more than the length along the longitudinal axis L. The second longitudinal section 4 has a conical taper 43 that extends from an upper end 431 to a lower end 432 of the second longitudinal section 4. The angle W1 of the conical taper 43 relative to the longitudinal axis is 12.15°. The taper 43 extends over the entire length of the second longitudinal section 4 at this angle. The taper 43 has a diameter D2 at its upper end 431, which corresponds to the diameter Z2 of the second cylindrical section 33. At its lower end 432, which adjoins a machining surface 41, the taper 43 has a diameter D3 of only 14.6 mm. The machining surface 41 protrudes approximately 0.8 mm relative to the second intermediate section 4 along the longitudinal axis L. Furthermore, the machining surface has a concave, inwardly curved machining area for rivet dome welding. A positioning pin 414 is centrally embedded in a recess 413 of the machining surface 41. Fig. 3A shows a longitudinal section of a portion of the second longitudinal section 4 of the sonotrode according to Fig. 2 with the machining surface 41. The machining surface 41 is convex inwards and the positioning pin 414 is arranged on the longitudinal axis L. The positioning pin 414 is pressed into the recess 413, so that the convex machining area 411 is suitable for forming or welding hollow rivets. Fig. 3B shows a perspective view of the machining surface 41 of the sonotrode according to Fig. 2. The machining surface 41 has a clean, rectangular shape that projects from the second longitudinal section 4 along a longitudinal axis of the sonotrode. The inwardly curved machining area 411 has a plurality of projections 412. The projections 412 extend wedge-shaped and parallel to the longitudinal direction in a star-shaped pattern within the machining area 411 to enable a more uniform pressure distribution. The projections 412 extend radially away from the recess 413 in which the positioning pin 414 is located. The machining surface 41 is formed integrally with the rest of the sonotrode. However, it can also be detachably connected to the second longitudinal section 4 of the sonotrode, allowing for low-maintenance replacement. Figures 3C and 3D show a cross-sectional view of the machining surface 41 of a sonotrode according to the invention during rivet dome welding of two workpieces 50 and 51. The first workpiece 50 is deformed by torsional ultrasonic action of the sonotrode at a frequency selected from 15 to 40 kHz and a displacement selected from 100 µm to 30 µm. This allows the first workpiece 50 to be joined to the second workpiece 51. The workpieces 50 and 51 in Figures 3C and 4D comprise plastic, in particular polyethylene, coated with a coating material in the form of a lacquer. The torsional sonotrode according to the invention allows the coating material to be subjected to less stress, since it is "rubbed" rather than "hammered." This ensures that the lacquered workpieces 50 and 51 remain intact without damaging the lacquer. Therefore, deformed painted surfaces may exhibit fewer optical and / or functional defects.Furthermore, this torsional sonotrode can also be used for foamed components or sensitive coverings such as leather. Figures 4A and 4B show a side view of the sonotrode 101 according to the invention and a sonotrode 301 not according to the invention, and the stress distribution of the sonotrodes under torsional ultrasonic stress. The scale is shown at the bottom of Figures 4A and 4B, represented by a black and a white line, each 25 mm long. Fig. 4A shows that the voltage distribution of the sonotrode 101 is distributed over a significantly wider band under the same ultrasonic stimulation of 30 kHz, so that it is subjected to less mechanical stress than the sonotrode 301 in Fig. 4B. This optimized stress distribution of the sonotrode 101 according to the invention can be achieved primarily by the fact that the second longitudinal section 4 has a conical taper 43 which extends in the longitudinal direction over at least one torsional wavelength λ of the sonotrode 101. Furthermore, the stress distribution can be improved by the narrowing through the tapering 31 of the intermediate longitudinal section 3 already in the upper area of ​​the sonotrode 101. The stress load scale in Fig. 4A ranges from 1.4 × 10⁵ Pa to 4.27 × 10⁸ Pa and in Fig. 4B from 1.77 × 10⁵ Pa to 6.63 × 10⁸ Pa. Thus, the peak load, particularly in the second longitudinal section 4 of the sonotrode, can be reduced by approximately 31.5 percent relative to a second longitudinal section 4' of the prior art sonotrode. As previously described, the sonotrode 101 according to the invention enables a transmission factor of three from an upper end to a lower end of the sonotrode 101 with a machining surface. The amplitude increases with the decreasing diameter of the sonotrode 101. Figures 5A and 5B show a side view of the sonotrode 101 according to the invention and a prior art sonotrode 301, and the amplitude in meters m under torsional ultrasonic stimulation at 30 kHz. The scale is shown at the bottom of Figures 5A and 5B, represented by a black and a white line, each 25 mm long. The amplitude scale in Figure 5A ranges from 0 to 127 µm and in Figure 5B from 0 to 150 µm. Fig. 5A shows that the amplitude increases under a first longitudinal section 2 in the region of a taper 31 of an intermediate longitudinal section 3 along a longitudinal axis of the sonotrode 101. Towards the end of the second longitudinal section 4, the amplitude increases again along the longitudinal axis with the smaller diameter due to a conical taper 43. Fig. 5B, however, shows that the amplitude increases particularly at the end of the sonotrode in the area of ​​a processing surface over a much shorter distance. Fig. 5C shows a longitudinal section of one end of the sonotrode according to the invention with the treatment surface 41 and the associated amplitude. The amplitude of a positioning pin 414 in the recess 413 on a longitudinal axis L of the sonotrode under torsional ultrasonic stimulation is essentially 0 m. The amplitude of the treatment surface increases continuously in the radial direction. QUOTES INCLUDED IN THE DESCRIPTION This list of documents cited by the applicant was automatically generated and is included solely for the reader's convenience. The list is not part of the German patent or utility model application. The DPMA accepts no liability for any errors or omissions. Cited patent literature EP 2 246 178

[0004] WO 2021 / 094329

[0053] WO 2022 / 194383

[0053]

Claims

A sonotrode (101) for forming or welding workpieces (50, 51) comprising: - a first longitudinal section (2) with a coupling area (21), wherein the first longitudinal section (2) has a first maximum cross-sectional dimension (D1), - a second longitudinal section (4) with a machining surface (41) for forming at least one workpiece (50, 51), which is arranged in particular at least partially perpendicular to the longitudinal axis (L) of the sonotrode (101), wherein the second longitudinal section (4) has a second maximum cross-sectional dimension (42) which is smaller than the first maximum cross-sectional dimension (D1), - an intermediate longitudinal section (3) which is arranged immediately adjacent to and between the first and second longitudinal sections (2, 4), characterized in that the second longitudinal section (4) has a taper (43), preferably a conical taper (43), which extends to the end of the second longitudinal section (4). Sonotrode (101) according to one of the preceding claims, wherein the tapering (43) extends over the entire second longitudinal section (4), wherein the second longitudinal section (4) has a length (L4) extending over at least along λ 4 extends along a torsional wavelength λ, and / or the length (L4) is selected from a range of 22 mm to 55 mm, in particular from 24 mm to 52 mm, preferably from 28 mm to 50 mm. Sonotrode (101) according to one of the preceding claims, wherein the conical taper (43) extends over the entire length of the second longitudinal section (4), and / or wherein the taper (43) has at least one angle (W1) relative to a longitudinal axis (L) of the sonotrode (101), which is selected from a range of 5° to 50°, in particular from 10° to 40°, preferably from 15° to 30°. Sonotrode (101) according to one of the preceding claims, wherein a first end (431) of the taper (43) has a cross-sectional dimension (D2) selected from a range of 15 mm to 45 mm, in particular from 20 mm to 40 mm, preferably from 25 mm to 35 mm, and a second end (432) of the taper (43) has a cross-sectional dimension (D3) selected from a range of 4 mm to 30 mm, in particular from 5 mm to 20 mm, preferably from 6 mm to 18 mm. Sonotrode (101) according to one of the preceding claims, wherein the first longitudinal section (2) comprises or consists of a cylindrical area (23) on a side facing the coupling area (21), wherein the cylindrical area (23) optionally comprises a first cross-sectional dimension (D1) selected from a range of 30 mm to 60 mm, in particular from 35 mm to 55 mm, preferably from 40 mm to 50 mm, and wherein the cylindrical area (23) optionally has a length (L2) extending over at least λ 4 the torsional wavelength λ extends, and / or the length (L2) is selected from a range of 15 mm to 45 mm, preferably from 20 mm to 40 mm, in particular from 25 mm to 35 mm. Sonotrode (101) according to one of the preceding claims, wherein the intermediate longitudinal section (3) comprises a taper (31), in particular a conical taper (31), which is arranged adjacent to the first longitudinal section (2), wherein the taper (31) optionally has at least partially an angle (W2) selected from a range of 15° to 45°, in particular from 20° to 40°, preferably from 22.5° to 30°, relative to the longitudinal axis (L). Sonotrode (101) according to one of the preceding claims, wherein the intermediate longitudinal section (3) comprises a first cylindrical intermediate section (32) on the side adjacent to the first longitudinal section (2) and a second cylindrical intermediate section (33) on the side adjacent to the second longitudinal section (4), wherein the first cylindrical intermediate section (32) has a smaller cross-sectional dimension (Z1) than the second cylindrical intermediate section (33), in particular in a ratio of the cross-sectional dimensions of the first cylindrical intermediate section (32) to the second cylindrical intermediate section (33) of 0.5 to 0.99, in particular of 0.65 to 0.95, preferably of 0.8 to 0.

9. Sonotrode (101) according to one of the preceding claims, wherein the intermediate longitudinal section (3) has a length (L3) extending over at least λ 2 the torsional wavelength λ extends, and / or the length (L3) is selected from a range of 45 mm to 85 mm, in particular from 50 mm to 80 mm, preferably from 55 mm to 75 mm. Sonotrode (101) according to one of the preceding claims, wherein the length (LT) of the sonotrode (101) is distributed over n λ 2 the torsional wavelength λ, wherein n is selected from the set of natural numbers, in particular n = 2, and / or the length (LT) of the sonotrode (101) is selected from a range of 100 mm to 180 mm, in particular from 105 mm to 170 mm, preferably from 110 mm to 150 mm. Sonotrode (101) according to one of the preceding claims, wherein the machining surface (41) has a concave machining area (411) for dome welding, and wherein the concave machining area (411) optionally has a cross-sectional dimension (B1) selected from a range of 5 mm to 20 mm, in particular from 5 mm to 18 mm, preferably from 6 mm to 15 mm. Sonotrode (101) according to one of the preceding claims, wherein the machining surface (41) has a plurality of longitudinally projecting protrusions (412), which are in particular wedge-shaped in the longitudinal direction, wherein the protrusions (412) extend in the radial direction. Sonotrode (101) according to one of the preceding claims, wherein a recess (413) for receiving a positioning pin (414) is arranged on the longitudinal axis (L) in the center of the machining surface (41), and the sonotrode (101) in particular comprises the positioning pin (414) which is inserted into the recess (413). Sonotrode (101) according to one of the preceding claims, wherein the sonotrode (101) is configured to achieve, when properly subjected to ultrasound at the coupling area (21) with an impingement frequency selected from a range of 10 kHz to 60 kHz, in particular from 15 kHz to 40 kHz, preferably from 20 kHz to 30 kHz, a maximum deflection measured at the outer edge of the processing surface (41) of 120 µm to 20 µm, in particular from 100 µm to 30 µm, preferably from 90 µm to 40 µm. Ultrasonic vibration system (201) comprising a sonotrode (101) according to one of the preceding claims, comprising: - an ultrasonic converter (5) which can be coupled to the coupling area (21). Ultrasonic vibration system (201) according to claim 14 comprising an ultrasonic generator which is coupled or can be coupled to the ultrasonic converter (5) in order to impart ultrasound to the sonotrode (101). Method for operating a sonotrode (101) according to one of claims 1 to 13, in particular an ultrasonic vibration system (201) according to one of claims 14 or 15, comprising the step:- forming a workpiece (50) for joining two workpieces (50, 51), wherein in particular at least one of the workpieces is coated with a coating material (51), preferably with a lacquer.

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