超聲變幅桿外文翻譯--超聲加工技術(shù)中超聲變幅桿的設計

1、<p>　　4300漢字，2500單詞，1.5萬英文字符</p><p>　　出處：Nad M. Ultrasonic horn design for ultrasonic machining technologies[J]. Západo?eská Univerzita, 2010, 4(1).</p><p>　　畢業(yè)設計（論文）外文文獻翻譯</p&

2、gt;<p>　　超聲加工技術(shù)中超聲變幅桿的設計</p><p><b>　　M Nad</b></p><p>　　摘要：許多工業(yè)的應用領(lǐng)域和生產(chǎn)技術(shù)都基于超聲波的應用。在許多情況下，超聲現(xiàn)象也運用于加工材料的工藝流程中。使超聲加工技術(shù)起作用的設備主要元件就是超聲變幅桿，也就是所謂的超聲波發(fā)生器。超聲波設備尤其是超聲加工設備的性能取決于超聲變幅桿外形的

3、合理設計。本文展示了不同幾何形狀的超聲變幅桿的動態(tài)特性。對各種不同形狀的超聲變幅桿的幾何參數(shù)特性（固有頻率、振動形態(tài)）都進行了分析。模態(tài)分析的模型是用有限元（FEM）數(shù)字模擬的方法實現(xiàn)的。本文也展示了各種超聲變幅桿的可比參數(shù)。</p><p>　　© 2010 西波西米亞大學 版權(quán)所有</p><p>　　關(guān)鍵詞：超聲波；超聲加工技術(shù)；模態(tài)特性；超聲變幅桿；縱向振動；有限元理論&

4、lt;/p><p><b>　　緒論</b></p><p>　　超聲現(xiàn)象的使用在許多工業(yè)應用中日趨增多。超聲波振動被應用于各個生產(chǎn)領(lǐng)域且有較好的效果，例如：超聲波清洗，塑料焊接，等等。且已經(jīng)被證明在其他許多應用中有很好的作用。這些應用包括汽車、食品加工、醫(yī)療、紡織和材料連接，且主要應用于加工制造業(yè)。性能和質(zhì)量的顯著提升是通過在加工工藝中使用超聲振動實現(xiàn)的。</p&

5、gt;<p>　　超聲振動能量在加工技術(shù)中的應用是由兩種不同的途徑實現(xiàn)的。第一種途徑，稱為超聲波加工，是基于材料去除的研磨原理。刀具的一端連接在變幅桿上，制成精確的外形對工件進行研磨。第二種方法是基于超聲波輔助加工的普通加工技術(shù)。超聲波振動被直接傳遞到切割工具上，直接運用于切割過程中。這些技術(shù)被應用于高精密加工和韌性材料還有難切削材料的加工，如高碳鋼，鎳基合金，鈦鋁—碳化硅金屬基復合材料。反復的高頻振動沖擊模式帶來了一些獨

6、特的性能并被改進成金屬切削工藝[2，5，9，10]，其中工件和刀具之間的相互作用被看成是一個微振動的過程。</p><p>　　超聲波振動能量在加工過程中的應用帶來了許多好處和切割工藝的改進。據(jù)報道，在最近公開的研究工作中，切削工具的高頻超聲波振動已顯著降低切削力和刀具磨損，表面光滑度達到了25—40%，圓度改善達到40—50%。在切割低碳合金鋼時，超聲波振動裝置中切削力降低了50%左右，并可生產(chǎn)出比傳統(tǒng)切削更小

7、和表面光潔度更高的芯片。</p><p>　　通常來說，所有使用超聲波振動的制造系統(tǒng)，都將電機換能器作為機械振動源，它將從發(fā)電機接收到的電能轉(zhuǎn)化為機械振動。電機換能器是基于磁性伸縮或壓電效應的原理。電機換能器產(chǎn)生的共振頻率為fres≈20KHz或其他頻率。其所得到的超聲振動振幅是不足以用于切割加工中的。為了解決這個問題，將超聲波加工設備中能放大機械波的元件連接到電機換能器，使其能夠達到需要的振幅大小。這種導波聚焦

8、裝置稱為超聲變幅桿（也成為集中器，超聲波變幅桿或刀具保持器），它被安裝在換能器的末端。超聲變幅桿從換能器的端部傳送縱向超聲波到切削道具的趾端，它將輸入的振幅放大，使之在輸出端的輸出振幅能滿足加工過程所需的大小。</p><p>　　超聲加工設備的切削性能主要取決于超聲變幅桿的精心設計[6]。超聲變幅桿是超聲波加工系統(tǒng)中唯一一個每個加工過程都獨一無二的部件。根據(jù)實際應用的需要，它們被加工成各種不同的形狀和大小，但和

9、其他部件一樣，必須是工作在諧振頻率上的。超聲變幅桿所使用的材料兼顧了超聲波的需求和應用——鈦合金，鋼，不銹鋼。如上文所述，其形狀取決于實際應用的工藝需求。</p><p>　　超聲變幅桿最常用的形狀是：圓柱形、圓錐形、指數(shù)形和梯形。為了使超聲加工系統(tǒng)達到最佳性能，就有必要考慮影響該系統(tǒng)動力特性的所有相關(guān)效應和參數(shù)[4]。超聲變幅桿作為超聲加工系統(tǒng)中最重要的元素之一，必須在設計階段就應該明確其所需要的動力特性。&l

10、t;/p><p>　　在最近的研究工作中，超聲變幅桿的合適外形及其相應的尺寸的選擇通常是通過有限元數(shù)值模擬來確定的[1，2，7，8，11]。由于筆者知識的局限，本文沒有提出對于不同超聲變幅桿的外形模態(tài)性能（固有頻率，放大系數(shù)）的相互比較，以上可在文獻中查閱。</p><p>　　本文對各種形狀的超聲變幅桿進行了動力學分析。超聲變幅桿的外形尺寸對固有頻率的影響是通過有限元法（FEM）分析的。此外

11、還對各種超聲變幅桿外形可比參數(shù)進行了相互比較。本文的主要目的是對超聲變幅桿合適外形及其動力學特性所需的幾何尺寸的選擇提出普遍適用的結(jié)論。</p><p><b>　　變幅桿的設計</b></p><p>　　超聲變幅桿的主要功能是把超聲波的振幅放大至刀具有效加工所需的大小。超聲變幅桿也可以看做是把換能器的振動能量傳遞到刀具的工具。它是通過與換能器的共振來是實現(xiàn)工作。超

12、聲變幅桿的設計制造過程需要格外注重。設計不合理的超聲變幅桿會有損設備的機械加工性能，會導致振動系統(tǒng)的破壞并對超聲發(fā)生器造成巨大的損壞。</p><p>　　通常，超聲變幅桿是由具有高疲勞強度和低聲損耗的金屬制成的。超聲變幅桿設計中最重要的部分就是其共振頻率和正確的諧振波長的確定。變幅桿的長度一般為其半波長的整數(shù)倍。簡單幾何形狀(圓柱形）的超聲變幅桿的諧振頻率是能夠確定的。對于復雜幾何形狀的變幅桿的諧振頻率通常通過

13、有限元的方法來確定。</p><p>　　超聲變幅桿的性能需有由放大系數(shù)估算</p><p>　　， （1）</p><p>　　式中 A0——超聲變幅桿輸入端振幅，</p><p>　　A1——超聲變幅桿輸出端振幅，</p><p>　　放大系數(shù)的基本要求是&

14、lt;/p><p>　　. (2)</p><p>　　2.1 超聲變幅桿自由端振動的解析</p><p>　　超聲變幅桿可變圓形截面縱向振動的基本方程如下，其適用于一維連續(xù)體（彈性細桿）</p><p>　　， (3)</p><p>　

15、　式中 ——在縱向方向上的坐標，</p><p>　　——橫截面的縱向位移，</p><p><b>　　——橫截面面積，</b></p><p>　　——圓形橫截面的半徑，</p><p>　　——縱波的一維連續(xù)速度，</p><p><b>　　——楊氏彈性模量，</b>

16、</p><p>　　——超聲變幅桿材料的密度</p><p>　　圓柱形超聲變幅桿自由振動的波動方程</p><p>　　. (4)</p><p>　　方程（4）的解假設為此形式。偏微分方程（4）可分為以下兩個常微分方程</p><p>　　，

17、 (5)</p><p>　　， (6)</p><p>　　式中 ——固有角頻率。</p><p><b>　　引入以下無量綱量</b></p><p>　　·縱坐標上的無量綱量：，</p><p>　　·橫

18、截面縱向位移的無量綱量：，</p><p>　　代入到方程（5）中，我們得到的無量綱方程</p><p>　　及其結(jié)果 ， （7）</p><p>　　式中 ——倍頻參數(shù)，</p><p><b>　　l0——變幅桿長度</b></p><p>　　超聲變幅桿的兩側(cè)都具有沿軸向運動的可能性。

19、其輸入端連接到產(chǎn)生超聲波軸向振動的電機換能器，其輸出端連接到振動刀具。超聲變幅桿自由振動的邊界條件被假設為兩端自由[7]，如下所示</p><p>　　， . (8)</p><p>　　然后將邊界條件（8）代入到（7）中 ，得到超聲變幅桿的模態(tài)參數(shù)如下</p><p>　　·第k個模式形狀的固有頻率（Hz）<

20、;/p><p>　　， (9)</p><p>　　·第k個模式形狀的無量綱波長</p><p>　　， (10)</p><p>　　式中是特征方程的第個特征根，其中=1，2，…</p><p>　　為

21、了達到超聲加工所需的效果，超聲變幅桿只使用前兩個模態(tài)即：對于= 1的“半波”形狀和= 2的“整波”形狀（圖1）。</p><p>　　如圖所示，分析確定圓柱形超聲變幅桿的模式形狀和固有頻率是相對簡單的。對于非圓柱形變幅桿的這些參數(shù)的分析測定更為復雜。因此，對幾何外形更復雜的超聲變幅桿的模態(tài)分析，使用有限元法（FEM）是更好的選擇。</p><p>　　圖一 圓柱超聲變幅桿的振動模態(tài)形狀&l

22、t;/p><p>　　2.2.自由振動超聲變幅桿的有限元分析</p><p>　　利用有限元法確定各種形狀的超聲變幅桿的模態(tài)特性和對模態(tài)特性相關(guān)幾何參數(shù)影響的評估。有限元建模和模態(tài)特性的計算都使用了軟件包ANSYS 。超聲變幅桿有限元模型的創(chuàng)建采用了元素SOLID45。</p><p>　　描述超聲變幅桿有限元模型自由振動的方程式是其模態(tài)特性決定的，運動方程如下<

23、/p><p>　　, (11)</p><p>　　式中 M（B、K）是質(zhì)量（阻尼和剛度）的矩陣，</p><p>　　ü（u、u）是節(jié)點的加速度（速度和位移）矢量。</p><p>　　假設該超聲變幅桿的材料具有低的阻尼能力（來自動力方面），其阻尼運動方程可以被忽略。當B = 0時，運

24、動方程（11）可以是改寫為如下形式</p><p>　　. (12)</p><p>　　超聲變幅桿的模態(tài)特性是由其特征值的解決定的</p><p>　　, (13)</p><p>　　式中 是第個特征值（模態(tài)形狀），</p&g

25、t;<p>　　是其在第個模態(tài)形狀下的固有頻率。</p><p>　　前文提出，變幅桿被制造成各種形狀和尺寸。在超聲波加工或超聲波輔助機加工中超聲變幅桿的橫截面大多為圓形。決定超聲變幅桿功能的縱向形狀可能是不同的。主要強調(diào)是獲得所需的動力學特性。超聲變幅桿幾何形狀的確定是基于有量綱參數(shù)和無量綱參數(shù)（表1）。</p><p>　　表1 超聲變幅桿的幾何參數(shù)</p>

26、<p>　　注：d0—超聲變幅桿輸入端直徑，d—階梯軸小端直徑，l0—超聲變幅桿長度。</p><p><b>　　3.數(shù)值模擬的結(jié)果</b></p><p>　　數(shù)值分析是針對表1中所定義的各種形狀和幾何參數(shù)超聲變幅桿進行的。數(shù)值模擬中超聲變幅桿所用的材料是鋼（E = 210GPa，= 7 800kg·m?3，= 0.3）。</p>

27、<p>　　在下文中，不同幾何形狀變幅桿的無量綱諧振頻率定義為</p><p>　　, (14)</p><p>　　式中——變幅桿第階模態(tài)下的固有頻率，</p><p>　　——長細比相同的圓柱形變幅桿的第階模態(tài)的固有頻率。</p><p>　　超聲變幅桿的動態(tài)分析

28、結(jié)果都取決于之前所提到的無量綱量。這種方法為提出普適結(jié)論提供了可能性，而且還可以將不同幾何形狀的超聲變幅桿進行相互比較。另外，此結(jié)果還可以對特定超聲變幅桿的尺寸與形狀的選擇提供參考。</p><p>　　不同幾何形狀的超聲變幅桿對應的諧振頻率的值是由以下等式來確定的</p><p>　　, (15)</p><p>

29、;　　式中 =1為半波型，=2為整波型。</p><p>　　在下文圖中（圖2—圖5），表示的是無量綱固有頻率和放大因素對超聲變幅桿設計的相關(guān)參數(shù)的決定性。</p><p><b>　　4.結(jié)論</b></p><p>　　超聲變幅桿是超聲波加工系統(tǒng)中最重要的元素之一，本文對幾種幾何形狀的超聲變幅桿進行了動力學分析。對特定的幾何形狀和尺寸的超聲

30、變幅桿，在諧振狀態(tài)下對其主要動力特性（固有頻率和放大系數(shù)）進行了分析。超聲波加工系統(tǒng)的效率和性能取決于具體的設計及其對應的大量參數(shù)。</p><p>　　超聲變幅桿幾何形狀的確定取決于其使用的技術(shù)要求。共振頻率的大小和超聲變幅桿輸出端振幅的放大倍數(shù)是選取適當?shù)某曌兎鶙U形狀的基本要求。圓柱形超聲變幅桿的形狀被用于比較表達其他形狀超聲變幅桿的性能。通常，可以認為超聲變幅桿的幾何形狀和尺寸影響著其剛度和質(zhì)量分布。當超

31、聲變幅桿輸出端的橫截面變化時，放大系數(shù)?也隨著橫截面的變化而變化（圖3b—圖4b）（?i >1.0時橫截面增大，?i <1.0時橫截面增量）。除了橫截面的變化，細長比對放大系數(shù)也具有顯著影響。當超聲變幅桿橫截面和細長比增大時，放大系數(shù)減小。橫截面等距增加，放大系數(shù)增加取決于長細比的增大（圖3b—圖4b）。超聲變幅桿細長比的變化和橫截面變化對固有頻率的影響示于圖2a-4a。超聲變幅桿的形狀在特殊情況下是梯形的。超聲變幅桿在這種

32、形狀下，和圓錐形和指數(shù)形超聲變幅桿具有相同的有效結(jié)論。對于這種超聲變幅桿形狀其固有頻率和放大系數(shù)在階梯改變直徑處變化的關(guān)系示于圖5a和圖5b 。</p><p>　　超聲變幅桿形狀的設計強調(diào)的是超聲加工技術(shù)中超聲變幅桿軸向形狀在實際應用中是首要考慮的。在技術(shù)加工中，當徑向載荷上升時，確保該幾何形狀的超聲變幅桿具有足夠的抗彎剛度的設計是必要的。</p><p>　　數(shù)值分析的結(jié)果已用無量綱量

33、和參數(shù)的形式表示。本文提供了可能的選擇和不同的形狀的比較，并提供了用于選擇合適超聲變幅桿的形狀與所需特性的有效工具。</p><p><b>　　圓柱形超聲變幅桿</b></p><p>　　圖2 圓柱形超聲變幅桿。細長比相同時，無量綱頻率（a）和放大系數(shù)（b）的關(guān)系</p><p><b>　　圓錐形超聲變幅桿</b>&

34、lt;/p><p>　　圖3 圓錐形超聲變幅桿。斜度相同時，無量綱頻率（a）和放大系數(shù)（b）的關(guān)系</p><p><b>　　指數(shù)形超聲變幅桿</b></p><p>　　圖4 指數(shù)形超聲變幅桿。指數(shù)函數(shù)參數(shù)“a”相同時，無量綱頻率（a）和放大系數(shù)（b）的關(guān)系</p><p><b>　　階梯形超聲變幅桿<

35、/b></p><p>　　圖5 指數(shù)形超聲變幅桿。階梯直徑無量綱量“”以及比例相同時，無量綱頻率（a）和放大系數(shù)（b）的關(guān)系。</p><p><b>　　致謝</b></p><p>　　一直支持本工作的補助機構(gòu)VEGA 1/0256/09 和 VEGA 1/0090/08。</p><p><b>

36、　　附錄：</b></p><p>　　Ultrasonic horn design for ultrasonic machining technologies</p><p><b>　　M. Nad</b></p><p><b>　　Abstract</b></p><p>　　M

37、any of industrial applications and production technologies are based on the application of ultrasound. In many cases, the phenomenon of ultrasound is also applied in technological processes of the machining of materials.

38、 The main element of equipments that use the effects of ultrasound for machining technology is the ultrasonic horn-so called sonotrode.The performance of ultrasonic equipment, respectively ultrasonic machining technologi

39、es depends on properly designed of sonotrode shape. The dyn</p><p>　　©2010 University of West Bohemia. All rights reserved.</p><p>　　Keywords:ultrasound；ultrasonic machining technologies；mo

40、dal properties； ultrasonic horn；longitudinal vibration；finite element method</p><p>　　1. Introduction</p><p>　　The use of ultrasound phenomenon is becoming increasingly used feature in many indu

41、strial applications. Ultrasonic vibrations have been harnessed with considerable benefits for a variety of production applications, for example, ultrasonic cleaning, plastic welding, etc. and has proved to offer advantag

42、es in a number of other applications. These applications include the automotive, food preparation, medical, textile and material joining and mainly applications in manufacturing industries. Signifi</p><p>　　

43、Applications of ultrasonic vibration energy in machining technologies are realized by two different approaches. The first approach, called as an ultrasonic machining, is based on abrasive principle of material removal. T

44、he tool which is shaped in the exact configuration to be ground in workpiece and it is attached to a vibrating horn. The second approach is based on the conventional machining technologies– ultrasonic assisted machining.

45、</p><p>　　The ultrasonic vibrations are transmitted directly on cutting tools, respectively directly to a cutting process. These techniques are used for high precision machining application and for non-britt

46、le materials and difficult-to-cut materials machining such as hardened steels,nickel-based alloys, titanium and aluminium-SiC metal matrix composites. The repetitive high-frequency vibro-impact mode brings some unique pr

47、operties and improvements into metal cutting process [2, 5, 9, 10], where the inter</p><p>　　The application of ultrasonic vibration energy in the machining process provides many benefits and improvements in

48、 the process of cutting. In recent published work is reported that the high-frequency ultrasonic vibration of the cutting tool has reportedly allowed a significant reduction of cutting forces and tool wear, a surface fin

49、ish improvement up to 25-40 %, as well as roundness improvements up to 40-50 %. When cutting low alloy steels, the ultrasonic vibration means a reduction in cutting f</p><p>　　Generally, in all manufacturing

50、 systems using ultrasonic vibrations, the electromechanical transducer acts as the source of mechanical oscillations, transforming the electrical power received from the generator into mechanical vibrations. The electrom

51、echanical transducers are based on the principle utilizing magnetostrictive or piezoelectric effects. The electromechanical ultrasonic transducers generate the vibration with resonant frequency and more.The amplitude of

52、the resulting ultrasonic vibr</p><p>　　The cutting performance of ultrasonic machining equipment primarily depends on the well-taken design of the sonotrode [6]. The sonotrode is the only part of the ultraso

53、nic machining system which is unique to each process. They are used in various shapes and sizes, according to the application, but like other components should be resonant at the operating frequency.The sonotrode materia

54、l used is a compromise between the needs of the ultrasonics and the application -titanium alloys, steel, stainle</p><p>　　The most frequently used shapes of ultrasonic horns are: cylindrical, tapered, expone

55、ntial and stepped. To achieve optimal performance of ultrasonic machining system is necessary to take into account all relevant effects and parameters that affect the dynamics of the system [4].One of the most important

56、elements of the ultrasonic system-sonotrode, must have the required dynamic properties, which must be determined already in design phase.</p><p>　　In the recent works, the selection of a suitable shape and c

57、orresponding dimensions of sonotrode are usually determined by numerical simulations using finite element method [1, 2, 7, 8, 11]. To the best author’s knowledge, no mutual comparison of the modal properties(natural freq

58、uencies, amplification factors) various sonotrode shapes, presented in this paper, is available in the literature.</p><p>　　In this paper, the dynamical analysis of various shapes of sonotrodes is presented.

59、 The effect of relevant sonotrode dimensions on natural frequencies and mode shapes is analyzed by finite element method (FEM). The mutual comparisons of the comparable parameters of the various sonotrode shapes are pres

60、ented. The main aim of this paper is to present generally valid results leading to the selection of suitable shape and corresponding geometrical dimensions of sonotrode with required dynamical prop</p><p>　　

61、2. The sonotrode design</p><p>　　The principal function of the sonotrode is to amplify the amplitude of ultrasonic vibrations of the tool to the level required to the effective machining. The sonotrode serve

62、s also as an element transmitting the vibration energy from the transducer towards to the tool interacting with workpiece. It does so by being in resonance state with the transducer. The design and manufacture of the son

63、otrode require special attention. Incorrectly manufactured sonotrode will impair machining performance and</p><p>　　Generally, the sonotrodes are made of metals that have high fatigue strengths and low acous

64、tic losses. The most important aspect of sonotrode design is a sonotrode resonant frequency and the determination of the correct sonotrode resonant wavelength. The wavelength should be usually integer multiple of the hal

65、f wavelength of the sonotrode. The resonant frequency of sonotrode, which has simple geometrical shape can by determined analytically (cylindrical shape). For complicated geometrical shape,</p><p>　　The requ

66、ired performance of sonotrode is assessed by an amplification factor</p><p>　　, (1)</p><p>　　Where A0-amplitude of input end of sonotrode,</p><p>　　

67、A1-amplitude of output end of sonotrode.</p><p>　　The basic requirement for the size of the amplification factor is</p><p>　　. (2)</p><p>　　2.1. Th

68、e analytical solution of the free sonotrode vibrations</p><p>　　The governing equation of longitudinally vibrating sonotrode with variable circular cross-section S(x), which is valid for 1D continuum (thin e

69、lastic bar), is expressed in the form</p><p>　　, (3)</p><p>　　Where x-coordinate in the longitudinal direction,</p><p>　　u(x, t)-longitudinal displacement of cross-s

70、ection,</p><p>　　S(x)=π(r(x))2-cross-section area,</p><p>　　r(x)-radius of circular cross-section,</p><p>　　-velocity of the longitudinal waves in 1D continuum,</p><p>

71、　　E-Young’s modulus of sonotrode material,</p><p>　　ρ- density of sonotrode material</p><p>　　The free sonotrode vibration of cylindrical shape (r(x)=r) is described by wave equation</p>

72、<p>　　. (4)</p><p>　　The solution of equation (4) is supposed in the form u(x,t)=U(x)T(t).Then partial</p><p>　　differential equation (4) is divided into following

73、 two ordinary differential equations</p><p>　　, (5)</p><p>　　, (6)</p><p>　　Where ω0-natural angular frequency.</p><p&

74、gt;　　Introducing the following non-dimensional quantities</p><p>　　?non-dimesional coordinate in the longitudinal direction:,</p><p>　　?non-dimesional longitudinal displacement of cross-section:

75、,</p><p>　　into the first of equations (5), we obtain non-dimensional equation</p><p>　　and its solution , (7)</p><p>　　where-frequency parameter,</p><p>　　l0-sonotr

76、ode length.</p><p>　　Both sides of sonotrode have the possibility of motion in the axial direction. To the input side is attached electromechanical transducer which generates ultrasonic axial vibrations and

77、to the output end is attached vibrating tool. Then the boundary conditions for free vibration of sonotrode are supposed as a free-free edge on both sides [7] in the form</p><p>　　, . (8)&

78、lt;/p><p>　　Then, after application of boundary conditions(8) into solution (7), the following modal parameters of sonotrode are obtained</p><p>　　?natural frequency (in [Hz]) of the kth mode shape

79、</p><p>　　, (9)</p><p>　　?non-dimensional wave length of the kth mode shape</p><p>　　, (10)</p><p>　　Where βk is

80、 kth root of characteristic equation and k=1,2,...</p><p>　　In order to achieve the desired effect on ultrasonic machining, only the first two mode shapes of sonotrode are used, i.e. For k=1 so-called”half

81、wave”shape and k=2”wave”shape (Fig. 1).</p><p>　　As it is seen, the analytical determination of mode shapes and the natural frequencies of cylindrical shape of sonotrode is relatively simple. Analytical dete

82、rmination of these parameters for non-cylindrical shapes is more complicated. Therefore, to the determination of modal properties for more complicated geometrical sonotrode shapes, the numerical method (FEM) is better to

83、 use.</p><p>　　Fig. 1. Mode shapes of cylindrical sonotrode vibrations</p><p>　　2.2. Finite element analysis of free sonotrode vibrations</p><p>　　The determination of modal propert

84、ies for various shapes of sonotrode and assessment of the effect of relevant geometrical parameters on the modal properties, the finite element method is used. The FEM modelling and calculation of modal properties was do

85、ne using the software package ANSYS. The element SOLID45 was used to the sonotrode FE model creation.</p><p>　　The equation of motion to description of free vibration of sonotrode FE model, by which the moda

86、l properties are determined, is expressed in following form</p><p>　　, (11)</p><p>　　Where M(B,resp.K) is mass (damping, resp. stiffness) matrix,</p><p&g

87、t;　　ü(u,resp.u) is vector of nodes acceleration (velocity, resp. displacement).</p><p>　　Since it can be supposed that the sonotrode materials have a low damping capacity (from dynamical aspect), the da

88、mping in equation of motion can be neglected. The equation of motion (11) can be for B=0 rewritten into the form</p><p>　　. (12)</p><p>　　The modal properties of s

89、onotrode are determined by the solution of eigenvalue problem</p><p>　　, (13)</p><p>　　Where Φi- ith eigenvector (mode shape),</p><p>　　Ωi-natural angu

90、lar frequency of ith mode shape.</p><p>　　As mentioned earlier, the horns are manufactured in various shapes and dimensions. The cross-section of sonotrodes for ultrasonic machining or ultrasonic assisted ma

91、chining has mostly circular shape. The functions that define the shape of longitudinal shape of sonotrode may be different. The main emphasis is to obtain the required dynamic properties. The geometries of considered son

92、otrode shapes are based on the dimensions and non-dimensional parameters (Table 1).</p><p>　　Table 1. Geometrical parameters of sonotrode shapes</p><p>　　Note: d0-diameter on input side of sonot

93、rode,d-step changed diameter, l0 – length of Sonotrode.</p><p>　　3. Results of numerical simulations</p><p>　　The numerical analyses are performed for various sonotrode shapes and geometrical pa

94、rameters defined in Table 1. The steel as a sonotrode material is used to numerical simulation (E= 210GPa,ρ= 7 800kg·m?3,ν=0.3).</p><p>　　In the following, the non-dimensional resonant frequencies for d

95、ifferent geometrical shapes of horn are defined as</p><p>　　, (14)</p><p>　　Where fk -kth natural frequency of analysed horn,</p><p>　　f0k–kth natur

96、al frequency of cylindrical sonotrode shape for corresponding slenderness ratio.</p><p>　　The results of dynamical analysis of considered sonotrodes are expressed in dependence to the above mentioned dimensi

97、onless quantities. This manner of presentation gives an opportunity for generalization of the results and also allows their mutual comparison for the various geometric shapes of sonotrodes.Moreover, this manner of result

98、s presentation can be used to the selection of shape and dimensions of sonotrode with the required properties.</p><p>　　The value of resonant frequency of corresponding geometrical horn shape is determined u

99、sing following equation</p><p>　　, (15)</p><p>　　Where k=1”half wave” shape,k=2“wave”shape.</p><p>　　In the next figures (Fig. 2–5), the dependences of no

100、n-dimensional natural frequencies and amplification factors on relevant parameters of sonotrode design are shown.</p><p>　　4. Conclusion</p><p>　　The dynamical analysis of the various geometrica

101、l shapes of sonotrodes as one of the most important elements of the ultrasonic machining systems is presented in this paper. The main dynamic characteristics (natural frequencies and amplification factors) of sonotrode i

102、n the resonant state were studied according to the geometric shape and dimensions. The efficiency and performance of ultrasonic machining systems depends on specific design and the relatively large number of parameters.&

103、lt;/p><p>　　Selection of a geometric shape of the sonotrode depends on technological operation for which the sonotrode will be used. The value of resonance frequency and amplitude amplification factor on the ou

104、tput side of sonotrode are fundamental requirements for the selection of appropriate sonotrode shape. The cylindrical sonotrode shape was used as a comparative geometric shape to the expressing of results of other sonotr

105、ode shapes. Generally, it can be said that the geometrical shapes and dimensions o</p><p>　　Cylindrical sonotrode shape</p><p>　　Fig. 2. Cylindrical sonotrode shape. Dependence of the non-dimesi