超声变幅杆的动力学分析

Sep.2015Hydromechatronics engineeringVol. 43 No. 18http://jdy.gkscqut.edu.cnE-mail;jdygcyw@126.comDynamic study on ultrasonic hornXing-hong ZHANG, Xin chen, Tao HE, Lei QitChongging Key Laboratory of Time-grating Sensing and Aduanced Testing TechnologChongqing University of Technology, Chongging 400054, China)Abstract: As an important part of power ultrasonic vibration system, ultrasonic horn can amplify the displacementof mechanical vibration and concentrate energy on a smaller radiating surface, and the amplification coefficient ofultrasonic horn is a vital parameter in vibration system. The longitudinal vibration wave equation of conical ultrasonic horn is established according to differential element method and solved by the method of separation of variables and boundary conditions, to attain the mathematical expression between amplification coefficient and threevariables called the ratio of the small end to the large end diameter, length and external excitation frequency. Twoof the three variables are set successively as constants, and then amplification coefficient can be achieved according to the third variable sampling value by the mathematical expression. The sample calculating data of ultrasonichorn is applied to get fitting curve by Matlab, and relationships between amplification coefficient and three variables are analyzed, qualitatively.Key words: Ultrasonic horn, Differential element method, Amplification coefficient, Curve fittingdoi:10.3969/j.issn.1001-3881.2015.18.013Document code. ACLC number. TB53mechanical vibration and concentrate energy on a1 Introductionsmaller radiating surface, and effectively delivers thesound energy to load by impedance matching betweenWhen machining with intractable materials with ul- transducer and acoustic load [4-8trasonic assisted turning, periodical separation of cut- Longitudinal vibration wave equation of conical ulting tool and processing materials, lead to the de- trasonic horn is established according to differential elcrease of cutting force, lower cutting temperature and ement method and solved by the method of separationPCompared with the traditional of variables and boundary conditions, to attain theturning processing ultrasonic assisted turning can not mathematical expression between amplification coeffiughness and roundness ofcient and three variables called the ratio of the smallimproved significantly, but can extend the life of cut end to the large end diameter, length and external ex-tool by hundreds or thousands times, and processing citation frequency. Two of the three variables are setcost greatly reduces 1-3]. The ultrasonic assisted successively as constants for better research, and thenturning technology has become one of important pro- amplification coefficient is achieved according to thecessing methods of complex parts, such as high hard- third variable sampling value. The sample calculatingigh data of ultrasonic hom is applied to get fitting curve bystrength glass and other hard processing material. Matlab, and relationships between amplification coefHorn is an important component of ultrasonic assisted fic中国煤化工 iables, are qualitativelyprocessing system, it can amplify the displacement ofTHCNMHGReceived 2 September 2014; revised 15 December 20142 Wave equation of ultrasonic hornaccepted 12 March 2015Corresponding author: Xin CHEN, E-mail: hetao814185754 2 I Longitudinal vibration of continuous elastomerVibrating particle can cause vibration of adjacentDynamic study on ultrasonic horn71particles in elastic medium. Assuming that the elasticmedium is divided into several layers, each layer iscomposed of many particles closely linked to each othOf which the force iser. Once one particle of the medium disturbed, themovement would deviate from its equilibriit will inevitably drive the neighboring particle beginThe equation of harmonic vibration [10] isto move, due to the contact exists between particles+ωδ(x,t)=0(4)In this way, the vibration of object is spread throughelastic medium. It's called fluctuation [6]. In order Take longitudinal vibration equation, and then getto be convenient for researchthat variablwave equation of variable cross-section horncross-section rod is composed of homogeneous and isoic materiallitting mechanical2+10s(x)axn+k28(x,1)=0(5)S(x)athe size of cross section is far less than wavelength itk=o6can be affirmed that stress distribution on cross sectionof rod is uniform when plane wave spreads along theEscrew axial [7](7)2.2 Establishment of wave equationWhere, h is the wave number, a is the frequency, cis the longitudinal wave propagation velocity in hornDifferential element method is a way of thinkinfrom local to overall, and we can use known physical 3 Solution of wave equationlaws to solve some complicated physical process quick-ly by this method. As shown in Fig. 1, a rectangularThe method of separating variables is aimed to dicoordinate system is established in the origin of coor- vide original equation into several simpler differentialdinates on the left side of horn. 8(x, t)is given for equations that containing only one independent variadisplacement with position x and time t, the dynamic ble by separating each variable apart. According toequation can be established by Newton's second the principle of linear superposition, split inhomogelaw as:neous equations into multiple homogeneous or easysolving equations. Using mathematic methods to aS(x)△x(x+△x,t)S(x+△r)and assem-δ(x,t)S(x)bled all the general solution"together"finallWhere, o(x)is the force with position x on the cross 3. 1 Longitudinal vibration of continuous elastomersection and time t, o(x+Ax) is the force with position x A on the cross section and time t, p is theThere exists two situations in boundary of continumaterial density of horn, S(x) is the cross-sectionalous longitudinal vibrating rod piece, one is that endfacforce free. foe lace is zero accordiareato Newton's third law: another is that end face is under fotablished in bothbased on Hooker's lahown in Fig. 2. force anda(r.)(x+△x,Ddisplacement of two end faces are given as f, F, andx+△tFig 1 Longitudinal vibration of variable crossrV凵中国煤化工F2CNMHG0s=im(x+△x,)(x+△x)-0(x)S(x)Fig 2 Diagram of stressn)S(x+Ax)-o(r, t)S(r+ Ari) When end face is force free, the axial force istakes for hooker’slaw72Xing-hong ZhANG. et alMA(x=)N(coshNklsink)=dx/ =0(9)(21)ii) When end face is under force, takes for Hook-er s aw4 Analysis and curve fitting of amplifi-(10)cation coefficientF,=-SEdsWhen designing. material of the horn should behosen after determining the maximum displacement3. 2 Solution of amplification coefficientThe material of horn is 1045. which is selected basedon the require of application. The physical parametersAccording to the characteristic of harmonic vibration of 1045 are shown in table 1equation, solution of wave equation can beexpressed asTable 1 Physical parameters of 10458(x, t)=A(x)sin(wt +(12)Modulus ofSpeed of SoundUsing wave equation(5) to achieveElasticity E/GPa kgm-3 c/(ms")2A(x),1,ds(x).d4(x)+k2A(x)=0d3)The given parameter of diameter of conical hornThen choose an initial value for tlwithL. are DI and 2quency, in order to produce high amplitude and totively. So the functions of cross sectional area and diminimize the effect of heat generation on performanceameter areof transducer. The initial value is chosen as 20 khwhich represents, theoretically, the low ultrasonicD(x)=D1(1-ax)(15)threshold 9Of which. s, is the cross sectional area of hornConsidering the relationships of amplification coeffiand parametersa, N are given ascient M and diameter ratio N. the length i and exterD,-Dnal excitation frequency f, set successively two of the(16) three variables as constants. and then achieved the reDlationship based olg calculations of the thirdD2(17)ification coefficientHere is the final wave equationSet l and f to fixed values. The frequency here was20 kHz. and the length of half wave ultrasonic hornd'A(x) 2a, dA(x)+kA(x)=0(18) was 0. 129 m Sampled value of N verified from 1. 0 toaxSo, displacement function7.9Then, the vibration frequency f= 20 kHz and MA(x)=(A, coskx B,sinkx) (19) 3. 8 were assumed. Sampled length of horn verifiedfrom 0. 114 m to 0. 160 m, and sampling interval wasAs shown in Fig. 2. the force on the end of horn is 0. 002 mFI and F2rely. When the end of horn isAssumed both l and w as constant values./=0.129force free the force is zero no matter what value t ism, N=3.8. Sampled frequency verified from 6 kHz toAccording to the boundary conditions and Hooker's 29中国煤化工 I was 1×10°kHzlaw mentioned before. constants can be determinedCNMHG Matlab. Approximate(20)growth of length. According to the Fig. 5, maximum ofBhamplification coefficient corresponded to frequency ofAnd then the amplification coefficient can be ex- fo( natural frequency of the ultrasonic horn)Dynamic study on ultrasonic hornproximate lielationship with the ratio of the smallend to the large end diameter.65432) When the ratio of the small end to the large endvalues it can be seen from the fitting curve that amplification coefficient decreases with the increase oflength of horn, especially when the length of horn isgreater than half wave length, amplification coefficientDiameter ratiodecreases faster with the increase of length3)When length of horn and diameter ratio of end3 Relationship between amplification coefficientand diameter ratioface are constant values, and vibration frequency as aalue of fo, the amplilargest. When external excitation frequency is lowerthan foquency increases. When external excitation frequencyis higher than fo, amplification coefficient decreases asthe freReferer[1]WANG Hongfei, Research of vibration assisted tur0lol15012012501301350140450150155016ting technology and its development [J]. Machinery DesignManufacture,2007(10):212-215Fig 4 Relationship between amplification coefficient [2 Brehl D E, Dow T A. Review of Vibration-assisted Machiand lengthning[ J]. Sciencedirect Precision engineering, 2008(32)153-172[3 PAN Hui. Design of Ultrasonic Horn and Its PerformancAnalysis [J]. Equipment Manufacturing Technology, 2009(8):69-72[4 ZHAO Li, WANG ShiYing. Dynamic Analysis of the Hornin Ultrasonic Machining [J]. Electromachining& Mould, 2005(2):34-38[5]HE XiPing, GAO Jie. A review of ultrasonic solid horn degm [J]. Technical Acoustics, 2006, 25(1): 82-87[6 CHU Tao. Finite element analyses of the ultrasonic ampli0.51.0tude transformer[J]. Mechanical Electrical EngineeringMagazine,2009,26(1):102-10[7 ZHU Yin. Resonant analysis of ultrasonic amplitude hornFig 5 Relationship between amplification coefficient with Finite Element Method [J]. Design and Research, 2005and external excitation frequency32(12):13-15[ 8FU Jin. The Design and Production of Transformers in the5 ConclusionsProcess of Micro-ultrasonic Machining[ J. Modern Machinery2010(2):33-38.Relationships of above variables can be[9]Sergei L P, Alexey S P. Matching a transducer to water atcavitation: Acoustic horn design principles [ J. Industrialconcludes as1)When length of horn and external excitation fre- [10中国煤化工fitting curve that amplification coefficient shares ap- Yai Yuan University of Technology, 2011, 42(6): 630-3 oquency are constant values, it can be seen from theCNMHGNGShiying. Numerical DeSolid Horn [J].Journal74Xing-hong ZhANG. et al超声变幅杆的动力学分析张兴红,陈鑫”,何涛,邱磊重庆理工大学时栅传感及先进检测技术重庆市重点实验室,重庆400054摘要:超声变幅杄是功率超声振动系统的重要组成部分,它的主要功能是把机械振动位移放大并把能量集中在较小的辐射面上,变幅杆的放大系数是超声振动系统中的重要参数。用微元法建立圆锥形超声变幅杆纵向振动的波动方程,用分离变量法和连续振动体的边界条件求解超越波动方程,得到圓锥形超声变幅杄放大系数与变幅杄的大小端直径之比、长度以及外激频率3个变量之间的数学表达式。为研究变幅杆的放大系教与变幅杆的大小端直径之比、长度以及外激频率的关系,依次把3个变量中的2个变量设为常量,对另外一个变量进行抽样,计算变幅杄的放大系数。通过 Matlab对超声变幅杄放大系数的样本计算数据进行曲线拟合,定性分析了圆锥形超声变幅杆放大系数随变幅杄大小端直径之比、长度、外激振动频率变化的关系关键词:超声变幅杆;微元法;放大系数;曲线拟合Introduction of the Fluid Control Engineering Institute ofKunming University of Science and TechnologyThe Fluid Control Engineering Institute of Kunming University of Science and Technology was set up in 1996. The researchesof institute concentrate on electro-hydraulic( pneumatic)servo/proportional control and hydromechatronics. The Institutted to research and development of electro-hydraulic control of high-end technical equipment in ferrous metallurgy refining production. Projects undertaken and participated by the copper electrolysis anode preparation equipment, lead residual anode washing pro-duction lines as a host device received the second prize of the National Science and Technology Progress Award in 2009, the firstprize and the third prize of the Yunnan Provincial Science and Technology Progress Award and many other awards. The institute hasdeveloped and put into operation more than a dozen sets of large equipment, and more than 20 national patents, which have beentransformed into related products, providing professional package services of technology and equipment for non-ferrous metallurgicalenterprises Address College of Mechanical and Electrical Engineering, Chenggong Campus of Kunming University, 727#, JingmingSouth road, Chenggong University City, Kunming City, Yunnan ProvinceZip Code: 650500Contact: Sun Chungeng. 1360885065Sen,13888749366H中国煤化工CNMHFig. I The copper electrolysis Fig. 2 The lead electrolysis Fig 3 Electro-hydraulicFig 4 The lead residualanode preparationwashing and rods drawingoperation specialanode washing