十字萬(wàn)向節(jié)傳動(dòng)軸輸出角速度的研究外文文獻(xiàn)翻譯、中英文翻譯、外文翻譯,十字,萬(wàn)向節(jié),傳動(dòng)軸,輸出,角速度,研究,外文,文獻(xiàn),翻譯,中英文 Received 04 March 2011Research on Angular Output Velocity of a Drive Shaftwith Double Cross Universal JointsMA Xiao- san，YU Zhi- fu，HAN YanSchool of Mechanical and Electrical Engineering，Hebei Universityof Engineering，Handan 056038，P R ChinaAbstract: The study object is the angular output velocity of the drive shaft which is made up of two series- woundcross universal joints We have deduced the function relation between the angular output velocity and initiative inputangle of the drive shaft with double cross universal joints that is based on the calculation formula of the angular out-put velocity of a single cross universal joint，and by analyzing the relation between the two input angles By usingthis function relation，the constant velocity condition of the drive shaft with double cross universal joints is verifiedThe step- by- step searching algorithm is adopted to obtain the optimal phase angle that leads to the minimum fluctu-ate index of the angular output velocity in the vary velocity condition At the same time， we worked out the maximaland minimum value of the angular output velocity， and their initiative input angle The correctness of the function ofthe angular output velocity and the step- by- step search algorithm are verified by an ADAMS simulation exampleKey words:drive shaft with double cross universal joints;angular output velocity;initiative input angle;phaseangle;step- by- step searching algorithm;ADAMS1IntroductionThe drive shaft with two series- wound cross universaljoints is widely used in the driveline and the steeringsystem of vehicles because of its advantages such asthe simple and compact structure，high transmissionefficiency ，torque and easy maintenance 1 It can beused to transmit torque or rotary motion between twoaxes which are not in the same line especially when thetwo- axis angle is large Compared with the initiativeangular input velocity，there is a periodic fluctuationin the angular output velocity of the drive shaft withdouble cross universal joints because of the vary veloc-ity of the single cross universal joint 2 The experi-ment shows that this periodic fluctuation will lead to anadditional load，mechanical vibrations and noises 3 So it is necessary to analyze the angular output velocityof the drive shaft with double cross universal joints Bythis analyzing，the optimal angle between the drivenfork of the first universal joint and the driving fork ofthe second universal joint that lead to the minimumfluctuation value of the angular output velocity，thenthe input angular velocity which matches for the outputwill be obtained Or find out the maximal and mini-mum values of the angular output velocity，and theircorresponding initiative input anglesBased on the calculation formula of the angular outputvelocity of a single cross universal joint，and by ana-lyzing the relation between two input angles， the func-tion relation between the angular output velocity andinitiative input angle of the drive shaft with doublecross universal joints has been deduced By using thisfunction relation，the angular output velocity of thedrive shaft with double cross universal joints is ana-lyzed The analysis results are verified by an ADAMSemulate example2Establishment of the angular outputvelocity functionThe operation principle of the drive shaft with doublecross universal joints is shown in Figure 1International Journal of Plant Engineering and ManagementVol 16No 2June2011Figure 1Operation principle drawing of the drive shaftThe mathematical relations of all the angular outputinstant velocities about the drive shaft with doublecross universal joints are as follows4 :2=1cos1 sin2cos2x1( 1)3=2cos1 sin2cos2x2( 2)Where:transmission angle is the angle between theinput shaft and the intermediate shat is the anglebetween the intermediate shaft and the output shaftInitiative input angle x1is the instant angle betweenthe plane including the fork A and the input andintermediate shaft x2is the instant angle between theplane including the fork C and the intermediateand input shaft The mathematical relation betweenx1and x2is as follows:x2= x1+ x + 90 + ( 3)Where: is the angle between the plane that includesboth the input shaft and the intermediate shaft and theplane that includes both the intermediate and the out-put shafts The phase angle is the angle between theplaneincluding the fork B and the plane includingthe fork C x is the difference of the instant anglesbetween the input shaft and the intermediate shafttanx =( 1 cos) tanx1cos + tan2x1( 4)The instant angular output velocity 3can be ex-pressed as follows according to formulas ( 1) ，( 2)and ( 3) :3=1cos1 sin2cos2( x1+ x + 90 + )cos1 sin2cos2x1( 5)with all the formulas， we find that in the process of op-eration of the drive shaft with double cross universaljoints， the instant angular velocities of the input shaft，the intermediate and output shaft 1，2and 3arenot equal，although their rotational speeds are equal( n1=n2= n3) In the condition that 1is a constantvalue，3is a periodic function of period 180which isinconsistent with 1 Its variation laws are determinedby the constant value， ， and Its fluctuation in-dex can be obtained by this formula: =3max 3min3=3max 3min1( 6)Where:3maxis the maximal value of the angular out-put velocity 3 3minis the minimum value 3is theaverage value The formula 3= 1is proper，be-cause of n1= n33Verification on the constant velocitycondition of the drive shaft with doublecross universal joints3. 1 In the condition that = and the threeshafts are in the same planeIn the condition that = and the input，interme-diate and output shafts are in the same plane，formulacos = cos and =0 can be obtained Then formula( 5)can be expressed as follows:3=1cos21 sin2cos2( x1+ x + 90 + )11 sin2cos2x1( 7)In the condition that = 0，formula ( 7)can be de-rived as follows:3=1cos21 sin2sin2( x1+ x)11 sin2cos2x1( 8)021International Journal of Plant Engineering and ManagementVol 16No 2June2011Another expression is like this:3=1cos21 sin2( sinx1cosx + cosx1sinx)211 sin2cos2x1( 9)Substitute formula ( 4)and formulas ( 10) ，( 11)( 12)as follows into formula ( 9) ，and make furtherderivations，then 3= 1and =0 can be obtainedsin2x =tan2x1 + tan2x( 10)cos2x =11 + tan2x( 11)sin( 2x)=2tanx1 + tan2x( 12)It shows that in the condition that = ，the threeshafts are in the same plane and = 0，the result of3= 1can be obtained，although 21，and thenthe ideal result that the instant angular output velocityis completely equal to the input velocity and =03. 2 In the condition of = and the threeshafts are not in same planeIn the condition of = and the input，intermedi-ate and output shafts are in the same plane，formulacos = cos and 0 can be obtained Then formula( 5)can be derived as follows:3=1cos21 sin2cos2( x1+ x + 90 + )11 sin2cos2x1( 13)In the condition that = ，formula ( 13)can be de-rived to ( 8) Make further derivations，3= 1canbe obtained too It shows that in the condition that = ，the three shafts are not in the same plane and = ，the result of 3= 1can be obtained although21， and then the ideal result that the instant an-gular output velocity is completely equal to its inputvelocity and =04Characteristic analysis of the angularoutput velocity in the condition of 4. 1 In the condition of and the threeshafts are in the same planeIn the condition that and the input，interme-diate and output shafts are in the same plane，formula =0 can be obtained Then formula ( 5)can be de-rived as follows:3=1cos1 sin2sin2( x1+ x + )cos1 sin2cos2x1( 14)Because the function expression of 3is very com-plex，and 3( )= 3( + 180) ，a step- by- stepsearching algorithm5 can be adopted to obtain theoptimal phase angle that lead to the minimum fluctua-tion index of the angular output velocity In the algo-rithm，the step length of is 180/mFigure 2 shows the flow chart of the step- by- stepsearching algorithm for obtaining the optimal phase an-gle In the flow chart，A is a number which is largeenough，usually it is ok if it is larger than two in orderto ensure the effectiveness of the algorithm process andresults In each step of the step- by- step search，eachfunction expression about ( k)3( x1)determined by ( k)matches the formula: ( k)3( x1)= ( k)3( x1+ 180) This step- by- step searching algorithm can be adoptedto obtain the maximal value ( k)3maxand minimum value( k)3minof ( k)3( x1)determined by ( k) In this step- by-step searching algorithm，the step length of x1in thealgorithm is 180/n Substitute ( k)3maxand ( k)3mininto for-mula ( 6) ，then the fluctuation index of the angularoutput velocity ( k)can be obtained The minimumvalue *of ( k)is the minimum fluctuation index ofthe angular output velocity，and the value *of ( k)corresponding *is the optimal phase angleIn the condition that the phase angle is determined，the step- by- step searching algorithm can be adopted toobtain the maximal and minimum value of 3andtheir initiative input angle x1is in accordance withformula ( 14)and Figure 2121Verification on the Constant Velocity Condition of the a Drive Shaft with Double Cross Universal JointsFigure 2Flow chart of a step- by- stepsearching algorithm4. 2 In the condition of and the threeshafts are not in the same planeIn the condition of and the input，intermedi-ate and output shafts are not in the same plane， and 0 in the formula ( 5) Order = ，then for-mula ( 5)can be derived as follows:3=1cos1 sin2sin2( x1+ x + )cos1 sin2cos2x1( 15)Consider formula ( 15)and ( 14)are in the sameform，the step- by- step searching algorithm accordingto Figure 2 can be adopted to obtain *correspondingthe minimum fluctuation index of the angular outputvelocity Then *can be obtained by the formula: *= *+ In the condition that the phase angle is determined，the step- by- step searching algorithm can be adopted toobtain the maximal value and minimum value of 3and their initiative input angle x1in accordance withformula ( 15)and Figure 24. 3 A example of the researchA drive shaft with double cross universal joints is re-quired for its transmission angle = 15， = 20 andthe angle =10 Its angular input velocity 1=60/sThe angular output velocity is analyzed as follows Bythis analyzing， the optimal phase angle *is obtained，and the maximal value，minimum value of the angularoutput velocity 3，and their initiative input angle x1will be determined in the condition that =100Substitute the values of ， and into formula( 15) ，we will get the results as follows:3=1cos201 sin220sin2( x1+ x + )cos151 sin215cos2x1( 16)Where:x can be obtained by the formula as fol-lows:x = arctan( 1 cos15) tanx1cos15 + tan2x1Program the step- by- step searching algorithm in C Lan-guage 6 in accordance with formula ( 16)and Figure 2The program ran the results that *= 0 and *=0.055 Then the optimal phase angle is *=10 Themaximal value of the angular output velocity 3is 3max=61.68/s，and its initiative input angle is x1=90The minimum value of 3is 3min=58.37/s，and itsinitiative input angle is x1=0When the phase angle is = 100， = 90 can beobtained By running the program of C programming221International Journal of Plant Engineering and ManagementVol 16No 2June2011language of the step- by- step searching algorithm，itshows that the maximal value of the angular output ve-locity 3is 3max= 66. 10/s with its initiative inputangle x1=0，and the minimum value of 3is 3min=64. 64/s with its initiative input angle x1=90Three fluctuating curves of the angular output velocity3in the three conditions that =10， =100，and = 55 can be produced by using Matlab 7 as inFigure 3Figure 3Three fluctuating curves of theangular output velocity5Simulation Verification of the exampleof the research5. 1 Simulation of the angular output velocity inthe condition of = The three- dimensional model of the drive shaft withdouble cross universal joints with its phase angle = =90 and = =15 is created by Pro/E Then thedriving model and virtual prototype is establishedbased on ADAMS as in Figure 4Figure 4The virtual prototype of the drive shaftThe angular input velocity of the drive shaft is set at1=60 r/min ( 60/s) Then the kinematics simula-tion based on the virtual prototype is carried out8 The fluctuating curve of the simulation angular outputvelocity is as Figure 5Figure 5The curve of the simulation angular output velocityunder the constant velocity conditionFigure 5 shows that simulation angular output velocity3is a constant value that is completely equal to 1It is consistent with the theoretical derivative resultsin the constant velocity condition5. 2 Simulation of the angular output velocity inthe condition of Three- dimensional models of the drive shaft withdouble cross universal joints with their =10 and = =15 under the three conditions that their phaseangle = 10， = 100 and = 55 are created byPro/E Their driving models and virtual prototypesare established based on ADAMS，and their angularinput velocities of the drive shaft are set at 1=60/s Then kinematics simulations are carried outThree fluctuating curves of the simulation angular out-put velocity in the three conditions are as Figure 6Figure 6Three curves of the simulation angular outputvelocity under the condition of It shows that the theoretical derivative results are con-sistent with the kinematics simulation by comparing321Verification on the Constant Velocity Condition of the a Drive Shaft with Double Cross Universal Jointsthe curves of Figure 6 and Figure 3 Then the correct-ness of the function of the angular output velocity andthe step- by- step search algorithm are verified6ConclusionsThe function of the angular output velocity of the driveshaft with double cross universal joints is derivedThis function provides the theory basis for the re-search on angular output velocity By using the step-by- step searching algorithm，the optimal phase anglethat leads to the minimum fluctuation index of the an-gular output velocity can be obtained With the step-by- step searching algorithm，the maximal value，theminimum value of the angular output velocity，andtheir initiative input angle can be determined Thecorrectness of the function of the angular output veloc-ity and the step- by- step search algorithm are verifiedby an example and its ADAMS simulation This paperprovides an algorithm workable and effectual for thedetermination And this algorithm can be as a refer-ence for research on angular output velocity of thedrive shaft with more than two series- wound cross uni-versal joints used in the shipbuilding industryReferences 1Ai W，F(xiàn)ang K X，Study on transmissioncharacteristic of shafting system with multiplecross universal joints Journal of Jiangsu Uni-versity of Science and Technology ( NaturalScience Edition) ，21( 2) :71 74，2007( In Chinese) 2Cai X W，Automobile structure and principle( volume 2 Chassis and Body) Beijing:Chi-na Machine Press， 2004 ( In Chinese) 3Ren S Y，Zhu Z L，ZHANG J W，Transmis-sion modeling and simulation of drive shaftwith double cross universal joints Journal ofShanghai Jiaotong University， 38( 11) : 19221927， 2004 ( In Chinese) 4Yang Z M，Drive shaft and universal jointBeijing:China Communications Press， 1986( In Chinese) 5Yan S M，Mechanical optimization designXuzhou:China University of Mining andTechnology Press， 2003 ( In Chinese) 6Wang B S，Li W Q，He H J，C program de-sign Beijing:Higher Education Press， 2004( In Chinese) 7Jiang S H，Matlab language and mathemat-ics experiment Beijing:Science Press， 2007( In Chinese) 8Zheng K，Hu R X，Chen L M，ADAMS2005advanced application examples of mechanicaldesign Beijing:China Machine Press， 2006( In Chinese)Brief BiographiesMA Xiao- san is a graduate student in the School ofMechanical and Electrical Engineering，Hebei Uni-versity of Engineering His research interests includemechanical optimization design， measurement andcontroltechnologyofelectromechanicalsystemsmmaxiaosan126 comYU Zhi- fu is a professor in the School of Mechanicaland Electrical Engineering，Hebei University of Engi-neering His research interests include mine electro-mechanical and safety engineeringHAN Yan is a graduate student in the School of Me-chanical and Electrical Engineering，Hebei Universityof Engineering Her research interests include meas-urement and control technology of electromechanicalsystems421International Journal of Plant Engineering and ManagementVol 16No 2June2011