Mathematical model for determination of strand twist angle and diameter in stranded-wire helical springs

Transcription

1 Journal of Mechanical cience an Technology 4 (6) (1) 13~11 DOI 1.17/s Mathematical moel for etermination of stran twist angle an iameter in strane-wire helical springs hilong Wang *, ong Lei, Jie Zhou an Hong Xiao The tate Key Laboratory of Mechanical Transmission, Chongqing University, Chongqinq 444, China (Manuscript Receive January 8, 9; Revise December 15, 9; Accepte March 8, 1) Abstract The twist angle of strans has a significant effect on the performance of strane-wire helical springs. This paper proposes two moels (a precise mathematical moel an an approximate mathematical moel) for calculation of the twist angle an the iameter of strans forme of an arbitrary number of wires. These two parameters can be erive from the screw pitch of strans an the iameter an number of their wires, regarless of the moel use. In comparative an analytical stuies, it was etermine that for strans with the same number of wires, the larger the ratio of the screw pitch to the iameter of the wires is, the smaller the relative errors of the two moels are. Only when the foresai ratio is two times larger than the number of wires, the results of the approximate mathematical moel can be acceptable. In other cases, the approximate mathematical moel cannot be use. Finally, application software was evelope for irect precise-mathematical-moel calculation of the twist angle an iameter of strans forme of arbitrary wires. Keywors: trane-wire helical spring; tran; Twist angle; Diameter; Design Introuction A strane-wire helical spring, first observe in Russian machine guns [1, ], is a unique cylinrically helical spring. It is reele by a stran that is forme of ~7 wires. The iameter of the wires is.5~3. mm, an their material is common carbon steel. It has been known for some time that strane-wire helical springs, as compare with conventional single helical springs, exhibit more amping an have longer fatigue lives in certain ynamic applications [3-5]. They have been use successfully in machine guns, small weapons, an stapling guns to ampen the high-velocity isplacement of the coils [6, 7]. For best strane spring performance, it is essential that strans be esigne an manufacture accurately [8]. As shown in Fig. 1, the twist angle of a stran, β, has a significant impact on strane spring performance. In the esign of strane-wire helical springs, the twist angle β an the iameter c are calculate accoring to the screw pitch of strans, the iameter of wires an the number of wires N. In Zhang, Liu an Wang s pring Hanbook [9], an in the paper by Maolin Wang [1], a metho of calculating stran This paper was recommene for publication in revise form by Associate Eitor Tae Hee Lee * Corresponing author. Tel.: , Fax.: (81) aress: slwang@cqu.eu.cn KME & pringer 1 iameter whereby the minimum istance of ajacent wires is to be equal to the iameter of wires was offere. In this metho, the twist angle of strans first is assume. Then, this angle is use to obtain the stran iameter in reference to a lookup table. However, as a matter of fact, the twist angle is etermine via the screw pitch of strans an the iameter of wires. Therefore, this metho oes not solve the problem of the calculation of β an c suitably. In the Hanbook of pring Design, New Technology of Manufacturing Process an Quality Control, Dongwen Zhang [11] propose a calculation metho in which the cross-section of a stran is compose of ellipses in contact with each other. However, it offere approximate calculation of strans of only three or four wires; for cases in which the number of wires excees four, the metho is infeasible. Uner some circumstances, the values of parameters can be acquire using a metho of interpolation, mentione in papers by Costello [1, 3], Clark [] an athikh [1]. However, these papers offer no explicit calculation process. Recently, hilong Wang an Jie Zhou [13] formulate a mathematical moel for strane-wire helical spring tension uring reeling. It is well known that in eriving such a moel, the most important parameter is the center istribution raius of strans r. From the structural imensions of a stran (see Fig. 1), two expressions (see Eqs. (1-)) can be obtaine:

2 14. Wang et al. / Journal of Mechanical cience an Technology 4 (6) (1) 13~11 Fig. 1. tructural imensions of stran. Fig. 3. Forming process of stran. Fig.. trans an their cross-sections: three strans; four strans. c = r+ (1) πr tan β = () In fact, the cross-section of strans is compose of irregular figures, not circles or ellipses, in contact with each other (see Fig. ). The purpose of the present stuy was to test two stran cross-section moels, a precise mathematical moel an an approximate mathematical moel. pecifically, the two methos calculations of the twist angle an iameter of strans of arbitrary wires were compare an analyze. Base on the results, the relationships among β,, an N were obtaine. Finally, application software was evelope for irect precise-mathematical-moel calculation of the twist angle an iameter of strans of arbitrary wires.. Mathematical moel of cross-section of strans.1 Precise mathematical moel of cross-section of strans Fig. epicts helical-spring strans forme of either three or four twiste wires, along with their cross-sections. For precise mathematical moeling of the irregular crosssection of strans, two Cartesian coorinate systems were erive accoring to the forming process for an arbitrary-wiretwiste stran (see Fig. 3). The first coorinate system is base on the original point O (confirme as σ = {O: X, Y, Z}). FF is the center line of an optional wire in the stran. OF is equal to the center istribution raius of the stran. The plane M P KH is the normal plane of the center line FF at point F. The point F is coincient with the original point O in the secon coorinate system; the cross-section of the wire is a circular wire section in the normal plane M P KH. The secon coorinate system was confirme as σ ' ( σ ' = {O : X, Y, Z }). By means of a series of translations an rotations, σ can be transforme to σ '. Accoring to the affine transformation of the Cartesian coorinate systems, there is a relation between σ an σ ' [14] (see Eqs. (3-4)). That is, or X X ' X Y A Y' Y = + Z Z' Z X ' X X Y' A' Y Y = Z ' Z Z A, an orthogonal matrix, is calle the transformation matrix. Accoring to the efinition of an orthogonal matrix, the 1 relation A = A' can be obtaine. The translation coorinates ( X, Y, Z ) are the coorinates of point O in the coorinate system σ. In the mathematical moel shown in Fig. 3, ( X, Y, Z ) is equal to ( rcos φ, rsin φ, φ/ π), via the equation of the spiral line. A ' is also the rotation matrix. In orer to impart a specific orientation to an object, it is subjecte to a sequence of three rotations etermine by Euler angles. This is to say, the rotation matrix can be ecompose into three elemental rotations. Assuming that the intersection of the OXY an O X Y coorinate planes is a line of noes (the -axis), φ is the angle between the X-axis an the -axis, θ is the angle between the Z-axis an the Z -axis, an ϕ is the angle between the -axis an the X -axis. The ranges of φ an ϕ are e- (3) (4)

3 . Wang et al. / Journal of Mechanical cience an Technology 4 (6) (1) 13~11 15 Combining Eq. (4) with Eq. (7), yiels X ' X X cosφ sinφ X rcosφ Y' A' Y Y sinαsinφ sinαcosφ cosα = = Y rsinφ Z' Z Z cosαsinφ cosαcosφ sinα Z φ π (8) As we have known, in the coorinate system σ ', the normal cross-section of a wire is a circle. Therefore, the basic equation of a circle can be use to represent the shape of the normal cross-section (see Fig. 3): (c) Fig. 4. Three rotations escribe by Euler angles: three Euler angles; rotation aroun Z-axis; (c) rotation aroun intermeiate - axis; () rotation aroun Z -axis. fine as moulo π raians. A vali range coul be [ ππ, ]. The θ range is a moulo π raian, for example, [, π ] or [-π/,π/] (see Fig. 4). Accoring to the above escription [14], A' = A' 3A' A' 1= cosϕ sinϕ 1 cosφ sinφ sinϕ cosϕ cosθ sinθ sinφ cosφ 1 sinθ cosθ 1 where matrix A ' 1 represents a rotation aroun the Z-axis of the original reference frame σ, matrix A ' is a rotation aroun an intermeiate -axis (i.e. the line of noes), an matrix A ' 3 is a rotation aroun the Z -axis of the final reference frame (see Eq. (5)). Carrying out the matrix multiplication, the result is cosϕcosφ cosθsinϕsinφ cosϕsinφ cosθsinϕcosφ sinθsinϕ + A' = sinϕcosφ cosθcosϕsinφ sinϕsinφ cosθcosϕcosφ sinθcosϕ + sinθsinφ sinθcosφ cosθ Because the X -axis is parallel to the OXY coorinate plane, in the mathematical moel (see Fig. 3), ϕ = an θ= π/+ αare obtaine. ubstituting these values, the simplifie version of Eq. (6) is () (5) (6) cosλ X ' Y ' = sinλ Z ' where λ is an intermeiate variable of Eq. (9). Combining Eq. (8) with Eq. (9), the result is ( r + cos λ)cosφ+ sinαsinλsinφ X Y = ( r+ cos λ)sinφ sinαsinλcosφ Z cosαsinλ φ + π (9) (1) This is the circle equation (see Eq. (9)) expresse for the coorinate system σ. Then, for the final cross-section equation of a wire in the coorinate system σ, where Z is zero, Eq. (1) becomes ( r cos λ)cosφ sinαsinλsinφ X + + Y = ( r + cos λ)sinφ sinαsinλcosφ where π φ= cosαsinλ πr cosα= sin β = ( πr) + sinα= cosβ = ( πr) + (11) cosφ sinφ A' = sin αsinφ sinαcosφ cosα cosαsinφ cosαcosφ sinα (7) The phase angle of the cross-section Eq. (11) is zero. The ifference in the phase angles of ajacent wires is π /N. Therefore, by means simply of changing the phase angle, it is

4 16. Wang et al. / Journal of Mechanical cience an Technology 4 (6) (1) 13~11 transcenental. Therefore, in orer to fin the final solution, the numerical metho has to be use. Take a wire for which the phase angle is π /3; the crosssection equation therefore is π π ( r cos λ)cos( φ ) sinαsinλsin( φ ) X Y = π π ( r + cos λ)sin( φ ) sinαsinλcos( φ ) 3 3 (1) Here, the bisection metho is use to get the numerical solution of r. By observation, the maximum value of Eq. (1) is zero when the cross-sections are tangential each other (see Fig. 5(c)). In the case of intersection (as confirme by a center istribution raius r a ) an separation ( a center istribution raius r b ), the maximum values are respectively larger (see Fig. 5) an smaller (see Fig. 5) than zero. The maximum values are inicate by MAX ( r a ) an MAX ( r b ), respectively. Given an absolute error, the bisection metho converges to a root of the maximum value very close to zero. An the value of the root is the numerical solution of r. Given = mm, =1 mm, ra = /1 =. mm an rb = 1 = mm, the center istribution raius of the stran r = mm, the iameter of the stran c = r+ = = mm, an the twist angle of the stran β = arctan( πr/ ) = arctan( /1) = Approximate mathematical moel of cross-section of strans (c) Fig. 5. Cross-sections of three wires: intersection of cross-sections; separation of cross-sections; (c) tangency of cross-sections. very convenient to obtain the equations for other crosssections of strans. As shown in Fig. 5, the stran is forme of three twiste wires. The positions of the cross-sections represent the fact that the phase angles are separate: π/3, π, π/3. With the values of the parameters, r,,, the cross-sections can be obtaine. Fig. 5 illustrates the three cases of intersection, separation an tangency. Usually, the values of an are alreay known. The center istribution raius r is etermine so that the crosssections of the stran are tangential to each other (see Fig. 5(c)). However, it is impossible to obtain the analytical solution of r from the known an, because Eq. (11) is In the preceing section, a precise mathematical moel of the cross-section of a stran was propose. However, this moel is very complex. In orer to obtain a precise center istribution raius solution, a program coe must be written. In this section, for the purposes of simplifying the calculation of the center istribution raius, an approximate mathematical moel is presente. Assume that the cross-section of a stran is compose of ellipses in contact with each other. The tangent lines at the contact points of the wires pass through the center of the stran. The center istribution circle passes through every tangent point (Actually, the center istribution circle in the precise mathematical moel oes not pass through every tangent point, as shown in Fig. 5(c). Here, an assumption is mae in orer to simplify the calculation.). Establish a plane coorinate system of which the original point is the center of a selecte wire, as shown in Fig. 6. No matter how many wires are in the same layer, the slope coefficient of the tangent line O M can be expresse with the equation K = tan(9 18 / N) (13) pecifically, for a stran of three an four wires, K is

5 . Wang et al. / Journal of Mechanical cience an Technology 4 (6) (1) 13~11 17 Y O a b Combining Eq. (16) an Eq. (17) yiels the expression β + + = (18) 4 (cos K ) x Krx ( r ) O' M X On account of the tangency between O M an the ellipse, there is a ouble root of Eq. (18). As such, it is easy to obtain the expression of the center istribution raius by K r = 1+ (19) cos β = πr +, the final ex- In accorance with pression of r is cos β / ( ) Y a r = K + 1 π K O b M X where O' 18 K = tan(9 ) () N Take the cross-section of a stran forme of three twiste wires; for the same values = mm, = 1 mm, N = 3, the results are K =.5774, r = mm, c = r+ = = mm, β = arctan( πr/ ) = arctan ( /1) = Fig. 6. Approximate cross-section of stran: forme of three wires; forme of four wires. equal to 3/3 an 1, respectively. The major raius of the ellipse, of which the center is the original point of the plane coorinate system, is expresse by a = (14) cosβ The minor raius is expresse by b= / (15) Combining Eq. (14) an Eq. (15), the cross-section equation of the ellipse is (cos β ) x + y = ( ) (16) The tangent line O M passes through the center of the center istribution circle, the equation of the tangent line O M can be expresse by y= Kx r (17) 3. Comparison an analysis The results for the approximate moel are smaller than those of the precise moel. In orer to verify the accuracy of the approximate mathematical moel, the twist angles an iameters of some strans to be use for strane-wire helical springs in automatic weapons were calculate using the two methos. Aitionally, the relative errors were obtaine by introucing the / ratio as an important parameter (see Table 1). Regarless of the mathematical moel use, precise or approximate, or the number of stran wires, the twist angle an stran iameter coul be obtaine irectly from the stran screw pitch, the wire iameter an the wire number. An analysis of the ata showe that the larger the / ratio is, the smaller the relative errors are, for the same number of wires in the stran. With an equal / ratio, a greater number of wires results in larger relative errors. There is no change in the relative errors an twist angles with the same ratio / an number of wires in the stran, regarless of the specific values of an. Fig. 7 illustrates the relationships between the twist angle β an the / ratio, an between the relative error of the two moels an the / N ratio, for a stran forme of three wires.

6 18. Wang et al. / Journal of Mechanical cience an Technology 4 (6) (1) 13~11 Table 1. Comparison of values for two moels. Weapon Names N /mm /mm M6 7.6 mm, Machine Gun, America 14.5 mm, Antiaircraft Machine Gun, mm, Aircraft Machine Gun, mm, Antiaircraft Machine Gun, 1977 MG4 7.9 mm, Machine Gun, Germany 1.7 mm, Antiaircraft Machine Gun, 1954 H-37, Aircraft Automatic Gun mm, Antiaircraft Automatic Gun AM-3, Aircraft Automatic Gun r/mm c /mm β / Ⅰ Ⅱ ε (%) Ⅰ Ⅱ ε (%) Ⅰ Ⅱ ε (%) / Fig. 7. Relationships between β an / / N an ε (%) (The relative error in Fig. 7 is that of the β of the two moels. The other two relative errors are consistent with it.). The relationship between β an / is almost inverse. This is consistent with the previous escription. When the / N range is in (1, ), the relative error increases very quickly with the ecrease of / N. In orer to obtain an acceptable relative error, the / ratio must be two times larger than the number of wires in the stran, that is, / > N. In such cases, the approximate mathematical moel can be use for convenience. Otherwise (if the / ratio is not two times larger than the number of wires in the stran), the approximate mathematical moel cannot be applie, owing to the high relative errors. In orer to verify our analysis, the values of some other strans were calculate (see Table ). Table confirms our analytical assumptions. pecifically, the twist angle β is close to 3 egrees when the / ratio is more or less equal to N. An when the / ratio is very close to N, the relative errors are extraorinarily high. Usually, the twist angle is about 5~3 egrees in orer to avoi higher stresses in the inner sies of the wires. Uner these circumstances, the approximate mathematical moel can be use, because the relative errors are acceptable. However, the application range of strane-wire helical springs is very wie. In some cases for example, when springs must have higher stiffness an absorb higher vibrations, the twist angles are require to be aroun 4 egrees or even to excee 4 egrees. In these situations, it is not appropriate to calculate the twist angle an iameter of strans with the approximate mathematical moel. 4. Applications For convenient esign of strane-wire helical springs, application software was evelope for irect calculation of the twist angle an iameter of strans forme of arbitrary wires, accoring to the precise mathematical moel. The software was esigne in orer to solve two problems: the ineffectiveness of the approximate mathematical moel in some cases; the inapplicableness of the existing spring hanbooks when

7 19. Wang et al. / Journal of Mechanical cience an Technology 4 (6) (1) 13~11 Table. ome other strans for strane-wire helical springs. r/mm c /mm N /mm /mm Ⅰ Ⅱ ε (%) β / Ⅱ ε (%) Ⅰ Ⅱ ε (%) Ⅰ Fig. 8. Program flowchart of software. the number of wires excees four. A program flowchart of the software is provie in Fig. 8. With the software, the precise twist angle an iameter of strans can be calculate irectly, simply by inputting the number of wires, the iameter of wires an the screw pitch of strans (see Fig. 9). Furthermore, the software inclues a numerical reference stanar for the manufacture of strane-wire helical springs. 5. Conclusions This paper proposes two moels for calculation of the twist angle an iameter of strans forme of arbitrary wires. For strans with the same number of wires, the larger the / ratio is, the smaller are the relative errors of the two moels. Only when the / ratio is two times larger than N, the results obtaine with the approximate mathematical moel are acceptable. In other cases, this moel cannot be employe. Aitionally, application software was evelope for precise irect calculation of the twist angle an stran iameter. ignificantly, with the software, there is no nee for researchers to look up ata in hanbooks. More specifically, the software overcomes the problems encountere with some applications of the approximate mathematical moel an reners irrelevant the limitation that existing spring hanbooks cannot be reference when the number of wires excees four. The next research step will be to consier the internal rela- Fig. 9. Calculation of twist angle an iameter of strans. tions between springback an springs an to optimize the twist angle an stran iameter values. Acknowlegement This research was supporte by the National Natural cience Funs for Distinguishe Young cholars (No ), the National cience Founation of China (No ), the National Key Project for High-en CNC Machine Tools an Basic Manufacturing Equipment (No. 9ZX41-81 an No. 9ZX411-41), the Key Project of the Chinese Ministry of Eucation (No. 1919), the Chongqing Key cientific an Technological Project (No. CTC9AC349), the Chongqing University Postgrauates cience an Innovation Fun (No. 911A1A318) an the Innovative Talent Training Project, the Thir tage of 11 Project, Chongqing University (No. -916). Nomenclature β c N α : Twist angle of stran : Diameter of stran : Diameter of wire : crew pitch of stran : Total number of wires in stran : Helix angle of stran

8 11. Wang et al. / Journal of Mechanical cience an Technology 4 (6) (1) 13~11 : Center istribution raius (i.e. istance between center point of wire an center point of stran) φ, θ, ϕ : Euler angles r References [1] G. A. Costello an J. W. Phillips, tatic Response of trane-wire Helical prings, International Journal of Mechanical ciences, 1 (1979) [] H. H. Clark, trane-wire Helical prings, pring Design an Application, McGraw-Hill Book Company, New York, UA, (1961) [3] J. W. Phillips an G. A. Costello, General Axial Response of trane-wire Helical prings, International Journal of Non-Linear Mechanics, 14 (1979) [4] H. Carlson, pring Manufacturing Hanbook, Marcel Dekker, New York, UA, (198) 141. [5] J. J. Min an. L. Wang, Analysis on Dynamic Calculation of trane-wire Helical pring, Chinese Journal of Mechanical Engineering, 43 (3) (7) [6] H. Carlson, pring Designer s Hanbook, Marcel Dekker, New York, UA, (1978) 3. [7] D. W. Yu, The Dynamic tress of trane-wire Helical pring an Its Useful Life, Journal of Nanjing University of cience an Technology, 75 (3) (1994) 4-9. [8] The pring Research Association, prings Materials/Design/Manufacture, J. W. Northen, heffiel, UK, (1968). [9] Y. H. Zhang, H. H. Liu an D. C. Wang, pring Hanbook, Mechanical Inustry Press, Beijing, PRC, (1999) [1] M. L. Wang, Analytic Calculation of Multi-wire Helical Compression prings With Multi-center-wires, Acta Armamentar Ⅱ, (1995) [11] D. W. Zhang, Hanbook of pring Design, New Technology of Manufacturing Process an Quality Control, Chinese cience an Culture Auio-vieo Publishing House, Beijing, PRC, (3) [1]. athikh, M. B. K. Moorthy an M. Krishnan, A ymmetric Linear Elastic Moel for Helical Wire trans uner Axisymmetric Loas, Journal of train Analysis, 31 (5) (1996) [13]. L. Wang, J. Zhou an L. Kang, Dynamic Tension of trane-wire Helical pring uring Reeling, Chinese Journal of Mechanical Engineering, 44 (6) (8) [14] D. R. Wu, An Introuction to pace Analytic Geometry, Higher Eucation Press, Beijing, PRC, (1989) hi-long Wang receive his B.., M.. an Ph.D. egrees in Mechanical Engineering from Chongqing University, China, in 1988, 1991 an 1995, respectively. Dr. Wang currently is a Professor at the chool of Mechanical Engineering, Chongqing University, Chongqing, China. He serves as a irector of the Chinese Journal of Mechanical Engineering. Dr. Wang s research interests inclue manufacturing automation, computer integrate manufacturing an enterprises informatization. ong Lei receive his B.. egree in Mechanical Engineering from Chongqing University, China, in 5. He receive his M.. egree in Robotics an Mechatronics from ant Anna University, Italy, in 7. Lei currently is a Ph.D. caniate at the chool of Mechanical Engineering, Chongqing University, Chongqing, China. Jie Zhou receive his B.. egree in Mechanical Engineering an his M.. egree in oftware Engineering from Chongqing University in 1988 an 7, respectively. Zhou currently is an Associate Professor at the chool of Mechanical Engineering, Chongqing University, Chongqing, China. Zhou s research interests inclue manufacturing automation an mechatronics. Hong Xiao receive her B.. egree in Mechanical Engineering from Yantai University, China, in 1. he receive her M.. egree in Mechanical Engineering from Chongqing University in 7. Xiao is currently a Ph.D. caniate at the chool of Mechanical Engineering, Chongqing University, Chongqing, China.