Analysis of prismatic springs of non-circular coil shape and non-prismatic springs of circular coils shape by analytical and finite element methods

Analysis of prismatic springs of non circular coil shape and non prismatic springs of circular coils shape by analytical and finite element methods Accepted Manuscript Analysis of prismatic springs of[.]

Accepted Manuscript Analysis of prismatic springs of non-circular coil shape and non-prismatic springs of circular coils shape by analytical and finite element methods Arkadeep Narayan Chaudhury, Debasis Datta PII: DOI: Reference: S2288-4300(16)30138-5 http://dx.doi.org/10.1016/j.jcde.2017.02.001 JCDE 81 To appear in: Journal of Computational Design and Engineering Please cite this article as: A.N Chaudhury, D Datta, Analysis of prismatic springs of non-circular coil shape and non-prismatic springs of circular coils shape by analytical and finite element methods, Journal of Computational Design and Engineering (2017), doi: http://dx.doi.org/10.1016/j.jcde.2017.02.001 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Analysis of prismatic springs of non-circular coil shape and non-prismatic springs of circular coils shape by analytical and finite element methods ∗ Arkadeep Narayan Chaudhury † and Debasis Datta ‡ January 2, 2017 Abstract This paper presents a methodology for designing prismatic springs of non-circular coil shape and non-prismatic springs of circular coil shape using analytical and numerical methods To start with, simple analytical formulations for obtaining the axial deformation of the springs under axial load have been demonstrated Next, the processes of obtaining CAD models of the springs and their subsequent finite element analysis (FEA) in commercial softwares have been outlined In the third part, the different springs have been compared with a common cylindrical spring and their merits compared to a common spring have been demonstrated Next, a fairly accurate analytical formulation (with maximum error of ∼ − 8%) for obtaining the value and location of maximum shear stress for all the springs has been demonstrated Next, two aspects of non-prismatic springs under dynamic loads, viz damping introduced in a vibrating system and contribution of the spring to the equivalent mass in an 1D vibrating spring mass system due to shape of the spring have been discussed The last part involves an analytical formulation for the linear elastic buckling of two springs with circular coil shapes For the majority of the work, emphasis has been on obtaining and using closed form analytical expressions for different quantities while numerical techniques such as FEA have been used for validation of the same Keywords: Prismatic springs of non-circular coil shape, Non-prismatic springs of circular coil shape, CAD modeling, FEA, design and selection of springs, equivalent damping, linear elastic buckling of springs Introduction The helical spring is one of the most fundamental flexible mechanical elements and mostly used in several industrial applications like balances, brakes, vehicles suspensions, engine valves etc to satisfy functions like applying forces, storing or absorbing energy, providing the mechanical system with the flexibility and maintaining a force or a pressure In addition, helical springs serve as the elastic member for most common types of vibration absorbers The most commonly known helical spring, used in these applications, is presented as a cylindrical three-dimensional curved beam, characterized by its spiral shape and its constant curvatures along the axis For these kinds of springs the demand of space in both lateral and vertical directions is undeniable But for some very specialized applications, where there are lateral and(or) vertical space constraints, common ∗ A part of the current work was published as [1] Undergraduate student and corresponding author, Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur, India Email: arkadeep@mecheng.iisc.ernet.in ‡ Professor, Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur, India Email: ddatta@mech.iiests.ac.in † springs may not be implemented with much success due to unwanted increase in stiffness mainly due to usage of multiple springs This can be avoided by the usage of two special kinds of springs, viz springs with non-circular shape to cater to restrictions in lateral space and springs of circular coil shape but non-prismatic profile to cater to restriction in vertical space Among the non-circular coil springs, the rectangular springs are used in light firearms Among the non-prismatic springs, conical springs are generally used in applications requiring low solid height and increased resistance to surging, like automotive engines, large stamping presses, lawn mowers, medical devices, cell phones, electronics and sensitive instrumentation devices and volute shaped springs offer more lateral stability and less tendency to buckle than regular compression springs Also, the possibility of resonance and excessive vibration (or surging) is reduced because volute springs have a uniform pitch, more damping due to coil structural (see section 6.1) and an increasing natural period of vibration (instead of a constant period of vibration as in a cylindrical spring) as each coil closes For design and selection of springs for practical purposes, the deflection of the spring under axial load and maximum stresses induced are two major factors Stress analysis is one of the main themes of research in helical springs Investigations in this area began with the pioneering works of Ancker and Goodier [2, 3], who used the boundary element method (not to be confused with the modern boundary element method) to apply theory of elasticity and to develop an approximate result to satisfy governing equations and boundary conditions along the surface of the coil For small deformations of the spring, Wahl [4] considered the wire of the spring as a round bar subjected to shear and torsion The coupling between axial and torsional deformations was neglected in Wahl’s approach and a correction factor was used to account for the curvature of the spring Nagaya [5] solved equations governing the distribution of stresses in the spring and developed an analytical approach but the aforesaid solution was applicable only for a few types of cross sections (circular, rectangular etc.) Kamiya and Kita [6] treated this problem also using boundary element method, and the analysis was limited to springs of small helix angle Also, Cook [7] analyzed the same type of springs by using finite element method and showed the limitation of the work associated with the methodology’s negligence to helix angle of the spring Haktanir [8] solved the same problem by an analytical method to determinate the static stresses in the spring Jiang and Henshall [9] developed an approach based on the finite element method to analyze the stresses in a circular cross section helical spring by developing accurate boundary conditions and using finite element analysis Fakhreddine et al [10] presented an efficient two-node finite element with six degrees of freedom per node, capable of modeling the total behavior of a helical spring In the approaches cited above, all the analyses were done considering only circular coil shaped prismatic springs of constant coil diameter And the analyses and methods cited, albeit accurate, may not be easily used in cases where the spring coil is non-circular or the coil dimensions vary axially But, as discussed before, springs of non-circular coil shape or non-prismatic springs find applications in practical cases when there is a limitation in space Therefore, in the current work, analytical methods of obtaining the stress and deflection characteristics, two main design checkpoints for springs, have been attempted and the results obtained through the methodologies so developed have been compared with an independent method, FEA, to validate them The organization of the current work is as follows Section gives the analytical formulation for the deflection of prismatic and non prismatic springs under axial loads and benchmarks them against FEA In section 3, a brief discussion is presented on CAD representation of the springs in commercial softwares and FE analysis of the same using commercial softwares In section 4, the various springs discussed in section have been compared with a common prismatic spring with circular coils with an aim to point out the merits of the different springs In section 5, analytical expressions for obtaining the maximum stresses in the different springs have been presented and compared with FE analyses done using commercial softwares The final section (section 6) deals with the properties of the non-prismatic springs under dynamic loads and comparison of linear elastic buckling strengths of conical and right cylindrical springs of equivalent mass Deflection analysis of springs In this section analytical methods for finding the deflections of different helical springs with constant pitch and wire diameter have been attempted The formulation involves the usage of basic equations of solid mechanics, equilibrium of forces, and basic geometrical relationships The results obtained from the formulations have been compared to those obtained from FEA of CAD models of the corresponding springs 2.1 Deflection Analysis of Prismatic Springs with Non-Circular Coil Shape In this section, the analytical formulation for two varieties of prismatic springs with non-circular coil shape have been attempted The prismatic springs have a uniform cross section through out length 2.1.1 Rectangular spring Mx = F r tan(θ) 2a N θ P Tx = F r r r Q N M C θ M r 2r Q J a (a) Representation of the spring coil shape x L F C (b) Explanation of the 4-fold symmetry of figure 1a Figure 1: Schematic representation of the rectangular spring In this section, a prismatic spring with a rectangular coil shape bounded by semicircles on the smaller sides (see figure 1a) has been attempted The spring, although having an uncommon shape finds application in various mechanical equipments like guns and rifles The basic dimensions of the profile of the spring is shown in figure The length is 2a and the center of the circular arcs on either sides are coincident with the midpoint of the corresponding sides The symbols as represented here will be followed throughout the section It is seen that the profile is symmetric about each of the quadrants of axes on the plane with the origin coinciding with the geometric center of the figure Advantage of this symmetry, shown in figure 1b, involving only the quarter of the coil shape is taken by deriving the relations for a quarter only and multiplying it by for each of the coils a The straight part of the spring, shown in figure 1b, subtends an angle φ = tan−1 at the center of r P N θ Q θ r p J M φ F a r C Figure 2: Analysis of the representative quarter of the spring under an axial force the coil The force F , acting vertically at the center, induces both bending and torsional moment on a section of the coil Expressions of moments in the circular and straight parts are different and are shown separately On a section of the spring at a distance x from the vertical center line (see figure 1b), the bending and torsional moments, Mx and Tx , induced by the force on the straight part are:  Mx = F r tan(θ) Tx = F r  (1) Also, from figure 2, p= p (a2 + r2 + 2ar sin(θ) r cos(θ) φ = sin−1 p (2) (3) Using above the values of bending and torsional moments, Mθ and Tθ , induced by the force on the curved part are:  Mθ = F p sin(θ + φ) Tθ = F p cos(θ + φ)  (4) The total strain energy of the section shown in figure is given by the sum of the strain energies due to the moments in the two separate sections M N and N Q Using equations (1) and (4) Usector = Za Mx2 dx 2EI + Za π Tx2 dx 2GJ + Z2 π Mθ2 dθ 2EI + Z2 Tθ2 dθ 2GJ (5) Where, I and J represent the bending and torsional moments of inertia of the section of the wire πd4 πd4 with diameter d I = , J = E and G represent the Young’s modulus and modulus 64 32 of rigidity of the spring wire material The total strain energy of the spring with Nr number of active coils1 may be given from equation (5) as UT otal = 4Nr Usector , and the axial deflection of ∂UT otal the spring due to the axial load F as shown in figure 2, may be given as δ = , following ∂F the well known Castigliano’s theorem A comparison of the above formulation and FEA of the same case is given below in table It has been assumed that E = 210 GPa for steel, the value of Poisson’s ratio has been taken as ν = 0.25 and wire diameter was taken as 3mm The spring under consideration has Nr = 7.5 for complete turns with ground ends, and is under 15N of axial load From table 1, it is seen that the analytical formulation for the deflection is in agreement # a in mm 13.26 14.30 15.53 17.02 18.66 r in mm 13.26 12.87 12.42 11.91 11.32 Analytically obtained deflection (mm) 10.47 10.75 11.17 11.56 12.23 Deflection from FEA (mm) 10.39 11.40 11.75 11.96 12.45 Table 1: Comparison of analytical formulation and FEA for the rectangular spring with the FEA Also, the closed form expression for the deflection can be attempted by symbolically differentiating equation (5) in this case However, this will not be possible for the next example 2.1.2 Triangular spring In this section an analysis regarding the deflection of a triangular profile with rounded edges under axial loading has been attempted Initially, a general geometric formulation of the spring profile has been done followed by FEA of the same Like the previous section the analytical and the numerical results have been collated The basic dimension of the spring profile is given in figure 3a In figure 3, a is the side of the main enveloping equilateral triangle, b is the length of the straight sides on the profile of the spring, C is the vertex of the triangle, G is the centroid of the triangle, r is the radius of the curved part of the triangle, O is the center of the curved part, b0 is the distance between G and O, the point O coincides with the vertices of the inner smaller triangle Similar to the previous section, a six fold symmetry is observed (see figure 3b) in the profile of the spring so analysis of one sixth of the coil given by the triangle 4CQG is considered The magnitude th and orientation of the local radius vector, a line connecting any point on the representative section and G, may be obtained from the following analysis: Referring to figure 3b, the following may be observed: The triangle is divided into identical parts 4CQG subtending an angle of 60o at the centroid G Only one of the representative parts have been analyzed The angles swept by the radius vector GP is denoted by θ0 about O and by θ about G ∠QGE = φ N N is the normal to the arc ~ and T T is the tangent UPE at the point P and is hence collinear with the radius vector OP _ _ to the circular arc UPE at the point P, ∠N P G = ξ Let OG be denoted by lc and OP by rc Therefore, from figure it is seen that the perpendicular distance between P and UG, P M is The number of active coils in a compression spring is generally less than the physical number of coils in the spring It depends on the end conditions of the spring and a few other factors For more details see the textbooks by Shigley [15] or Bhandari [18] C T’ ξ N 30o U P θ E rc O C T U N’ E r b lc Q θ b’ L F G 600 90o Q G a (a) Representation of the spring coil shape (b) Explanation of the 6-fold symmetry of figure 3a Figure 3: Schematic representation of the triangular spring rc sin(θ0 ) = (lc + rc cos(θ0 )) tan(θ) Therefore, θ = tan−1 rc sin(θ0 ) lc + rc cos(θ0 ) (6) b a a Also, EQ = , GQ = √ and CQ = 2 ∠OP G = 180o − [(180o − θ0 ) + θ] = θ0 − θ = ξ (7) ~ = l is given as And for the arc UPE, the radius vector GP ~ = GM ~ + M~P = lc cos(θ) + rc cos(ξ) GP (8) The moment on a point on the curved section U P E, due to the centrally applied moment is given as M = F l The bending and torsional moments on the point P as shown in figure is given as,   Mθ = F l sin(θ0 ) (9) Tθ = F l cos(θ0 ) However, the same for the straight part QE are easily calculated as   aF tan(θ)   √ Mx =     Tx = aF  √ (10) C T’ U M ξ N θ0 Mθ O P E rc Tθ N’ T lc θ Mx Tx F G Figure 4: Analysis of the representative a sixth of the spring under an axial force The total strain energy for the section shown in figure due to the axial force F may be given as, Usection = θ−30 Z o Mθ2 ldθ0 + 2EI θ−30 Z o Tθ2 ldθ0 + 2GJ Zξ Mx2 ldθ + 2EI Zξ Tx2 ldθ 2GJ (11) In equation (11), the first two terms denote the energy in the curved section and the last two terms denote the energy in the straight section It may be noted that the first two integrals are easily solved, symbolically, by changing the variable to θ0 from θ This is easily done by the following substitution: dθ = lc2 rc (rc + lc cos(θ)) dθ0 + rc2 + 2lc rc cos(θ) For the total spring with Nt effective number of coils, the strain energy is given as UT otal = ∂UT otal 6Nt Usection , and using Castigliano’s theorem, the deflection may be given as, δ = It ∂F may be noted that due to the complicated expressions of the forces and moments, obtaining the analytical expressions is very hard Therefore, the operations were performed numerically For comparison, the following spring dimensions have been used: a = 0.06m, b = 0.04m, Nt = 7.5, r = 0.00577m, lc = 0.2309m and the value of the parameter c (see section 4.1.2) was selected as 5.0 Table shows that the analytical formulation is in good agreement with the finite element analysis 2.2 Analysis of non-prismatic springs with circular coil shape The springs analyzed in this section consist of circular coil shape of varying coil diameter across the length of the spring The general analytical formulation for obtaining the deflection of springs with circular coil shape is given following Timoshenko [12] In figure 5, it may be observed that, # Axial force (N) 10 20 50 75 Deflection obtained analytically (mm) 9.2 18.2 46.0 69.0 Deflection obtained by FEA (mm) 8.9 17.8 44.5 66.24 Table 2: Comparison of analytical formulation and FEA for the triangular spring center of any coil dα m 2R (a) Elevation view of the spring coil shape m0 n0 n (b) Plan view of the spring coil shape Figure 5: Schematic representation of the non-prismatic spring the coil radius is R An arbitrary section of the wire is considered The wire diameter is d and the section of the spring being symmetric about the central axis on all planes, the axial compressive force doesn’t generate any bending moment on the section, only torsional moment is generated The angle of twist due to this moment is given by dφ = F R2 dα GJ (12) Where, in figure 5, dα is the angle subtended by coil element under consideration, at the coil center Again, from figure 5a, the center of the wire section O and the point of application of force P are joined by drawing a straight line at B and OB = a Due to the twist the lower portion of the section rotates with respect to The point O and the point of application of force F describes a _ small BB0 = adφ In the following expressions, δ is the deflection in the direction of the application of the force and αis the angle subtended by the differential element at the center of the coil The deflection of whole spring due to the differential element is given in equation (13) B 0~B 00 = ~ 0R BB F R3 dα = Rdφ = α GJ (13) and, the total deflection due to the total loaded spring is obtained by integrating the right-hand side of equation (13) The following observations may be made from the process outlined above, • Any assumption about the constancy of the coil radius along the spring has not been assumed.Therefore, the term R in equation (13) may be replaced by a function R(x) describing the radius of a coil at any axial distance from base or top, wherever the force is being applied • The section mn − m0 n0 in figure 5b is under torsional moment only, and as it is known that deflection of springs with non-circular shaped coils is influenced by both bending and torsional moments, therefore this method is not applicable to springs described in section 2.1 in its native form as in equation (13) 2.2.1 Non-prismatic spring with conical shape Y A R(x) = ax2 + bx + c B R C R r X O 2l (b) Schematic of volute spring {coils not shown} (a) Schematic of conical spring Figure 6: Schematic representation of conical and volute springs In this part, an analytical formulation for obtaining the deflection of non-prismatic springs wound around a conical profile has been attempted The dimensions of the spring profile and its representation is given in figure 6a R1 and R2 are the minimum and maximum radii of the frustum about which the conical profile is described The radius of a coil at any axial distance subtending an angle α at the center of the spring may be given by, R(α) = R1 − (R2 − R1 )α 2πNco (14) Where, Nco is the effective number of spring coils It is easily seen that this is an integrable function in α Using equation (14) in equation (13), the deflection of the spring is obtained as, δco = 16F Nco (R12 + R22 )(R1 + R2 ) Gd4 (15) Calculations were done for a conical spring with R1 = 8.65mm, R2 = 30mm, wire diameter d = 6mm, Uncompressed length 80mm, effective number of coils Nco = 5.34 made of steel with E = 210GPa, ν = 0.27, under a compressive load of F = 250N Results obtained through equation (15) and FEA were compared, as shown in table It is observed that the results obtained through FEA approaches the analytically obtained result with successive mesh refinement 15 Axial deflection in mm 14 13 12 11 10 0.4 0.6 0.8 1.2 k 1.4 1.6 1.8 (a) Variation in axial deflection of rectangular spring of various parameter k Maximum von-Mises stress (in MPa) 220 200 180 160 140 120 0.4 0.6 0.8 1.2 1.4 1.6 1.8 k (b) Variation in maximum von-Mises stress (from FEA) of rectangular spring of various parameter ’k ’ Figure 8: Variation in axial deflection and maximum von-Mises stress of rectangular spring of various parameter ’k ’ 4.2 Comparison between non-prismatic springs of circular coil shape and cylindrical springs The motivation behind the study of the non-prismatic springs with circular coil shape was to cater to design requirements where an elastic element is to be accommodated within a space having non-uniform space about the axial directions The comparison between these springs and a normal cylindrical spring having the same base area is not of much practical value and it is therefore prudent to compare them for same mass or same free height In this case, the springs are subjected to have same mass, same pitch and a relation between the ratio of number of coils of the springs under comparison and the major dimensions of the non-prismatic spring has been obtained 4.2.1 Study with conical springs The conventional notations from section 2.2.1 and figure 6a are being used R is the coil radius for the cylindrical spring Ncy is the number of coils of the cylindrical spring and Nco is the number 14 10.8 Axial deflection in mm 10.6 10.4 10.2 10 9.8 9.6 9.4 9.2 2.5 3.5 4.5 5.5 c (a) Variation in axial deflection of triangular spring of various parameter c Maximum von-Mises stress (in MPa) 200 190 180 170 160 150 2.5 3.5 4.5 5.5 c (b) Variation in maximum Von-Mises stress (from FEA) of triangular spring of various parameter ’c’ Figure 9: Variation in axial deflection and maximum Von-Mises stress of rectangular spring of various parameter ’c’ of coils of the conical spring And γ is the weight of the wire per unit length The mass of the cylindrical spring is Mcy = 2πRγNcy The mass of a non-prismatic spring with N coils may be obtained as, M= 2πN Z γR(θ)dθ (20) Using equation (14) into equation (20) and simplifying, the mass of the conical spring, Mco may be obtained as, Mco = 2πNco R1 + R2 γ (21) Therefore, for a conical and cylindrical spring with equal mass, the expression of Mco and equation (21) can be compared and simplified to obtain, R= Nco R1 + R2 Ncy 15 (22) A standard cylindrical spring is compared with a conical spring having a larger radius of 25mm and smaller radius of 15mm under 15N axial load The following results may be obtained Cylindrical spring Number of coils Deflection (mm) Max von-Mises stress (MPa) values 3.00 95.41 Conical spring Number of coils Deflection (mm) Max von-Mises stress (MPa) values 4.1 (analytical) 120 Table 6: Comparison between cylindrical and conical springs 4.2.2 Study with volute shaped springs For the volute spring, the mass of the spring may be calculated by using equation (17) in equation (20) Equating the masses for the volute and the cylindrical spring and simplifying the resulting expression is, R= Nv 11R + 8r Ncy 12 (23) The standard cylindrical spring was compared with a volute spring having R = 25mm and r = 15mm From tables and and preceding analysis the following may be concluded for non-prismatic Cylindrical spring Number of coils Deflection (mm) Max von-Mises stress (MPa) values 3.00 95.41 Volute spring Number of coils Deflection (mm) Max von-Mises stress (MPa) values 1.3 (analytical) 120 Table 7: Comparison between cylindrical and volute springs springs, • Maximum stress is found to be on the higher side for non-prismatic profiles with circular coil shape, independent of the profile (either of conical or volute) for the cases concerned • For the particular selection of radii for non-prismatic profiles, the equivalent springs have same number of turns, hence same length for the conical configuration, however the volute profile has much less no of turns, 5, thus the free length is reduced • Nothing conclusive can be said about the deflection but it is noticed that for the same weight, under the same force, the deflection of the conical spring was more than that of the volute spring by a factor of about 2.18 Which implies that for the same load and spring mass, to restrict deflection, volute spring is a better choice • With the availability of equations (22) and (23) a problem of optimizing the shape of the springs can be attempted by considering general deflection or maximum stress or weight as the objective function • For the suitable choices of smallest and largest coil radii the number of coils can be reduced/altered for the non-prismatic springs as compared to the cylindrical spring Therefore a possibility of changing the free length of the non-prismatic springs over cylindrical springs can be seen 16 In summary, from the results given in the sections 4.1 and 4.2, it is seen that mass equivalent non-circular springs had incurred more maximum von-Mises stress of the order of 195-200 MPa as compared to 95-120 MPa for the circular coil springs with the maximum von-Mises stress for the cylindrical spring being as low as 65.5 MPa It is also seen that the deflection of the mass equivalent springs are different with the non-circular springs (across varying coil circularities) deflect more on average while, their circular counterparts are more rigid, with the volute spring being the most rigid and conical spring being the least rigid Therefore, barring a few cases with special lateral space requirements, it is almost always reasonable to choose springs with circular coil shape for a given design scenario Analysis of stresses in springs So far stresses have been obtained by FE analysis of springs as and when required But for design and selection of springs, CAD modeling and subsequent FEA of springs to find out deflection and stresses for every spring under consideration is an elaborate and cumbersome approach so, an expression for finding those might come in handy during design and selection Analytical formulations for obtaining deflections has already been obtained in section 2, while an analytical formulation remains to be obtained for stresses It may be observed that general design and selection procedures for springs require only the maximum values of stresses, most importantly the maximum shear and von-Mises stresses, which are the two decisive criteria for failure of ductile materials [15] Also, the point of their occurrence is important too In the following section an analytical formulation for obtaining the maximum shear stress of the spring and its location on the spring has been attempted Exact formulation for obtaining stresses in springs is a problem best solved by theory of elasticity The problem was taken up by Timoshenko and Goodier [16] and involves complicated expressions of stresses using potential functions, which yield simple closed form expressions for a few special cases, however, another method was proposed by Wahl [4] which is much simpler and has been shown to have an error of − 3% for practical springs with spring indices2 C ≥ and small helix angle The method may be summarized as, • The formulation considers that the section of the spring is loaded under torsional and direct shear stresses only 16M 4C − , πd3 4C − where C is the spring index of the spring and M is the torsional moment on the wire section s • Considering only torsional effect, the maximum shear stress is obtained as τmax = • The effect of direct shear can be obtained from theory of elasticity Following [16], the value + 2ν P d2 Where, ν is the Poisson’s of stress due to direct shear was obtained as, τd = 4(1 + ν) 4I ratio, P is the direct shear force and I is the section modulus of the wire 2R(α) spring coil diameter C can be defined locally as C(α) = , where R(α) may be taken spring wire diameter d from equation (14) or equation (17) Spring index(C)= 17 s • Adding τmax and τd and putting ν = 0.3, a commonly accepted value for spring steel, the well known equation from Wahl [4] is obtained as   8P 4C(α) − 0.615 τmax (α) = + πd2 4C(α) − C(α) (24) It is easily seen that the value and location of the maximum stress in a spring with circular coil shape can be analytically obtained using equation (24) 5.1 Analysis of stresses in non-prismatic springs with circular coil shape With the expressions for R(α) at hand from equation (14) and equation (17), the location and the magnitude of the maximum coil stresses may be obtained for the non-prismatic springs It may also be noted that for these springs, the assumptions in deriving equation (24) are not violated when the coils are open3 Therefore, equation (24) and the conclusions obtained therefore are applicable for conical and volute springs For the conical spring with largest coil radius R = 25mm, smallest coil radius r = 15mm, free length l = 100mm and wire diameter of 5mm under an axial load of 15N the following results (see table 8) were obtained The location of the maximum stress was correctly predicted to be at the edge of the largest coil by equation (24) and validated by FEA Stress obtained from equation (24) 17.49 MPa Stress obtained by FEA ∼ 18MPa % error over FEA 2.8 % less Table 8: Comparison of equation (24) and FEA for the conical spring For a volute shaped spring with largest coil radius R = 25mm, smallest coil radius r = 15mm, free length l = 80mm and wire diameter of 5mm under an axial load of 50N the following result (see table 9) were obtained The location of the maximum stress was found to be at 80mm from base i.e the edge of the first (largest) coil This was also observed from FEA It is also observed from Stress obtained from equation (24) 88.69 MPa Stress obtained by FEA ∼ 89MPa % error over FEA 0.384 % less Table 9: Comparison of equation (24) and FEA for the volute spring tables and that equation (24) yields a constant error of less than 3% which is in compliance with the claim made by Wahl[4] 5.2 Analysis of stresses in prismatic springs with non-circular coil shape It is seen from figures 1a and 3a that the sections at Q and U respectively are free form bending moment due to the axially impressed force F This follows from the fact that for the said sections, the local radius vector and the line joining the section to center of the geometry are coincident This also means that the direction of the resultant moment impressed due to the axial force F The coils with larger diameters deflect more, therefore with excessive axial force, they can close upon their neighboring coils and decrease the effective number of coils, thus introducing non-linear force deflection behavior and impressing various other forces upon the neighboring coils 18 has no component along the direction of the radius vector It is acting totally along the direction of the local tangent thus impressing a torsional moment only Apparently, it may seem useless to concentrate on the stress analysis of these sections but FE analyses reveal that the maximum shear stress also occurs at these points, which is also understood from common intuition However, unlike Wahl’s formulation considering a circular coil for which equation (24) is valid throughout the coil, for the current case equation (24) is valid only locally at a point and therefore some stress raising effects of the surrounding sections loaded with both torsion and bending moments will be seen for the section under consideration Also, this might be reckoned from the fact that the state of stress in the material is described as a continuum and therefore the stress at a section adjacent to the concerned sections (points Q and U in figures 1a and 3a respectively) will contribute to the stress at the concerned section as well The error obtained in using equation (24) over actual or FEA in this case will depend on the curvature of the section under study It is clearly Maximum shear stress in MPa 120 Analytical FEA 100 80 60 0.4 0.6 0.8 1.2 1.4 1.6 1.8 k (a) For rectangular spring Maximum shear stress in MPa 90 80 70 60 Analytical FEA 50 2.5 3.5 4.5 5.5 c (b) For triangular spring Figure 10: Comparison of analytical formulation and FEA for maximum shear stress in rectangular and triangular springs spring seen from figure 10 that with decrease in circularity of the spring coil shape, the error in prediction of maximum stress using equation (24) increases, while it is fairly accurate when the coil shape is 19 .. .Analysis of prismatic springs of non -circular coil shape and non -prismatic springs of circular coils shape by analytical and finite element methods ∗ Arkadeep Narayan Chaudhury † and Debasis... methodology for designing prismatic springs of non -circular coil shape and non -prismatic springs of circular coil shape using analytical and numerical methods To start with, simple analytical formulations... used for validation of the same Keywords: Prismatic springs of non -circular coil shape, Non -prismatic springs of circular coil shape, CAD modeling, FEA, design and selection of springs, equivalent