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

DOI: http://dx.doi.org/10.1016/j.jcde.2017.02.001

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 andnon-prismatic springs of circular coils shape by analytical and finite element methods, Journal of ComputationalDesign and Engineering (2017), doi: http://dx.doi.org/10.1016/j.jcde.2017.02.001

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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.

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 tosatisfy functions like applying forces, storing or absorbing energy, providing the mechanical systemwith the flexibility and maintaining a force or a pressure In addition, helical springs serve as theelastic member for most common types of vibration absorbers The most commonly known helicalspring, 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 ofsprings the demand of space in both lateral and vertical directions is undeniable But for somevery specialized applications, where there are lateral and(or) vertical space constraints, common

springs may not be implemented with much success due to unwanted increase in stiffness mainlydue 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 circularcoil shape but non-prismatic profile to cater to restriction in vertical space Among the non-circularcoil 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 resis-tance to surging, like automotive engines, large stamping presses, lawn mowers, medical devices,cell phones, electronics and sensitive instrumentation devices and volute shaped springs offer morelateral 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 uniformpitch, more damping due to coil structural (see section 6.1) and an increasing natural period ofvibration (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 underaxial load and maximum stresses induced are two major factors Stress analysis is one of the mainthemes 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 themodern boundary element method) to apply theory of elasticity and to develop an approximate re-sult to satisfy governing equations and boundary conditions along the surface of the coil For smalldeformations of the spring, Wahl [4] considered the wire of the spring as a round bar subjected toshear and torsion The coupling between axial and torsional deformations was neglected in Wahl’sapproach 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 analyticalapproach 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 withthe 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 circularcross section helical spring by developing accurate boundary conditions and using finite elementanalysis Fakhreddine et al [10] presented an efficient two-node finite element with six degrees offreedom 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 shapedprismatic 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 varyaxially But, as discussed before, springs of non-circular coil shape or non-prismatic springs findapplications 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 check-points for springs, have been attempted and the results obtained through the methodologies sodeveloped have been compared with an independent method, FEA, to validate them

The organization of the current work is as follows Section 2 gives the analytical formulationfor the deflection of prismatic and non prismatic springs under axial loads and benchmarks themagainst FEA In section 3, a brief discussion is presented on CAD representation of the springs incommercial softwares and FE analysis of the same using commercial softwares In section 4, thevarious springs discussed in section 2 have been compared with a common prismatic spring withcircular coils with an aim to point out the merits of the different springs In section 5, analyticalexpressions for obtaining the maximum stresses in the different springs have been presented andcompared with FE analyses done using commercial softwares The final section (section 6) dealswith the properties of the non-prismatic springs under dynamic loads and comparison of linearelastic buckling strengths of conical and right cylindrical springs of equivalent mass

In this section analytical methods for finding the deflections of different helical springs with constantpitch 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 obtainedfrom the formulations have been compared to those obtained from FEA of CAD models of thecorresponding 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-circularcoil shape have been attempted The prismatic springs have a uniform cross section through outlength

2.1.1 Rectangular spring

Cr

r

P

θ

M N

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 thefigure 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 4 for each of the coils.The straight part of the spring, shown in figure 1b, subtends an angleφ = tan−1a

φ J

P

p

θ θ

r

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 andare shown separately On a section of the spring at a distance x from the vertical center line (seefigure 1b), the bending and torsional moments, Mx and Tx, induced by the force on the straight

Using above the values of bending and torsional moments,Mθ and Tθ, induced by the force on the

π 2

32 . E and G represent the Young’s modulus and modulus

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 = 4NrUsector, and the axial deflection ofthe spring due to the axial load F as shown in figure 2, may be given as δ = ∂UT otal

∂F , followingthe well known Castigliano’s theorem A comparison of the above formulation and FEA of thesame case is given below in table 1 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 springunder consideration has Nr= 7.5 for 8 complete turns with ground ends, and is under 15N of axialload From table 1, it is seen that the analytical formulation for the deflection is in agreement

# a in mm r in mm Analytically obtained deflection (mm) Deflection from FEA (mm)

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 symbolicallydifferentiating 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 underaxial loading has been attempted Initially, a general geometric formulation of the spring profile hasbeen done followed by FEA of the same Like the previous section the analytical and the numericalresults 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 straightsides 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 distancebetween 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 thespring so analysis of one sixth of the coil given by the triangle4CQG is considered The magnitudeand orientation of the local radius vector, a line connecting any point on the representative 1

6

th

section and G, may be obtained from the following analysis: Referring to figure 3b, the followingmay be observed:

The triangle is divided into 6 identical parts4CQG 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 NN0 is the normal to the arcUPE_ at the point P and is hence collinear with the radius vector OP and T T~ 0 is the tangent

to the circular arc UPE_ at the point P, ∠N P G = ξ Let OG be denoted by l

c and OP by rc.Therefore, from figure 4 it is seen that the perpendicular distance between P and UG, P M is

1 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]

P

N’

N T’

~

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 4 is given as,

Tx= aF

2√3





PE

UC

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 4 due to the axial force F may be givenas,

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) wasselected as 5.0 Table 2 shows that the analytical formulation is in good agreement with the finiteelement 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 acrossthe length of the spring The general analytical formulation for obtaining the deflection of springswith circular coil shape is given following Timoshenko [12] In figure 5, it may be observed that,

Table 2: Comparison of analytical formulation and FEA for the triangular spring

(a) Elevation view of the spring coil shape

of the force and αis the angle subtended by the differential element at the center of the coil Thedeflection of whole spring due to the differential element is given in equation (13)

• Any assumption about the constancy of the coil radius along the spring has not been

as-sumed.Therefore, the termR 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 − m0n0 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

(a) Schematic of conical spring

R(x) = ax2+ bx + c Y

(b) Schematic of volute spring {coils not shown}

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 andR2 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−(R22πN− R1)α

co

(14)

Where,Ncois 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(R2

1+R2

2)(R1+R2)

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 3 It is observed that the results obtained through

FEA approaches the analytically obtained result with successive mesh refinement

Table 3: Comparison of the performance of FEA and equation (15)

2.2.2 Non-prismatic spring with volute spring shape

In this section, an attempt has been made to obtain the deflection of a non-prismatic spring wound

on a volute shaped profile The profile of the spring (see figure 6b) is a closed parabolic curve formed

by revolving a parabola around a straight line to form a non-prismatic shape The dimensions ofthe spring profile and its representation is given in figure 6b R and r are the maximum andminimum distance of the describing parabola from the axis of revolution, andl is the free length ofthe spring The radius of a coil at any axial distance may be given byR(x) = ax2+bx + c, where,

a = R− 2r2l2 ; b = 4r− R

The total deflection of the spring may be obtained by substituting equation (17) in equation (13)

It may be noted that the expression of the integral obtained by integrating equation (17) is veryinvolving, hence it is easy to seek a numerical value of integrand instead of a closed form equation

as in equation (15)

A similar exercise of comparing analytical and FEA results for the deflection of the springunder axial load under F = 19.6N was undertaken with r = 40mm, R = 60mm, l = 40mm,

p = 26.667mm/coil, effective number of spring coils Nv = 6, wire diameter d = 6mm, along with

E = 200GPa and ν = 0.27 The following results in table 4 shows that with successive refinement

of the mesh, the FEA results converge to the analytically obtained result

Mesh size Deflection Nature of deflection

Table 4: Comparison of the performances of FEA and analytical expression for deflection

A detailed account on modeling the springs and their FE analysis using commercial softwares

therefore, a few key points regarding the same are presented in this section The CAD models ofsprings discussed so far (in sections 2.1.1, 2.1.2, 2.2.1 and 2.2.2) are shown in figures 7a to 7d Themodels were produced by using a commercial CAD environment (SolidWorks [13]) Subsequently,the FEA of the springs under axial load was performed by importing the models into ANSYS [14]Workbench environment

(a) CAD model of rectangular

spring

(b) CAD model of triangular spring

(c) CAD model of conical spring (d) CAD model of volute spring

Figure 7: CAD models of the springs studied so far

3.1 FEA of spring-plate assembly

All the springs discussed in section 2 were subjected to FEA to obtain the deflection of the springsunder axial load, maximum shear and maximum von-Mises stresses The springs were assembledwith two base plates on two ends to transmit forces The following points are of paramount im-portance regarding the assembly and subsequent FEA of the springs FEA of the springs werecarried out by importing the assembled CAD part from SolidWorks environment to Ansys Work-bench Static Structural module [14] The choice of the module was justified because all the stressesencountered were below the elastic limit, causing only elastic deformations of the spring The mesh-

ing was done by selecting 8 node brick element (classified as SOLID187 in [14]) and tetrahedralelements with straight edges (classified as SOLID285 in [14]) The elements used were inferior tothose developed by Fakhreddine et al [10], the GCSCC and GLSCC elements by Choi et al [11]

or the elements developed by Jiang et al [9], but their simplicity, commercial availability and sibility of very high h-refinement without much addition computational load (simulations with up

pos-to 200,000 elements have been carried out) justified their usage For a typical case of a rectangularspring with r = 9.78mm and a = 24.45mm, (see section 2.1.1), numbers of elements and nodeswere 50515 and 191320 respectively All the forces considered in the analysis were axial, thereforeuniformly distributed pressures were applied on the backing plates attached to the ground ends

of the springs, instead of point loads for better representation of the loading During the FEAanalysis, thick plates were used to back the springs The thick plates were used to minimize theeffect of denting of the plates due to the mode of loading, to be added to the observed deflection ofthe spring However, the weight of the plate does not come into play as the gravity loading optionwas not activated

The most common spring used in mechanical components is the constant pitch cylindrical springbut so far the discussion has included two springs of non-circular coil shape and two springs ofcircular coil shape but non-prismatic profile In the following section a comparison is done betweensprings discussed so far and a cylindrical compression spring of constant pitch The cylindricalspring used for comparison is a standard spring with coil diameter of 40mm, wire diameter of 3mmand having 8 coils and 80mm free length The material of the spring was considered to be steelwithE = 200GPa and ν = 0.25, and it has ground ends The coil area is 1256.64mm2 This springwas subjected to an axial load of 15N and the following data relating to the maximum shear stressgenerated on the spring and it’s axial deflection was obtained

Axial Load (N) Deflection (mm) Maximum Stress (von-Mises) in MPa

Table 5: Stress-deflection characteristics of a standard cylindrical spring

4.1 Comparison between prismatic springs of non-circular coil shape and drical springs

cylin-The motivation behind the study of the prismatic springs with non-circular coil shape was tocater to design requirements where an elastic element is to be accommodated within a non-circularspace and where using more than one cylindrical spring would increase the stiffness of the system.Therefore the springs are subjected to have same base area but different coil shapes in terms ofvarying circularity of the coil, same pitch and same free length, which essentially means that theyhave same effective number of turns

4.1.1 Study with rectangular springs

The area enclosed by the spring coil in figure 1a may be given as A = πr2 + 4ar A furthersimplification on the expression of r can be obtained by substituting a = r

k,k6= 0

r =

sA

In equation (18) the base area A is expressed in terms of only one variable r for a particularvalue of the newly introduced parameter k Using equation (18), many springs having the samebase area A but of different coil shape, in terms of circularity, may be obtained by varying thevalue of k and subsequently obtaining different values of r and a This concept was used to make

a SolidWorks design table feature for obtaining the various springs The results obtained fromsubjecting rectangular springs of varying parameter value k The spring parameters are same as

in section 2.1.1 Figures 8a and 8b show the variation in axial deflection and maximum von-Misesstress of rectangular spring of various values of the parameter k The following were observed fromfigure 8

• Deflections for the springs were almost same with deflection slightly increasing for lower values

of k

• Stress values of the spring are considerably higher than that of the cylindrical spring and thevalue of maximum stress increases for lower values of k This is attributed to the curvatureeffect of the spring The higher value of k signifies approach towards a circular profile andthus the stress increasing effect of the curvature effect is less pronounced

4.1.2 Study with triangular springs

In this section, a case study is attempted to compare the axial compression of triangular springswith a circular coil cylindrical spring having the same coil area A parameter c (previouslyreferred to in the last part of section 2.1.2) is defined as c = b

0+rc

rc It may be observed that c

is a measure of circularity of the profile of the spring The higher the value of c the more is thedeparture of the profile of the spring from a circular shape and with the decrease in the value of

c the coil shape approaches a circle A perfect circle is obtained at c = 1 and a perfect triangle

at c =∞ From figure 3b it is clear that b0 = b

The following observations may be made from figure 9:

• With the departure of the spring profile from a circular one, i.e with increasing value of cthe maximum stress induced in the coil increases

• The spring also loses rigidity as there is departure from a circular coil shape