International Journal of Engineering & Technology, 3 (2) (2014) 268-278

©Science Publishing Corporation

www.sciencepubco.com/index.php/IJET doi: 10.14419/ijet.v3i2.2492 Research Paper

Compression and impact characterization of helical

and slotted cylinder springs

Ahmed Ibrahim Razooqi *, Hani Aziz Ameen, Kadhim Mijbel Mashloosh

Technical College- Baghdad- Dies and Tools Eng. Dept.

*Corresponding author E-mail: ahmed_air24@yahoo.com

Copyright © 2014 Ahmed Ibrahim Razooqi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Helical and slotted cylinder springs are indispensable elements in mechanical engineering. This paper investigates

helical and slotted cylinder springs subjected to axial loads under static and dynamic conditions. The objective is to

determine the stiffness of a circular cross-section helical coil compression spring and slotted cylinder springs with five

sizes and dynamic characteristics. A theoretical and finite element models are developed and presented in order to

describe the various steps undertaken to calculate the spring’s stiffnesses. Five cases of the spring’s geometric are

presented. A finite element model was generated using ANSYS software and the stiffness matrix evaluated by applying

a load along the spring’s axis, then calculating the corresponding changes in deformation. The stiffness is obtained by

solving the changes of load and deformation. The natural frequencies, mode shapes and transient response of springs are

also determined. Finally, a comparison of the stiffnesses are obtained using the theoretical methods and those obtained

from the finite element analysis were made and good agreement are evident and it can be found that the stiffness of

spring for the slotted cylinder spring is much larger than that for helical spring and the stiffness for slotted cylinder

spring increases with the number of slots per section. Natural frequencies, mode shape and transient response of helical

spring and slotted cylinder spring have been represented in ANSYS software and results have been compared and it

found that the natural frequency has also increased in the same proportion of stiffness because the natural frequency is

directly proportional to the stiffness for all the cases that have been studied.

Keywords: ANSYS, Finite Element Analysis, Helical Spring, Slotted Cylinder Spring, Stiffness.

1. Introduction

A spring is an elastic member used to connect two parts of machine to form a flexible joint which is used to produce

constant forces as in brake, to adjust machine members as in value, to accumulate elastic strain as in watch spring and

for vibration isolation, it can secure against propagation of vibrations between joined elements and used as shock-

absorbers. The analysis is extended 2 to the round wire cross-section spring and the slotted cylinder spring. Extensive

study has been done by Wahl [1] and others in the design of circular cross-section helical compression springs. In the

dynamic system, the system consists of helical coil compression spring or slotted cylinder spring that is fixed rigidly at

the base and allowed to oscillate about the spring’s three orthogonal axes x, y and z. This system can be considered a

multiple-degree-of-freedom system that allows six-degrees-of-freedom (three translations and three rotations) about the

spring’s x, y and z axes. The objective of this study is to determine the stiffness of the compression springs (helical and

slotted cylinder) that is loaded by axial forces. A model of the system was created using ANSYS software, and static

and dynamic analyses were performed. These analyses resulted in stiffness terms, natural frequencies, mode shapes and

transient response of the springs.

2. Literature review

A review of available literature was done in the area of helical compression spring design and slotted cylinder spring.

W.G. Jiang and J.L. Henshall, 2000 [2] developed a general and accurate finite element model for helical springs

International Journal of Engineering & Technology 269

subject to axial loads (extension or/and torsion). Due to the establishment of precise boundary conditions, only a slice of

the wire cross-section needs to be modelled; hence, more accurate results can be achieved. Merville K. Forrester, 2001,

[3] studied the three - dimensional stiffness matrix of a rectangular cross-section helical coil compression spring. The

stiffnesses of the spring are derived using strain energy methods and Castigliano’s second theorem. S. S. Gaikwad and P.

S. Kachare, 2013 [4] have attempted to analyze the safe load of the helical compression spring. A typical helical

compression spring configuration of two wheeler horn is chosen for study. This work describes static analysis of the

helical compression spring is performed using NASTRAN solver and compared with analytical results. The pre

processing of the spring model is done by using HYPERMESH software. Anis Hamza et al, 2013, [5] in this study, the

vibrations of a coil, excited axially, in helical compression springs such as tamping rammers are discussed. The

mathematical formulation is comprised of a system of four partial differential equations of first-order hyperbolic type,

as the unknown variables are angular and axial deformations and velocities. The numerical resolution is performed by

the conservative finite difference scheme of Lax–Wendroff. The impedance method is applied to calculate the

frequency spectrum. C. J. Yang et al, 2014, [6] this paper investigates helical springs subjected to axial loads under

different dynamic conditions. The mechanical system, composed of a helical spring and two blocks, is considered and

analyzed. Multibody system dynamics theory is applied to model the system, where the spring is modeled by Euler–

Bernoulli curved beam elements based on an absolute nodal coordinate formulation. Compared with previous studies,

contact between the coils of spring is considered here. A three-dimensional beam-to-beam contact model is presented to

describe the interaction between the spring coils.

The report of Wilhem A. Schneider, 1963 [7], discussed the characteristics of slotted cylinder spring with high load

capacity and low deflection in extremely small size, the experimental and theoretical investigation are reported.

Krzysztof Michalczyk, 2006, [8] studied the modern construction of slotted springs. It was proven that maximal stresses

in such springs under load have higher values than the stresses with previous method. Viatcheslav Gnateski, 2012, [9],

using the slotted cylinder spring as a vibration damping device adapted to receive an electronic component and reduce

vibration. The vibration damping device includes a spring body extending along an axis, one or more slots formed in the

spring body, and a shaft extending substantially coaxially within the spring body. Ahmed Ibrahim Razooqi et al, 2013,

[10] studied the characteristics of such a spring under both static and dynamic loading using a finite element method

with the aid of ANSYS11 program. Two cases were studied, one with three slots and the other with four slots. Five

modes shapes of each case were employed. The transient analysis due to impact and static loadings were presented. In

the present paper, an investigation theoretically and numerically via ANSYS software to predict the design

characterization of the helical and slotted cylinder springs with different sizes in which the stiffness of springs, natural

frequencies, mode shapes and transient responses are evident.

3. Theoretical analysis

The theoretical methods used were formulated in terms of scalar quantities in which the applied forces, displacements

and positions along the spring’s helix are defined. The spring is considered linearly elastic and undergoes small

deflections. This analysis stated that an element of an axially loaded helical spring of circular cross-section behaves

essentially as a straight bar in pure torsion. The deflection of the spring will be [11]:

(1)

Where:

P = load

D = mean coil diameter

n = Number of active coils

d = bar diameter

G = rigidity

The total deflection of a slotted cylinder spring is reported by Wilhem A. Schneider, 1963 [7] as follows:

(2)

Where

P = total compression load

E = Yong’s modulus of elasticity

Ls = length of slot

nss = number of slot section

ns = number of slots per section

b = wall thickness

h = height of horizontal path

Total deflection

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3. Finite element method

In this study, the finite element analysis (FEA) was performed using ANSYS11. Finite element analysis was applied to

determine the stiffness, natural frequencies, mode shapes and transient responses of the springs. This method is based

on the solution of differential equations with imposed boundary conditions. The system under investigation is an

assembly of nodes that serve to connect elements together. All elements used in ANSYS11 have two defined sets of

property tables; material and physical tables. The finite element method is an approximate technique used to obtain a

solution to a specific problem. The following procedure was used in obtaining the finite element solution:

a) Generate a solid model of the springs.

b) Create a grid of nodes connected by elements.

c) Apply boundary conditions.

d) Solving of static and dynamic models.

e) Model updating.

f) Display and interpreting of results.

The geometry for the helical coil compression spring was modeled in the AutoCAd14 software and export as a SAT file,

then imported the SAT file in ANSYS11 code. While the geometry of slotted cylinder spring plotted section by section

according to the each case’s size. Element selected for this analysis is SOLID45. SHELL element is used for meshing

the cross section of helical spring then dragged by solid45. SOLID92 is a pyramid element that increases time of

calculations and it has error in nonlinear complex models. Therefore, a cubic SOLID45 element has been used in the

stress analysis for both helical spring and slotted cylinder spring. This element is defined by eight nodes having three

degrees of freedom at each node: translations in nodal x, y and z directions. For the design of the slotted cylinder spring,

the design in Fig.(1-A) is fail under static and dynamic load, so the slots in the section is design as a verse versa, as

shown in Fig.(1-B) similar to Ref.[7]. The finite element method is an approximate numerical technique for solving

structural problems. It must also be remembered that inaccuracy may arise from the fact that the finite element model is

rarely an exact representation of the physical structure. The element mesh may not exactly fit the structure’s geometry.

In addition, the actual distribution of the load and possibly elastic properties may be approximated by simple

interpolation functions. Boundary conditions simulating the rigid base may also be approximated.

A B

Fig. 1: Slotted Cylinder Spring (A) Worst Design (B) Good Design

Case A Case B Case C Case D Case E

Helical Coil Compression Spring

Case A Case B Case C Case D Case E

Slotted Cylinder Spring With Three Slots Per Section

Case A

Case B

Case C

Case D

Case E

Fig. 2: Shows the Meshes of the Helical and Slotted Cylinder Spring with three Slots and Four Slots Per Section for the Cases Studied.

Slotted cylinder spring – with four slots per section

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4. Static analysis

In the static modeling of the system, three dimension solid elements were used to represent the helical coil compression

spring and slotted cylinder spring. These elements mathematically modeled the overall deflection of the spring. At the

base of the model, the nodes representing the inactive coil were completely restrained. This condition created the fixed

base associated with the real system. The applied load was considered to be concentrated at the centerline of the spring.

The elements of the stiffness matrix were determined based upon the linear load deformation relationship

P = [K] δ (3)

Where [K] is the stiffness matrix of the spring. As the spring is deformed, the spring exerts a force that is proportional

to the displacement, the resulting stiffness of spring was determined. Tables(1) to (5) indicated deflection of the helical

and slotted cylinder springs with 3-slots and 4-slots for cases A,B,C,D and E. Fig.(3) explained the ANSYS and

theoretical stiffness of springs (Helical and slotted cylinder spring with 3-slots and 4-slots).

Table 1: Deflection [M] of the Springs: Case-A-

Load

[N]

Helical Spring Slotted cylinder spring

ANSYS Theory

Eq.(1)

ANSYS

(3-slots)

ANSYS

(4-slots)

Theory

(3-slots)

Eq.(2)

Theory

(4-slots)

Eq.(2)

10 0.431e-3 9.9e-4 0.119e-6 0.572e-7 3.44472e-8 1.08993e-8

20 0.862e-3 1.98e-3 0.293e-6 0.114e-6 6.88944e-8 2.17986e-8

30 0.001336 2.97e-3 0.358e-6 0.171e-6 1.03342e-7 3.26979e-8

40 0.00188 3.96e-3 0.477e-6 0.229e-6 1.37789e-7 4.35972e-8

50 0.002425 4.95e-3 0.597e-6 0.286e-6 1.72236e-7 5.44965e-8

60 0.002975 5.94e-3 0.716e-6 0.343e-6 2.06683e-7 6.53958e-8

70 0.003525 6.93e-3 0.836e-6 0.4e-6 2.4113e-7 7.62951e-8

80 0.004076 7.92e-3 0.955e-6 0.457e-6 2.75577e-7 8.71944e-8

90 0.004627 8.91e-3 0.107e-5 0.514e-6 3.10025e-7 9.80937e-8

100 0.005178 9.9e-3 0.119e-5 0.572e-6 3.44472e-7 1.08993e-7

Table 2: Deflection [M] of the Springs: Case-B

Load

[N]

Helical Spring Slotted cylinder spring

ANSYS Theory

Eq.(1)

ANSYS

(3-slots)

ANSYS

(4-slots)

Theory

(3-slots)

Eq.(2)

Theory

(4-slots)

Eq.(2)

10 0.896e-3 1.17e-3 0.537e-6 0.205e-6 4.6368e-7 1.46711e-7

20 0.001793 2.34e-3 0.107e-5 0.409e-6 9.27361e-7 2.9342e-7

30 0.002689 3.51e-3 0.161e-5 0.614e-6 1.39104e-6 4.40134e-7

40 0.003586 4.68e-3 0.215e-5 0.818e-6 1.85472e-6 5.86846e-7

50 0.004638 5.85e-3 0.268e-5 0.102e-5 2.3184e-6 7.3355e-7

60 0.005773 7.02e-3 0.322e-5 0.123e-5 2.78208e-6 8.80268e-7

70 0.00691 8.19e-3 0.376e-5 0.143e-5 3.24576e-6 1.02698e-6

80 0.008051 9.36e-3 0.430e-5 0.164e-5 3.70944e-6 1.17369e-6

90 0.009196 1.05e-2 0.483e-5 0.184e-5 4.17312e-6 1.3204e-6

100 0.0010344 1.17e-2 0.537e-5 0.205e-5 4.6368e-6 1.46711e-6

Table 3: Deflection [M] of the Springs: Case-C

Load

[N]

Helical Spring Slotted cylinder spring

ANSYS Theory

Eq.(1)

ANSYS

(3-slots)

ANSYS

(4-slots)

Theory

(3-slots)

Eq.(2)

Theory

(4-slots)

Eq.(2)

10 0.001392 1.2e-3 0.543e-6 0.180e-6 6.94611e-7 2.19779e-7

20 0.002784 2.4e-3 0.109e-5 0.36e-6 1.38922e-6 4.39558e-7

30 0.004176 3.6e-3 0.163e-5 0.541e-6 2.08383e-6 6.59338e-7

40 0.005568 4.8e-3 0.217e-5 0.721e-6 2.77844e-6 8.79117e-7

50 0.007129 6e-3 0.272e-5 0.901e-6 3.47305e-6 1.0989e-6

60 0.008894 7.2e-3 0.326e-5 0.108e-5 4.16767e-6 1.31868e-6

70 0.010662 8.4e-3 0.380e-5 0.126e-5 4.86228e-6 1.53845e-6

80 0.102432 9.6e-3 0.435e-5 0.144e-5 5.55689e-6 1.75823e-6

90 0.014204 1.08e-2 0.489e-5 0.162e-5 6.2515e-6 1.97801e-6

100 0.015983 1.2e-2 0.543e-5 0.18e-5 6.94611e-6 2.19779e-6

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Table 4: Deflection [M] of the Springs: Case-D

Load

[N]

Helical Spring Slotted cylinder spring

ANSYS Theory

Eq.(1)

ANSYS

(3-slots)

ANSYS

(4-slots)

Theory

(3-slots)

Eq.(2)

Theory

(4-slots)

Eq.(2)

10 0.662e-3 8.66e-4 0.503e-6 0.164e-6 7.51568e-7 2.37801e-7

20 0.001324 1.73e-3 0.101e-5 0.328e-6 1.50314e-6 4.75602e-7

30 0.001986 2.6e-3 0.151e-5 0.492e-6 2.2547e-6 7.16402e-7

40 0.002648 3.46e-3 0.201e-5 0.656e-6 3.00627e-6 9.51203e-7

50 0.00331 4.33e-3 0.251e-5 0.82e-5 3.75784e-6 1.189e-6

60 0.003972 5.2e-3 0.302e-5 0.984e-5 4.50941e-6 1.4268e-6

70 0.004634 6.06e-3 0.352e-5 0.115e-5 5.26098e-6 1.66461e-6

80 0.005296 6.93e-3 0.402e-5 0.131e-5 6.01254e-6 1.90241e-6

90 0.005958 7.79e-3 0.452e-5 0.148e-5 6.76411e-6 2.14021e-6

100 0.00662 8.66e-4 0.503e-5 0.164e-5 7.51568e-6 2.37801e-6

Table 5: Deflection [M] of the Springs: Case-E

Load

[N]

Helical Spring Slotted cylinder spring

ANSYS Theory

Eq.(1)

ANSYS

(3-slots)

ANSYS

(4-slots)

Theory

(3-slots)

Eq.(2)

Theory

(4-slots)

Eq.(2)

10 0.003588 1.87e-2 0.493e-5 0.157e-5 1.39996e-5 4.42957e-6

20 0.007175 3.75e-2 0.987e-5 0.314e-5 2.79993e-5 8.85915e-6

30 0.010763 5.62e-2 0.148e-4 0.471e-5 4.19989e-5 1.32887e-5

40 0.015007 7.49e-2 0.197e-4 0.627e-5 5.59986e-5 1.77183e-5

50 0.019573 9.37e-2 0.247e-4 0.784e-5 6.99982e-5 2.21479e-5

60 0.024145 1.12e-1 0.296e-4 0.941e-5 8.39978e-5 2.65774e-5

70 0.028728 1.31e-1 0.345e-4 0.11e-4 9.79975e-5 3.1007e-5

80 0.033321 1.5e-1 0.395e-4 0.125e-4 0.000111997 3.54366e-5

90 0.037915 1.69e-1 0.444e-4 0.141e-4 0.000125997 3.98662e-5

100 0.042511 1.87e-1 0.493e-4 0.157e-4 0.000139996 4.42957e-5

International Journal of Engineering & Technology 273

Fig. 3: ANSYS and Theoretical Stiffness of Springs ( Helical and Slotted Cylinder Spring with 3-Slots and 4-Slots)

Case-A- Helical Case-A- 3slots Case-A-4slots

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Case-B- Helical Case-B- 3slots Case-B-4slots

Case-C- Helical Case-C- 3slots Case-C-4slots

Case-D- Helical Case-D- 3slots Case-D-4slots

Case-E- Helical Case-E- 3slots Case-E-4slots

Fig. 4: Equivalent Stresses for Helical and Slotted Cylinder Spring with 3-Slots and 4-Slots

Numerical analysis provides details such as spring stiffness, static stresses for helical spring and slotted cylinder spring

under compression. All these results are available in design of helical and slotted cylinder springs.

5. Dynamics analysis

In this system, energy is transformed from kinetic energy to potential energy and back again. This results in a vibrating

system. A vibrating system dissipates energy in the form of damping and the governing equation of motion representing

this system is written in matrix form as:

(4)

Where

F = a vector force on each DOF in the system

[M] = mass matrix

[K] = stiffness matrix

International Journal of Engineering & Technology 275

= displacement, velocity and acceleration of each DOF

Respectively

The ANSYS software solves for the modes of vibration (natural frequencies and modes shapes) and uses these to

calculate dynamic responses. Fig. (5) Illustrated the natural frequencies of springs (Helical and slotted cylinder spring

with 3-slots and 4-slots). Fig. (6) Illustrated the mode shape of springs (Helical and slotted cylinder spring with 3-slots

and 4-slots). Fig.(7) explained the transient response of springs (Helical and slotted cylinder spring with 3-slots and 4-

slots) in three position, bottom of spring , middle of spring and top of spring.

Fig. 5: Natural Frequencies of springs (Helical and Slotted Cylinder Spring With 3-Slots and 4-Slots)

276 International Journal of Engineering & Technology

Case-A- helical Case-A- 3slots Case-A-4slots

Case-B- helical Case-B- 3slots Case-B-4slots

Case-C- helical Case-C- 3slots Case-C-4slots

Case-D- helical Case-D- 3slots Case-D-4slots

Case-E- helical Case-E- 3slots Case-E-4slots

Fig. 6: Mode Shape (Number Five) of Springs ( Helical and Slotted Cylinder Spring with 3-Slots and 4-Slots)

International Journal of Engineering & Technology 277

Case-A- helical Case-A- 3slots Case-A-4slots

Case-B- helical Case-B- 3slots Case-B-4slots

Case-C- helical Case-C- 3slots Case-C-4slots

Case-D- helical Case-D- 3slots Case-D-4slots

Case-E- helical Case-E- 3slots Case-E-4slots

Fig. 7: Transient Response of Springs ( Helical and Slotted Cylinder Spring with 3-Slots and 4-Slots)

6. Discussions

The main objective of spring design is to obtain a spring which will be most economical for a given application and will

have satisfactory life in service. Equally important is the spring’s resistance to deformation under a given load. In the

preliminary investigation it was evident that extensive study has been done (Wahl [1]) in the design of circular cross-

278 International Journal of Engineering & Technology

section springs, specifically in determining the stiffness due to various loading conditions. This investigation focused on

determining the stiffness for a circular cross-section helical compression spring and slotted cylinder spring and utilized

theoretical methods and Finite element analysis. The finite element analysis using ANSYS which may be resulted in the

stiffness, natural frequencies, mode shapes and response of the modeled springs. The analysis is also based on the

linearity between load and deformation and the results of the analysis must be accepted subjected to the validity of this

assumption. The work done due to the deformation of coil of the spring is stored as strain energy. For a unit volume of

the linear elastic spring, it was determined by comparison of results that the strain energy in the finite element model

adequately represented the strain energy determined theoretically.

7. Conclusion

ANSYS results have been compared with theoretical results and found to be in good agreement. Compared the helical

and slotted cylinder springs, the slotted cylinder spring has been found to have lesser stress and has the most stiffness

value. It can be found that the stiffness of spring for the slotted cylinder spring is much larger than that for helical spring

and the stiffness for slotted cylinder spring increases with the number of slots per section. Also including the stiffness

note that it is increasing the natural frequency has also increased in the same proportion of stiffness because the natural

frequency is directly proportional to the stiffness for all the cases that have been studied.

References

[1] Wahl, A.M., “Mechanical Springs”, 2nd edition, McGraw-Hill, Inc., New York, 1979.

[2] W.G. Jiang and J.L. Henshall, “A novel finite element model for helical springs”, Finite Elements in Analysis and Design, Volume 35, Issue 4, Pages 363–377, July 2000.

[3] Merville K. Forrester, “Stiffness Model of a Die Spring”, M.Sc. thesis, Blacksburg, Virginia, 2001.

[4] S. S. Gaikwad and P. S. Kachare, “Static Analysis of Helical Compression Spring Used in Two-Wheeler Horn”, International Journal of Engineering and Advanced Technology (IJEAT), Volume 2, Issue 3, February 2013.

[5] Viatcheslav Gnateski, “Slotted Spring Vibration Isolator”, United States Patent Application Publication, Pub. No.: US 2012/0049422 A1, Pub.

Date: Mar. 1, 2012. [6] C. J. Yang, W. H. Zhang, G. X. Ren, X. Y. Liu, “Modeling and dynamics analysis of helical spring under compression using a curved beam

element with consideration on contact between its coils”, Meccanica, Volume 49, Issue 4, pp 907-917, April 2014.

[7] Wilhelm A. Schneider, “Design and application of slotted cylinder springs”, United States Army, USAELRDL Technical Report 2327, 1963. [8] Krzysztof Michalczyk, “Stress Analysis in Slotted Springs”, Mechanics, Vol.25, No. 3, 2006.

[9] Anis Hamza, Sami Ayadi, Ezzeddine Hadj-Taïeb, “The natural frequencies of waves in helical springs”, Comptes Rendus Mécanique, Volume

341, Issues 9–10, September–October 2013, Pages 672–686. [10] Ahmed Ibrahim Razooqi, Hani Aziz Ameen, Kadhim Mijbel Mashloosh, “Static and Dynamic Characteristics of Slotted Cylinder Spring”,

International Journal of Engineering Research & Technology (IJERT) Vol. 2 Issue 12, December – 2013.

[11] Shigley J.E., Mischke C.R., “ Mechanical Engineering Design”, 5th edition, McGraw Hill Inc., New York, 1989.