Ahmed I.razooqi - Compression and Impact Characterization of Helical | Finite Element Method

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Ahmed I.razooqi - Compression and Impact Characterization of Helical
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    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 srcinal 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 var  ious 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 c alculating 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 L s = length of slot n ss = number of slot section n s = number of slots per section  b = wall thickness h = height of horizontal path    Total deflection  270  International Journal of Engineering & Technology 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 f  or 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   International Journal of Engineering & Technology 271 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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