DESIGNING A FLEXIBLE BELLOWS COUPLING MADE FROM COMPOSITE MATERIALS USING NUMERICAL SIMULATIONS SVOČ FST 2018 Ing. Frantisek Sedlacek, University of West Bohemia, Univerzitni 8, 306 14, Pilsen, Czech Republic ABSTRACT This paper deals with the design of a composite bellows coupling. The mechanical properties of the composite material were determined using experimental tests according to ASTM standards. A geometric optimization was done in connection with structural analysis to find the optimal design and layout of the laminate of the composite flexible bellows coupling. An advanced finite element model was used to verify the coupling using advanced intra-laminar and interlaminar failure criterions. The parameters for these numerical simulations were obtained using a series of experimental tests according to ASTM standards. KEYWORDS Flexible coupling, composite materials, numerical simulations, FEM, structural optimization, cohesive elements INTRODUCTION The main function of the coupling is to transfer torque from a master mover (e.g. electric or combustion motor, steam turbine) to a functional machine (e.g. gear, compressor, pump, generator) [1]. In most practical applications, the perfect alignment of couplings to machines, and/or shafts is impossible [2]. The conventional way of connecting is by using rigid couplings. However, if there are some misalignments, a rigid coupling can generate high reaction forces. These reaction forces may cause failure of bearings, induce noisy operation, increase vibrations or even cause breakage. Flexible couplings exist for these reasons. Elastomer flexible couplings are most widely used if there is a requirement for multiaxial compliance or transfer of loads. But their construction has many disadvantages and problems, such as very high weight, low durability (service life), high maintenance requirements, they can be used only at low speeds, rubber degradation, their stiffness is dependent on temperature, complicated design and they are expensive. The goal of this paper is to create a highly flexible coupling from composite materials which can transmit the required torque with the possibility of specific angular and axial deformation and all this with the desired stiffness. It must also have: low weight (ability to work at high rotational speeds), high eigenvalues, high durability, low maintenance requirements and a simple design. The basic function schema is shown in Fig. 1. Fig. 1 Schema of loadcases of flexible bellows coupling. However, composite material was not only used for reasons such as high strength and low weight (compared to conventional spring steel up to 80%), low friction, high eigenvalues, acid and weather (corrosion) resistance, high fatigue strength, and many other positive properties of composite materials. The main factor for its selection is its very high elastic deformation energy per unit of weight U (kj/kg). This parameter is the main factor when selecting the material for a spring, and it can be generally expressed as: U = 1 2 σ f 2 ρe [kj kg ]. (1) It means that the material with a lower modulus of elasticity E or density ρ has a relatively higher elastic deformation energy per unit of mass (EDEPM) within the same maximum allowed stress of the material σ f. Fig. 2 gives the EDEPM for common engineering materials. Carbon Fibre Reinforced Polymers (CFRP) have more than seven times higher EDEPM than standard spring steel. Only elastomers are better than CFRP, but they have a significant loss factor and many other issues mentioned above.
Material Elastic deformation energy per unit weight U (kj/kg) Rubber 18 45 CFRP (Carbon Fibre Reinforced Polymer) 3.9 6.5 Ti alloys 0.9 2.6 Nylon 1.3 2.1 GFRP (Glass Fibre Reinforced Polymer) 1.0 1.8 Spring steel 0.4 0.9 Wood 0.3 0.7 Fig. 2 Schema and main values of elastic deformation energy per unit of weight of engineering materials [3]. The design methodology for the composite flexible bellows coupling (CFBC) was created based on specified goals and general requirements - see Fig. 3. It comprises four main stages; the first deals with creation of a generic CAD model and basic FEM model of the coupling. The second is intended to find the required mechanical properties of the coupling (the required stiffness in the main directions/angles). The third stage verifies the design proposal using advanced numerical simulation and experimental tests. The last stage deals with the final design and validation of the coupling. Fig. 3 Flowchart of design of composite flexible bellows coupling. BASIC DESIGN OF THE COUPLING First, it was necessary to choose the main shape of the coupling. This was very limited due to the restricted installation dimensions (especially the maximum width b = 80mm). Furthermore, the manufacturing technology and mounting options were taken into account. Several variations of the basic geometry of the CFBC were created. The most suitable design variant was the W-shaped cross-section, see Fig. 1. A generic 2D CAD model of the CFBC was created based on the main shape and FEM was applied to this model. The structural analysis was done in Siemens Simcenter NX 12 with non-linear multiphysics solver NX Nastran 12 SOL 401 (based on first-order shear deformation theory). A 2D mapped mesh with parabolic quad elements (CQUAD8) was used (see Fig. 4). The orthotropic physical properties of the layup of the composite were entered using the special NX Laminate Composites module. The solution was solved with six load subcases: the load from max. torque moment (M 0 = 12 knm), centrifugal force (n max = 4000 rev.min -1 ), max. required positive axial deformation (Δx a = 5.1 mm) and angular deformation (Δα 0 = 1.5 deg); and the load from max. torque moment, centrifugal force, maximum required negative axial deformation (Δx a = -5.1 mm) and angular deformation (Δα 0 = -1.5 deg), see Fig. 1.
Fig. 4 2D FEM model of CFBC. DETERMINATION OF MECHANICAL PROPERTIES OF COMPOSITE Before the numerical simulation itself, the correct mechanical and strength parameters of the laminate were determined. High strength 200gsm 2x2 twill carbon fabric prepreg gg200t (with 3k T700 toray fibres) was chosen for the coupling. The carbon fibre sheet was fabricated in laboratories at the University of West Bohemia. It was cured in an autoclave with an 8 hour cycle at 110 C. Eight different series of laminate specimens were made according to ASTM standards (a total of 118 specimens). Tensile testing the composite Steel fittings were glued to the carbon sheets because of the high strength of the specimens (to avoid slipping out of the jaws). The tensile properties of the laminate were determined according to ASTM standard D3039 to find the tensile modulus (E 1, E 2, E 3) and stress limits (X T, Y T, Z T ) of the laminate [4]. Experimental tensile tests were carried out by quasistatic load (2 mm/min) on the Zwick/Roell Z050 machine. Subsequently, failure modes of the specimens were evaluated (according to mode code, see Fig. 5) and the individual mechanical parameters were calculated using a sub-routine in Python. The final values are given in Tab. 1. Fig. 5 Experimental tensile measurement of the composite. Compressive test of the composite The compressive properties of the laminate were determined using experimental testing according to ASTM standard D3410 to find stress limits (X C, Y C, Z C ) of the prepreg in compression for all the main directions [5]. Experimental compressive tests were carried out by quasi-static load (1.5 mm/min) on a Zwick/Roell Z050 machine. Subsequently, failure modes of the specimens were evaluated (according to mode code, Chyba! Nenalezen zdroj odkazů.) and parameters were calculated with a subroutine in Python 2.71. The final values are given in Tab. 1. Fig. 6 Experimental compressive measurement of the composite.
Objective function [N] In-plane shear test of the composite A standard test method for in-plane shear response of polymer matrix composite materials by tensile testing at ±45 was done according to ASTM standard D3518 [6]. Experimental tests were carried out using quasi-static load (2 mm/min) on the Zwick/Roell Z050 machine. Final parameters of the in-plane shear modulus (G 12, G 23, G 31) and shear stress limits (S 12, S 23, S 31) are given in Tab. 1. Mechanical properties Fig. 7 Experimental in-plane shear measurement of the composite. Strength properties ρ (kg/m 3 ) 1820 Laminate density X T (MPa) 693 Tensile strength (0 ) t (mm) 0.208 Ply thickness Y T (MPa) 610 Tensile strength (90 ) E 1 (GPa) 55.8 Young s Modulus 0 Z T (MPa) 67 Tensile strength (N90 ) E 2 (GPa) 53.7 Young s Modulus 90 X C (MPa) 552 Compression strength 0 E 3 (GPa) 6.4 Young s Modulus N90 Y C (MPa) 558 Compression strength 90 G 12 (GPa) 3.12 In-plane Shear Modulus 12 Z C (MPa) 268 Compression strength N90 G 23 (GPa) 2.89 In-plane Shear Modulus 23 S 12 (MPa) 109.1 In-Plane Shear Strength in 12 G 31 (GPa) 2.89 In-plane Shear Modulus 31 S 23 (MPa) 121 In-Plane Shear Strength in 23 ν 12 (-) 0.28 Poisson s ratio in plane 12 S 31 (MPa) 121 In-Plane Shear Strength in 31 k I (GPa/m) 10e3 Interface stiffness for Mode-I G IC (J/m 2 ) 695.7 Fracture toughness for Mode-I k II (GPa/m) 14e3 Interface stiffness for Mode-II G IIC (J/m 2 ) 1408 Fracture toughness for Mode-II Tab. 1 Mechanical and strength properties of CFRP (gg200t prepreg). FINDING THE MOST SUITABLE GEOMETRY AND COMPOSITE LAYOUT OF THE CFBC The geometric optimization was combined with the structural analysis to find the best shape and layup of the CFBC that meets all the required mechanical properties of the coupling (stiffness in main directions/angles). The optimization was done using NX Optimizer 12.0.1. The optimization algorithm implemented in NX Optimizer belongs to a class of methods called gradient methods [7], [8]. Finding the minimum weight of the coupling was chosen as the objective function of the optimization. The cross-section of the 3D model was divided into five main zones. The physical properties for individual plies (thickness of the plies and main angle of the fibres) were applied on these sections (Fig. 9). These forty-two parameters with four geometric parameters of the coupling were set as design variables of the geometric optimization. 65 55 45 35 25 15 0 5 10 15 20 25 30 35 40 45 50 55 60 65 Iteration of geometry otimization [-] Fig. 8 Change of objective function across iterations of geometry optimization (left) and change of the geometry of the coupling across iterations of the geometric optimization of the coupling (right). The maximum allowed stresses in all main normal (X T, Y T, Z T, X C, Y C, Z C ) and shear (S 12, S 23, S 31) directions were used as constrains of the optimization. The maximum rotation in the direction of the axis of the coupling (minimum desired torsional stiffness) was set as the last constrain of the geometry optimization. The progress of the objective function across iterations of the geometric optimization and change of the geometry of the CFBC are given in Fig. 8. A design weighing 2.53 kg was found with axial stiffness 410 kn/m, torsional stiffness 11 kn.m/deg and twisting stiffness 34.5 kn.m/deg. The final layup of the composite is given in Fig. 9.
Fig. 9 Final values of layup of the composite (from geometric optimization). VERIFICATION OF FINAL DESIGN USING ADVANCED NUMERICAL SIMULATION An advanced numerical model was created to verify the mechanical properties of the coupling and to determine the strength of the final design of the coupling from geometric optimization. An advanced FEM model was created from a basic 2D model using the NX Laminate Composites module. The 2D FEM model of the CFBC was filled into a 3D mesh with respect to the manufacturing technology for all plies, such as; cuts of the individual plies, distortions of the main directions of the fibres of the plies, resin drops and pockets and overlaps of the individual plies. A total of 168 plies were manually set using the special draping function based on the Fish-net algorithm, see Fig. 10. Fig. 10 Advanced FEM model of the coupling with details of the layup. Intra-laminar (strength properties in plies) and inter-laminar strength (strength properties between plies) was evaluated using a 3D FEM model, see Fig. 11. The inter-laminar strength of the laminate was determined because the coupling is highly flexible and works with large displacements. The numerical simulation was done using NX Nastran 12 SOL401 Multi-step Non-linear Multiphysics solver (100 steps). The flanges of the shafts and bolted connections were included with a full contact condition in the simulation. Fig. 11 Schema of Intra- and Inter-laminar failure of fibre composite.
DETERMINATION OF THE INTRA-LAMINAR STRENGTH The interactive Tsai-Wu failure criterion was used to determine the intra-laminar strength of the laminate [9]. The Tsai- Wu failure criterion can be written as: σ 1 2 ( 1 X T 1 X C) σ 1 + ( 1 Y T 1 Y C) σ 2 + X T X C + σ 2 Y T Y C + σ 12 (S L ) 2 + 2F 12 σ 1 σ 2 X T X C = 1, (2) Where F 12 is the coefficient of interaction and it can be wrriten as: F 12 = 2 1 2σ 2 {1 [XC X T + XT X C Y T Y C (YC Y T )] σ + (1 + XT X C Y T Y C ) σ2 }. (3) The individual ultimate stress parameters were determined using the experimental tests listed earlier and they are given in Tab. 1. DETERMINATION OF INTER-LAMINAR STRENGTH For highly flexible composite parts, it is recommended to determine the inter-laminar strength of the laminate too. A cohesive elements in the form of 2D layers between all the plies of the laminate were created (a total of 92 layers). Special cohesive parabolic elements were used with a cohesive damage interface approach according to Cachan, Allix and Ladevèze. It was necessary to find the fracture toughness (G IC, G IIC) and stiffness (k I, k II) of the interface for the first two modes of the cohesive failure (Mode-I is normal strength and Mode-II is the shear strength of the inter-laminar interface). Experimental tests for both modes of inter-laminar failure were carried out. Determination of Mode-I of the inter-laminar strength Special specimens for the double cantilever beam test (DCB test) were created from material gg200t according to ASTM standards D5528-01 [11]. Experimental tensile DCB tests were carried out using a quasi-static load (2 mm/min) on the Zwick/Roell Z050 machine, see Fig. 12. 2 Fig. 12 Experimental double cantilever beam test for finding parameters of Mode-I. The parameters of the fracture toughness (G IC) and stiffness (k I) for Mode-I were found using a special Python 2.73 subroutine in combination with a FEM model of the DCB in Abaqus 6.13 that allows fitting of the parameters. The fitting process was done with an accuracy of 4%, see Fig. 13 (red line in the chart). The final values are given in Tab. 1. Fig. 13 Fitting of fracture toughness and stiffness of interface for mode-i
Determination of Mode-II of the inter-laminar strength Special specimens for end-notched flexure (ENF) were fabricated from material gg200t according to ASTM standards D7905 [12]. Experimental three point bending ENF tests were carried out using a quasi-static load (0.5 mm/min) on the Zwick/Roell Z050 machine, see Fig. 14. Fig. 14 Experimental end-notched flexure test to find parameters of Mode-II. The parameters of the fracture toughness (G IIC) and stiffness (k II) for Mode-II were found in the same way as the DCB test. The fitting process was done with an accuracy of 3%, see Fig. 15 (red line in the chart). The final values are given in Tab. 1. Fig. 15 Fitting of fracture toughness and stiffness of interface for mode-ii. RESULTS OF NUMERICAL SIMULATION Fig. 16 shows the results of the displacement and the most critical normal and shear stress for the most critical loadcase (combination of: the load from max. torque moment, centrifugal force, maximum required positive axial deformation and angular deformation). Displacement (mm) Normal stress (MPa) direction 11 and 22 Shear stress (MPa) direction 12 and 23 Fig. 16 Results of the displacement and critical normal and shear stress for most critical loadcase. Fig. 17 shows the results of the intra- and inter-laminar strength of the CFBC for the same most critical loadcase. The results indicate that the design is satisfactory for both intra- and inter-laminar strength.
Failure index of intra-laminar strength (Tsai-Wu) Fig. 17 Results of the intra- and inter-laminar strength of CFBC. Failure index of inter-laminar strength (cohesive) CONCLUSION The optimal design of the CFBC was created using the proposed methodology (Fig. 3). The main shape and composite layup was found using geometric optimization and verified using advanced FEM numerical simulation. The intra- and inter-laminar strengths were determined based on the parameters from eight different experimental tests carried out on 118 specimens. The weight of the coupling (2.51 kg) was found to be in compliance with all the mechanical requirementsit was more than 70% lighter than a conventional steel/elastomer flexible coupling. Currently, I am working on the validation of the CFBC. A function sample of the CFBC has already been laminated. A divided positive mould was used for fabrication. The core of the mould was created using additive manufacturing technology (FDM 3D printer) from a heat-stable copolymer PA6 with short carbon fibres. The manufacturing process is shown in Fig. 18. In the future, the validation of the CFBC will be carried out using experimental testing (modal and multiaxial durability tests). Fig. 18 Laminating of function sample of CFBC. REFERENCES [1] C. M. Johnson, An introduction to flexible couplings, World Pumps, vol. 1996, no. Volume 1996, pp. 38 43. [2] Anon, Coupling & U-joints, Motion System Design, no. Sv. no. 12, pp. 1187 1192. [3] Materials Selection in Mechanical Design - 4th Edition. [Online]. Available: https://www.elsevier.com/books/materials-selection-in-mechanical-design/ashby/, [Accessed: 23-Oct-2017]. [4] ASTM D3039 / D3039M-17, Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials. West Conshohocken, PA, 2017. [5] ASTM D3410 / D3410M-16, Standard Test Method for Compressive Properties of Polymer Matrix Composite Materials with Unsupported Gage Section by Shear Loading. West Conshohocken, PA, 2016. [6] ASTM D3518 / D3518M-13, Standard Test Method for In-Plane Shear Response of Polymer Matrix Composite Materials by Tensile Test of a ±45 Laminate. West Conshohocken, PA, 2013. [7] Siemens PLM, NX Nastran Design Sensitivity and Optimization User s Guide. Siemens AG., 2016. [8] P. Goncharov, Engineering analysis with NX advanced simulation, vol. 2014. Raleigh: Lulu Press. [9] W. Van Paepegem and J. Degrieck, Calculation of damage-dependent directional failure indices from the Tsai Wu static failure criterion, Composites Science and Technology, vol. 63, no. 2, pp. 305 310, Feb. 2003. [10] Calculation of damage-dependent directional failure indices from the Tsai - Wu failure criterion, ResearchGate. [Online]. Available: https://www.researchgate.net/publication/234025973 [11] A. ASTM, D5528-13, Standard test method for mode I interlaminar fracture toughness of unidirectional fiberreinforced polymer matrix composites. West Conshohocken, PA, 2013. [12] A. Standard, D7905/D7905M 14, Standard Test Method for Determination of the Mode II Interlaminar Fracture Toughness of Unidirectional Fiber-Reinforced Polymer Matrix Composites, 2014.