Shift Fork Design Optimisation using Nonlinear FEA Methods

Shift Fork Design Optimisation using Nonlinear FEA Methods 1209885 April 28, 2016 ES4B5 Finite Element Analysis - Dr. K. Mao

Executive Summary The shift fork is an essential component in a racing car s transmission by actuating a shift sleeve to engage or disengage gear pairs to and from the output shaft. There are multiple shift forks within a transmission and thus the part must be lightweight yet durable with high axial stiffness. This design report comes from a necessity to refine the initial design of a shift fork for a racing car application within a strict specification on stiffness and weight. An iterative design improvement methodology was employed thorough linear and nonlinear numerical analysis of design iterations in Dassault Systèmes Abaqus CAE software suite. Firstly, mesh quality; mesh density and grid dependency were investigated with respect to analysis accuracy and consistency. Linear analysis was conducted on the initial design and subsequently it was deemed necessary to conduct nonlinear analysis due to the complex nature of the fork s contact with the shift sleeve. Further design iterations were developed by critically analysing and evaluating the results of previous iterations. In addition, linear dynamic analysis of the final design iteration was computed to assess modal vibrations of the part with respect to resonance in the axial direction. The results of nonlinear and linear dynamic analysis on the final design are validated with simple bending and simple harmonic motion theory to give assurance to the findings of Abaqus CAE. Manufactured from titanium, the resulting final design achieved a reduction in weight of 89.8% from the initial design manufactured in mild steel whilst achieving a yield stress factor of safety of 11.8 and remaining inside specified deflection limits. When successfully implemented in place of the initial shift fork design, the final design in a manual or sequential five speed transmission containing three shift forks would result in an overall weight reduction of 3.47kg making the design highly suited to light weight racing applications.

Contents 1 Design Problem and Objectives 1 1.1 Shift Fork Function............................................ 1 1.2 Specification................................................ 1 1.3 Generating Geometry.......................................... 2 2 Mesh Improvement 2 2.1 Partitioning................................................ 2 2.2 Grid Dependency............................................. 3 3 Design Documentation - Linear Analysis 3 3.1 Results................................................... 3 3.2 Evaluation of Initial Design....................................... 4 4 Design Documentation - Nonlinear Analysis 4 4.1 Analysis of Initial Design........................................ 5 4.1.1 Results.............................................. 5 4.1.2 Evaluation............................................ 5 4.2 Analysis of Design Iteration One: Improved Sleeve Contact..................... 6 4.2.1 Results.............................................. 6 4.2.2 Evaluation............................................ 7 4.3 Analysis of Design Iteration Two: Revised Material Choice..................... 8 4.3.1 Results.............................................. 8 4.3.2 Evaluation............................................ 8 4.4 Analysis of Design Iteration Three: Revised Geometry........................ 8 4.4.1 Results.............................................. 9 4.4.2 Evaluation............................................ 9 5 Final Design 10 5.1 Nonlinear Analysis............................................ 11 5.1.1 Results.............................................. 11 5.1.2 Static Validation......................................... 11 5.2 Linear Dynamic Analysis........................................ 12 5.2.1 Results.............................................. 13 5.2.2 Dynamic Validation....................................... 13 5.3 Evaluation Against Specification.................................... 14 5.4 Conclusion................................................ 14 6 Final Design General Arrangement (GA) 15 List of Figures 1 Annotated rendering of the output and lay shaft of a manual transmission. Adapted from http://www.grabcad.com........................................ 1 2 (a) An effective partitioning solution to remedy (b) a poor mesh, producing results shown in (c). 2 3 Grid dependency plot showing stress and displacement dependence on mesh density........ 3 4 Results of linear analysis with applied force of 1000 N over the entire contact face......... 4 5 Results of nonlinear analysis carried out on initial design....................... 5 6 (a) Nonlinear analysis reveals that the initial design imposes a moment about the sleeve. (b) Improved design would produce only an axial force........................... 6 7 Adaptations made during transition from the initial design to design iteration one......... 6 8 Results of nonlinear analysis carried out on design iteration one................... 7 9 Adaptations made during transition from design iteration two to design iteration three...... 9 10 Results of nonlinear analysis carried out on design iteration two................... 9 11 Photo-realistic renders of the final design................................ 10 12 Results of nonlinear analysis carried out on the final design...................... 11 13 Diagrammatic breakdown of static validation.............................. 12 14 Bending theory for a cantilever beam. Adapted from Uni. of Warwick Engineering Data Book.. 12 15 Results of linear dynamic analysis reveal modes 4 and 12 to be critical................ 13 16 S-N curves of the β-cez, Ti-6246, and Ti-6Al-4V microstructures (Peters et al. 2002)....... 14 17 General Arrangement Engineering Drawing.............................. 15

1 Design Problem and Objectives There is a need to improve the initial design of a transmission shift fork with the global aim of maximising axial rigidity whilst keeping the component lightweight, due to racing application for which the shift fork is intended. Transmissions represent an essential part of automotive design since they allow alterations in the output torque for given vehicle speeds, maintaining optimum power output. Transmissions have been in development since Louis René Panhard and Emilie Levassor developed and demonstrated a three speed manual transmission in 1894 (VanGelder 2014). Modern computing power, Computer Aided Engineering (CAE) and advancing Finite Element Analysis (FEA) methods are necessarily involved in the refinement and optimisation of transmissions due to the increasing demands placed on such systems. 1.1 Shift Fork Function The shift fork is an essential component of the transmission. Its overall function is to facilitate engaging gear pairs connecting the lay shaft,which is driven by the engine, to the output shaft and subsequently the wheels of the car. The shift fork sits atop of the sleeve, annotated in figure 1, and is actuated by the driver in a manual transmission car. The sleeve is mated to a hub which is permanently connected to the output shaft. As the sleeve is actuated by the shift fork towards the gear pair, the sleeve engages with the gear, physically joining the layshaft to the output shaft, driving the wheels. Figure 1: Annotated rendering of the output and lay shaft of a manual transmission. Adapted from http://www.grabcad.com 1.2 Specification The shift fork design is intended for a race car transmission and as such the final design proposed by this project should conform to the following set of specifications: The design must not exceed the physical size boundary of the initial design. The design must not exceed a maximum deflection of more than 0.25mm in the axial direction up to loads of 1000N. The highest peak stress in the design must not exceed the yield stress of the material chosen when axially loaded with 1000 N. The design is to have a yield stress factor of safety (FoS) of 6-8; in line with IMechE recommendation for engine components (Matthews 2012). The design shall be durable for at least 10 7 load cycles. 1

1.3 Generating Geometry The geometry of the initial design and subsequent design iterations is drawn in Dassault Systèmes Solidworks and imported into Abaqus CAE as an Initial Graphics Exchange Specification (.iges) file. 2 Mesh Improvement An essential part of numerical analysis methods such as finite element analysis is the deconstruction of complex geometries into discrete cells for computational input as well as display output. In three dimensions, discrete cell meshes take the form of tetrahedral or hexahedral elements. It is faster to compute tetrahedral meshes (algorithms can calculate in excess of 400 thousand tetrahedra per minute (Loriot 2006).) but meshing hexahedral elements is beneficial for numerical analysis: tetrahedral meshes must contain 4-10 times as many elements to achieve the same level of accuracy as hexahedral meshes (Weingarten 1994). Additionally, tetrahedral elements are more mathematically stiff due to fewer degrees of freedom and thus create a problem known as tet-locking (Benzley et al. 1995). 2.1 Partitioning During the meshing phase of analysis, Abaqus denotes the quality of the mesh by colour. Orange indicates that Abaqus cannot mesh the part using its algorithms. Partitioning the part into more computationally easy blocks allows Abaqus to successfully apply it s meshing algorithms. As depicted in figure 2(a), green indicates that the part may be entirely meshed using structured (predominantly hexahedral) cells. (a) Effective Partitioning Solution (b) Before strategic Meshing (c) After strategic Meshing Figure 2: (a) An effective partitioning solution to remedy (b) a poor mesh, producing results shown in (c). Figure 2(b) shows the quality of mesh achieved after partitioning the shift fork design into three partitions, indicating that additional partitions are required. Partitioning shown in figure 2(a) produced the mesh result shown in figure 2(c), which is of satisfactory quality for linear analysis. It is notable that mesh quality is affected by cell size and design geometry. Different partitioning is required for different geometries with the aim of hexahedral elements where possible. Mesh density is considered by means of grid dependency analysis in the following section. 2

2.2 Grid Dependency Grid dependency is a measure of how the results are influenced by the number of nodes used to perform an FEA task. It is used to establish a mesh density whereby a finer mesh would not lead to an improvement in result accuracy. This ensures accurate results without compromising on long processing periods during analysis. 18 Stress (MP a) 16 14 12 10 Stress Displacement 7.5 7 6.5 Displacement ( 10 6 m) 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Node Number 10 5 Figure 3: Grid dependency plot showing stress and displacement dependence on mesh density. Figure 3 reveals that no substantial further improvement in accuracy can be observed past 4 10 4 which corresponds to a global seed size of 1.5mm. This mesh density is used in subsequent analysis iterations. Additionally a partitioning strategy has been established to produce optimum element types and structure. 3 Design Documentation - Linear Analysis A detailed numerical investigation is undertaken to improve and optimise the initial design to meet the design specifications. Dassault Systemes Abaqus CAE software is used for accurate finite element analysis of proposed design iterations. Linear analysis may be though of as a proportional analysis. Whilst not particularly accurate, it is carried out on the initial design of shift fork to establish the efficacy of the meshing procedure carried out in section 2 and to gain an early insight into the initial design and its conformity to the specification. The procedure for carrying out linear analysis is relatively simple. Boundary conditions are applied to the part replicating the rail fixing on which the fork is to be fixed to. A force of 1000N is applied over the circular contact face using a pressure condition in Abaqus. Pressure is given by P = F A 1 where F is the force of 1000N and the area A, is expressed A = π 2 ( α 2 β 2), (1) where α and β are the external and internal radii of the contact face respectively. Given α = 48mm and β = 44mm, this yields an applied pressure of 1.73MP a to the contact face in Abaqus. The material properties inputted to Abaqus CAE for linear analysis were that of low carbon steel. 3.1 Results Figure 4(a) shows the meshing structure of the undeformed part for comparison. Figure 4(b) depicts a plot of part deformation gradated by displacement. Deformation becomes most prominent towards 3

to fork prong ends where a maximum displacement of 0.0124mm is observed. Figure 4(c) shows a gradated plot of stress across the part. Stress appears most prominent around the fork rail connection as a result of reaction forces. The fork prongs, despite being heavily displaced relative to the rest of the part exhibit stresses in the 10 3 range. Linear analysis indicates a maximum stress observed in the part of 29.3MP a. (a) Meshed Part (b) Displacement Plot (c) Stress Plot Figure 4: Results of linear analysis with applied force of 1000 N over the entire contact face. 3.2 Evaluation of Initial Design As a result of linear analysis the following points are to be considered during design progression: Due to uneven displacement of the part it is noted that nonlinear contact with the shift sleeve is likely to significantly affect the accuracy of numerically derived results. Nonlinear analysis should be conducted to the initial design before design changes may be proposed. The fork rail interface appears is an area of concentrated stress. Future design iterations should aim to reduce and distribute such stress. Linear analysis shows that the design made from low carbon steel falls well within specification for maximum stress: 7.9% of yield stress, and deformation: 5.0% of allowable deformation. Manufactured from low carbon steel, the initial design has a mass of 1288 grams. 4 Design Documentation - Nonlinear Analysis Nonlinear analysis is more appropriate for the optimisation of the shift fork design because it accounts for changing geometry and contact points as the part deforms under an applied load. More accurate results are typically extract from nonlinear analysis when multiple parts are in contact and interacting in a complex manner. The procedure for carrying out nonlinear analysis in Abaqus CAE differs somewhat from the procedure for linear analysis. Unlike for linear analysis, where a uniform pressure is applied across the contact face of the fork, nonlinear analysis necessitates the addition of a new part in Abaqus. This part takes the form of a massless disk which is constrained in five degrees of freedom, allowing movement only in the axial direction - like the sleeve component in a real transmission. This disk part is constrained and assembled with contact interactions within Abaqus and a load of 1000 N is applied as per the specification requirements. 4

4.1 Analysis of Initial Design The material properties inputted to Abaqus CAE for nonlinear analysis of the initial component design were that of low carbon steel with a Young s Modulus of 210GP a and a density of 7850Kgm 3, giving the part a total mass of 1288 grams. The part geometry under investigation is identical to the part analysed in section 3. 4.1.1 Results Figures 5(a) and (b) shows the deformation geometry of the initial component design and a maximum deformation of just 0.006mm, approximately half the maximum deformation achieved in the linear analysis. Figures 5(c), (d) and (e) show the distribution of stresses in the design. Much like the results of linear analysis, concentrated stress is observed around the fork rail connection, with figure 5(e) revealing that stress is further concentrated near the surface of the design. A maximum stress of 15.1M P a is observed: approximately half of the peak stress observed in linear analysis. Crucially the fork prongs, gradated in blue, exhibit negligible stress (less than 10P a), indicating that they are either superfluous to the design, or that the design does not perform its function correctly. Figure 5(f) shows the concentration of stress as a result of surface contact and reveals information that linear analysis could not: during deformation of the part, contact shifts to a single concentrated point closer to the fork rail connection. Peak contact stress is observed to be 22.1MP a. (a) Side view of Displacement (b) Front View of Displacement (c) Isometric View of Stress (d) Stress Plot (e) Stress Cut Plot (f) Contact Pressure from Normal Force Figure 5: Results of nonlinear analysis carried out on initial design. 4.1.2 Evaluation As a result of nonlinear analysis the following points are to be considered during progression to the first design iteration. 5

Much like the results from linear analysis: the fork rail interface appears to be an area of concentrated stress. Future design iterations should aim to reduce and distribute such stress. Nonlinear analysis reveals that due to part deformation, contact shifts to a single point near the fork rail connection, explaining why the fork prongs appear unstressed. It is suggested that this shift to a single point will not only lead to an increase in wear, but will additionally compromise the efficacy of the design, as the fork will tend to apply a turning moment as it actuates the shift sleeve. Figure 6(a) depicts how the current design of fork will apply load to the shift sleeve. Further iterations must be adapted to apply even load to the sleeve, so as to apply a purely axial force resulting in no moment applied to the sleeve. Load should be applied as 6(b) shows. (a) Single Contact Point on Sleeve (b) Two Contact Points on Sleeve Figure 6: (a) Nonlinear analysis reveals that the initial design imposes a moment about the sleeve. (b) Improved design would produce only an axial force. Nonlinear analysis shows that the initial design made from low carbon steel falls well within specification for maximum stress: 4.1% of yield stress, and deformation: 2.4% of allowable deformation. Manufactured from low carbon steel, the initial design has a mass of 1288 grams. 4.2 Analysis of Design Iteration One: Improved Sleeve Contact The material properties inputted to Abaqus CAE for nonlinear analysis of the first design iteration were that of low carbon steel with a Young s Modulus of 210GP a and a density of 7850Kgm 3 giving the part a total mass of 998 grams. Figure 7 shows geometric changes to the part design. Principally the design change is a removal of material near the shift fork rail connection shifting contact to two regions further down the fork legs. This design is proposed for better functionality of the fork and sleeve interaction. Figure 7: Adaptations made during transition from the initial design to design iteration one. 4.2.1 Results Figures 8(a) and (b) shows the deformation of the first design iteration under load, with a maximum deformation of 0.021mm: approximately 3.5 times the deformation of the initial design under the same load conditions. Figures 8(c), (d) and (e) show the distribution of stresses in the design. The body of the design exhibits a maximum stress of approximately 28M P a focused around the fork rail connection. 6

An improvement of the stress distribution is observed as the stress concentration seen in figure 5(c) and (d) disappears as stress is transferred to the prongs of the fork where 16.0MP a of stress is observed. The bottom of the fork prongs exhibit little stress which is explained by figure 8(f) which indicates that contact occurs in a focused region at the top of the contact area. This focused contact region results in a local stress raiser, with a maximum stress of 185MP a. (a) Side view of Displacement (b) Front View of Displacement (c) Isometric View of Stress (d) Stress Plot (e) Stress Cut Plot (f) Contact Pressure from Normal Force Figure 8: Results of nonlinear analysis carried out on design iteration one. 4.2.2 Evaluation As a result of nonlinear analysis of the first design iteration, the following points are to be considered during progression to the second design iteration. The design displays more evenly distributed stresses and has reduced stress concentrations. Deformation has more than tripled and peak stress in the majority of the design has doubled from the previous design iteration. Contact stresses due to the interation between the fork and the sleeve are high but do not exceed yield stress. These contact stresses are likely to result in component wear but will not contribute towards component failure. Manufactured from low carbon steel, the design has a mass of 998 grams, a reduction in weight of 290 grams over the initial design manufactured from low carbon steel. As concluded from the analysis of the initial design, the component falls well within specification for maximum stress: 7.57% of yield stress, and deformation: 8.4% of allowable deformation. To achieve a lightweight component, as per the specification, it is suggested that a more lightweight material is utilised to further optimise the design. 7

4.3 Analysis of Design Iteration Two: Revised Material Choice Titanium Ti-6AI-4V (Grade 5) is was the revised material choice for the second design iteration; its material properties were inputted to Abaqus CAE for nonlinear analysis of the second iteration. Titanium s high yield stress; low density and corrosion resistance make it a popular choice in aerospace, military and performance automotive applications. For modelling, the following material parameters were used: a Young s Modulus of 113.9GP a and a density of 4430Kgm 3. With the revised material choice, the part with the same geometry as design iteration one has a mass of 563 grams. 4.3.1 Results Figures are not included as deformation and stress distributions are nearly identical to those shown in figure 8. Due to the material change, measured values of stress and deformation differ, however. Design iteration two manufactured with Titanium Ti-6AI-4V (Grade 5) exhibits a maximum deformation of 0.039mm, approximately double the deformation observed in design iteration two made from low carbon steel. The design shows a maximum stress in the main body of the part of 28MP a and a localised stress raiser with a peak stress of Titanium 189MP a, similar to the values observed in the first design iteration. 4.3.2 Evaluation As a result of nonlinear analysis of the first design iteration, the following points are to be considered during progression to the second design iteration: Modifying material choice from low carbon steel to Titanium Ti-6AI-4V (Grade 5) results in a design of mass 563 grams, a reduction in weight of 725 grams over the initial design manufactured from low carbon steel. It is suggested that Titanium is a suitable material choice. Despite having a lower Young s modulus than low carbon steel, it s yield stress of 880MP a will contribute to a longer working life and less susceptibility to fatigue failure. As concluded from the analysis of the initial design, the component falls well within specification for maximum stress: 3.18% of yield stress, and deformation: 15.6% of allowable deformation. Since a suitable material choice has been made, it is suggested that adaptations are made to the geometry of the design to further reduce weight. As observed in section 4.2, stress appears to be low towards the bottom of the fork prongs and low around the top of the fork rail connection. Additionally, deformation is well within specification, suggesting that a thinner design is more appropriate to achieve the required specification of a lightweight component. 4.4 Analysis of Design Iteration Three: Revised Geometry Iteration three sees alterations to the design in three areas: Slots are added to the fork prongs to reduce part mass whilst maintaining rigidity. The prongs and fork rail connection hole have been thinned down for further weight reduction. The depth of the part is reduced by 7mm. Given a part made from Titanium Ti-6AI-4V (Grade 5) with a Young s Modulus of 113.9GP a and a density of 4430Kgm 3, the part weight comes in at 217.8 grams. The material properties of Titanium Ti-6AI-4V (Grade 5) are inputted to Abaqus CAE for nonlinear analysis of the third design iteration. 8

Figure 9: Adaptations made during transition from design iteration two to design iteration three. 4.4.1 Results Figures 10(a) and (b) shows the deformation of the third design iteration under load, with a maximum deformation of 0.232mm: approximately 11 times the deformation of design iteration two under the same load conditions. Figures 10(c), (d) and (e) show the stress distribution in the part, indicating a maximum stress of 84.3MP a which occurs in the inside the inner most portion of the prongs, as located by the stress cut plot in figure 10(e). Contact stresses exceed 450MP a. The upper most region of the part surrounding the top of the fork rail connection remains negligibly stressed as a result of the applied 1000N load, as are the bottom of the shift fork prongs. (a) Side view of Displacement (b) Front View of Displacement (c) Isometric View of Stress (d) Stress Plot (e) Stress Cut Plot (f) Contact Pressure from Normal Force Figure 10: Results of nonlinear analysis carried out on design iteration two. 4.4.2 Evaluation As a result of nonlinear analysis of the first design iteration, the following points are to be considered during progression to the fourth design iteration: 9

Modifying material choice and geometry results in a design of mass 218 grams, a reduction in weight of 1070 grams over the initial design manufactured from low carbon steel. The introduction of the slots into the prongs of the fork design has created a disparity in stress. Figure 10(e) highlights how the inside spline of the prong experiences around 2.5 times the stress of the outside spline. It is suggested that a redesign bias the thickness towards the inner most spline of the fork prong. The fork-rail connection appears unstressed towards the top of the part. It is suggested that material may be removed from this region without introducing additional stress. The bottom of the prongs appear unstressed. It is suggested that the part geometry is altered to minimise material in this area. It is observed that contact stresses scale with angle of deformation, however the current surface stress of 450M P a does not exceed the yield stress of Titanium Ti-6AI-4V (Grade 5). The component falls well within specification for maximum stress: 9.58% of yield stress, and deformation: 92.8% of allowable deformation. 5 Final Design The final design is the result of in-depth linear and nonlinear analysis of the initial design proposal and three subsequent iterations, the final design being the fourth. Nonlinear analysis as well as linear dynamic analysis is to be conducted on the final design and the results validated with conventional engineering theory to demonstrate its design efficacy. The design will be critically evaluated against the specification laid out in section 1.2. (a) Dimetric View (b) Offset Side Angle View Figure 11: Photo-realistic renders of the final design. As per recommendations made in section 4.4.2, the following design changes have been made for the final design: The slots have been augmented so as to create a thicker spline on the inner-most part of the prongs in response to a high build up of stress in this area. The prongs have been tapered to a width of 9.5mm at the bottom. Material has been removed around the fork rail connection. The resulting final design has a mass of 131.8 grams when manufactured from Titanium Ti-6AI-4V (Grade 5) with a Young s Modulus of 113.9GP a and a density of 4430Kgm 3. The material properties of Titanium Ti-6AI-4V (Grade 5) are inputted to Abaqus CAE for nonlinear analysis of the final design. 10

5.1 Nonlinear Analysis 5.1.1 Results Figures 12(a) and (b) show a gradated plot of the predicted deformation of the final design when under load. A maximum deformation of 0.24mm is observed placing the deformation narrowly within specification at 96% of allowable deformation. Stress plots in figures 12(c), (d) and (e) show that efforts to distribute stress evenly across the design and avoid stress raisers has been effective. Figure 12(d) highlights how stress is distributed along the prongs of the fork. As suggested in the evaluation for design iteration three in section 4.4.2, the inner spline is thickened and it is clear from figure 12(e) that this has more evenly distributed load between the inner and outer splines. Peak stress in the main body of the part is found in the fork prongs at 74.4MP a. Contact stress is still expected to be high at 645MP a but this does not exceed the yield stress of Titanium Ti-6AI-4V (Grade 5). Simple stresses in the design fall well within specification at 8.45% of the yield stress of Titanium (880MP a), giving the design a stress factor of safety of 11.8. High contact forces are predicted to contribute to wear of the fork and sleeve, but are not expected to contribute towards its failure. (a) Side view of Displacement (b) Front View of Displacement (c) Isometric View of Stress (d) Stress Plot (e) Stress Cut Plot (f) Contact Pressure from Normal Force Figure 12: Results of nonlinear analysis carried out on the final design. 5.1.2 Static Validation Validation of the results in section 5.1 is carried out to ensure that the results from Abaqus CAE are sensible and are to a reasonable degree of accuracy. Figure 13(a) shows the cut plot used to generate the geometric estimations for validation: the combined profile highlighted in blue produces a rectangle of width 21.5mm and height 17.1mm. Figure 13(b) indicates the lengths used for validation: principally force is applied at a distance of 85.8mm and peak deflection occurs at a distance of 104.6mm. 11

(a) Cut Plot of Fork Prongs (b) Breakdown of Distances Figure 13: Diagrammatic breakdown of static validation. Given a force W, acting at a distance a, along a uniform beam of length L composed of material of Young s modulus E with a second moment of area I, the maximum deflection observed y max may be expressed y max = W a3 6EI (3 La 1 ). (2) Second moment of area I, may be expressed I = bh 3 /12 where b is the base length of the rectangular cross section and h the height. The fork prongs, therefore, have a combined second moment I = 9.0 10 9 m 4. Titanium Ti-6AI-4V (Grade 5) has a Young s modulus E = 113.8GP a and the force applied to the part W = 1000N. Inputting the following data into equation (2) gives y max = (1000)(85.5mm) 3 ( 6(113.8GP a)(9.0 10 9 m 4 3 104.6mm ) ) 85.8mm 1, (3) yielding a deflection y max of 0.27mm, a 12.5% difference from the actual result. This might be explained because the validation method doesn t account for the more mechanically stiff section around the fork rail intersection. Validation reveals that the Abaqus results are within a reasonable degree of accuracy. Figure 14: Bending theory for a cantilever beam. Adapted from Uni. of Warwick Engineering Data Book. 5.2 Linear Dynamic Analysis The final design undergoes linear dynamic analysis to establish modes of vibration that may introduce unwanted or potentially design compromising effects, particularly in the axial direction, that is, co-axial with loading. Abaqus CAE is set up much like in the linear analysis phase in section 3. Steps are modified to first calculate the natural frequencies and mode shapes of the part and then the modal 12

dynamic response of the part. Abaqus makes use of Lanczos algorithms to extract the eigenvalues and modes for linear dynamic analysis. The following analysis is the result of a 0.2 second impulse of 1000N of force. 5.2.1 Results Linear dynamic reveals two modes of vibration of concern. Mode 4 corresponds to a frequency of 790.1Hz and gives the component an effective mass of 54.0 grams axial to the shift fork rail. Mode 12 too causes unwanted resonance in the axial direction at a frequency of 2755.0Hz leading to an effective mass in the axial direction of 36.5 grams. Effective mass indicates the amount of mass active in each degree of freedom for a given mode. Figure 15(a) shows the un-deformed final design. Figures 15(b) and (c) show the effects of the modal vibrations on the part. Mode 4 displaces the prongs back and forth almost entirely in the axial direction, explaining why this mode of vibration is of biggest concern. Vibration of the fork prongs in mode 12 leads to vibrations axially as well as twisting of the part. (a) Unloaded Component (b) Fourth Mode of Vibration (c) 12th Mode of Vibration Figure 15: Results of linear dynamic analysis reveal modes 4 and 12 to be critical. Principally the design of the shift fork to shift sleeve interaction should be such that it avoids resonating the shift fork in modes 4 and 12 in particular. Resonance in these modes may lead to increased wear and premature fatigue failure of the part. 5.2.2 Dynamic Validation Nonlinear validation provides a deflection y max = δ = 0.27mm for a given force F = 1000N. The stiffness k, therefore, is given by k = F/δ = 3.7 10 6 Nmm 1. Frequency f may be expressed f = 1 k 2π m, (4) where m represents the mass of the part. Given the mass of the part m = 0.1318kg, the results from section 5.2.1 may be validated. The result of equation (4) is expected to correspond to the mode 4 frequency because the value of k is derived from displacment in the axial direction and it is this mode which corresponds also to vibrations in the axial direction. Indeed, equation (4) evaluates to f = 1 3.7 106 Nmm 1 = 843.4Hz. (5) 2π 0.1318 13

This value represents a 6.31% difference from the actual value, validating the dynamic analysis result of section 5.2.1 to a reasonably degree of accuracy. 5.3 Evaluation Against Specification The following final design is critically assessed in relation to the specification: Modifying material choice and geometry results in a design of mass 131.8 grams, a reduction in weight of 1156.2 grams over the initial design manufactured from low carbon steel. The component falls well within specification for maximum stress: 8.45% of yield stress The component falls withing specification for deformation: 96% of allowable deformation. Peak stress observed in the component is associated with contact stresses measured at 645M P a, which falls within specification since this value does not exceed the yield stress of Titanium Ti- 6AI-4V (Grade 5) with a yield stress of 880MP a. The yield stress factor of safety (FoS) for normal stress (excluding contact stresses) is 11.8. The part will be durable for in excess of 10 7 load cycles, as indicated by figure 16. According to data collected by Peters et al. (2002), the part manufactured from Ti-6Al-4V should withstand approximately 400MPa for 10 7 load cycles under a load ratio R = 1, noting that load ratio R = σ min /σ max. Figure 16: S-N curves of the β-cez, Ti-6246, and Ti-6Al-4V microstructures (Peters et al. 2002). 5.4 Conclusion The result of this iterative process is a validated final design. The design fulfills the specification whilst achieving a lightweight, racing-specific design application. When successfully implemented in place of the initial shift fork design, the final design in a typical manual or sequential five speed transmission containing three shift forks would result in an overall weight reduction of 3.47kg. It is suggested that this design be pushed forward for prototype testing to validate the findings of Abaqus CAE. 14

6 Final Design General Arrangement (GA) Figure 17: General Arrangement Engineering Drawing 15

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