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T804 End-of-Module Assessment
Structural Analysis of Machine Shaft
This document has been prepared by and represents his own work.
This report contains the finite element analysis of a rotating cutting shaft from a roller cutting machine such as that used in the mass production of insulating slab materials.
The report is an evaluation of the existing design against higher cutting loads for the purposes of alternative application. Therefore, this report includes recommendations on improvements to the shaft in order that is should better handle increased loading.
Figure 1 Error Message 9
Figure 2 Boundary Conditions 11
Figure 3 Loading 12
Figure 4 Vector Sum Displacement 13
Figure 5 X-displacement 13
Figure 6 y-Displacement 14
Figure 7 Z-displacement 14
Figure 8 Stress intensity 15
Figure 9 X-stress 15
Figure 10 y-stress 16
Figure 11 Z-stress 16
Figure 12 Vector sum 17
Figure 13 X displacement 17
Figure 14 Y-displacement 18
Figure 15 Z-displacement 18
Figure 16 Stress intensity 19
Figure 17 x-stress 19
Figure 18 y-stress 20
Figure 19 z-stress 20
Figure 20 Nodal Frequencies 21
Figure 21 Deflection at 220GPA 24
The purpose of this report is to evaluate the machine shaft for alternative cutting applications. The existing shaft design is to be evaluated for stress and deflection under normal operating loads. It is also to be evaluated under modal analysis to determine the first four normal modes of natural vibration in the shaft under no load conditions. Under normal loads the shaft will be subject to three vertically orientated (-Y) loads to represent each cutting point. The side cutting loads are of magnitude 400N. The Centre Cutting load is to be of magnitude 380N. Each side of the shaft is constrained by bearings. As the Z-Y bending is unknown both fully constrained and simply supported models must be considered. Rotation speed is 2500RPM, Axis of rotation is parallel to the Z axis and the only unconstrained rotational axis is the Z-axis. The outputs from the analysis will be Maximum deflection under normal operating loads, Maximum stress under normal operating loads and first four normal modes of natural vibration under no load conditions. Based on the results of the normal load analysis optimisation recommendations have been made in line with the following design criteria. Outer diameter to remain at 80mm, Maximum tensile stress to be no greater than 60MPa and Maximum deflection to be no greater than 0.5mm
⦁ Analysis Approach
The process for analysing the shaft design cosseted consisted of 5 phases
Perform hand calculations
At this point the variables given in the evaluation criteria are evaluated by hand in order to give a guideline as to model performance.
In this phase the model is designed in ANSYS Mechanical including mechanical variables and physical dimensions.
Apply loads and boundary conditions
In applying load and boundary conditions the restrictions which form the basis of the load case will be applied to the model. These represent the forces the shaft will be subjected to and any restrictions in movement.
Solve and interpret data
In solving the model, the software will calculate the deformation of the model based on the data input previously. This data is then available to be interpreted by the software user. From this data it will be possible to determine the viability of the model under the load case in question. The data should also be compared against the hand calculations to determine validity.
Refine the model
In this final phase the results from phase four are considered and the model adjusted towards optimisation. This phase will require re-solving of the adjusted model and interpretation of the new results. The model should be repeatedly optimised until it meets the design criteria.
⦁ Theory and Validation Calculations
Nodal Calculations for the shaft:
The following table is correct:
Pinned-Pinned 616.066 1232.132 1848.198 2464.265
Clamped-Clamped 927.553 1539.97 2156.31 3388.41
Bending Moment Diagram
The analysis was conducted using ANSYS Mechanical APDL to model the shaft, apply loads and boundary conditions. Choice of Element:
To make a decision about element in FE analysis, one needs to consider following factors:
• Family of Element:
Family of element is the most fundamental property of an element. Solid, rigid, and truss are few of the families of element.
• Number of Nodes:
It is also imperative to consider number of nodes of elements. In linear elements, nodes are taken at edges only, while, quadratic elements have nodes on their mid-points as well.
• Degrees of Freedom:
Output variables to be calculated for FE analysis define degrees of freedom. For instance, a solid box cannot rotate, it possesses translational motion only, while a beam contains translational and rotational motions both.
Size of Mesh:
For FEA, continuous shape has to be discretized, the size of discrete mesh significantly determines the degree of accuracy of FE analysis, therefore, it is critical to choose an appropriate mesh size. Different values of accuracy will be obtained for every different size of mesh.
A model with correct boundary conditions can have a significantly high value of accuracy, while an error in boundary conditions will dramatically affects accuracy. Furthermore, wrong boundary conditions can lead to application of force on undesired area and can alter the stiffness of surface.
⦁ Analysis description
The model was produced as a volume and meshed to Solid 285, which is based upon 4 node tetrahedrons. This mesh was selected due to the great performance of Brick and tetrahedral meshes for volumes. Tetrahedral was selected over brick as ANSYS was unable to mesh the model after cooling holes were introduced in a brick mesh. The 4 node was selected due to the limitation of elements in the academic version of ANSYS. This precluded the use of 10 node tetrahedrons it generated too many nodes for the licence in use.
Figure 1 Error Message
Several assumptions have been made in the production of this model.
The effect of gravity on the shaft can be ignored.
The weights associated with the cutting blades and the restraining bearings can be ignored.
The shaft is homogenous continuous and free of voids or cracks
Weight of cooling liquid is negligible and can be ignored
Temperature variations within the shaft due to cutting and cooling are negligible and can be ignored.
Residual stresses introduced to the shaft through temperature variation are negligible and can be ignored.
The drive assembly for the shaft produces a consistent speed of 2500rpm
The resistance of the material being cut is such that the 400N & 380N forces are representative.
A safety factor is not to be included in optimisation.
⦁ Material Properties
Material properties to be stated here and referenced.
The model comprises the following components:
Table 1 Component materials
Shaft Structural Steel
The Structural Steel properties are taken from Ref.
Table 2 Material properties of Structural Steel for analysis
Material Structural Steel
Young’s Modulus (GPa) 206
Poisson’s ratio 0.29
Density (kg/m3) 7860
Table 3 Material properties of Structural steel from Eurocode Materials Database
Material Structural Steel
Young’s Modulus (GPa) 210
Poisson’s ratio 0.30
Density (kg/m3) 7850
From the caparison between the EMA material properties and those supplied by Eurocode it can be seen that the EMA values are approximately correct, although denser and with a lower Poisson’s ratio and modulus of elasticity. This will lend the model a degree of tolerance as the real-world material values would perform better.
⦁ Boundary Conditions (extremely important section)
The cutting shaft is located and supported by two roller bearings, B1 and B2, fitted at the ends. The bearings allow the shaft and cutters assembly to rotate about its longitudinal axis, which is parallel to the z direction, but prevent it moving radially in the x–y plane.
The shaft is also restrained by circlip type retaining rings, which prevent it moving in the z direction but with sufficient play longitudinally to allow for any expansion and avoid hindering the rotation in all operating conditions.
The bearings can be regarded as having infinite radial stiffness in comparison with the cutting shaft.
The information given with regards to the bearings requires that for one analysis the shaft is restrained in all rotational movement except about the z axis at both ends. In this same analysis it must also be construed in all translational movements. In this analysis the shaft can be considered to be fully pinned or encastré. This is representative of the shaft having utilised all play in the bearings and roller rings. This is potentially due to thermal expansion during the cutting process. Where the shaft is of hollow construction it lacks a central key point to apply boundary conditions to. Boundary conditions were therefore applied to the key points at the 3 & 9 o’clock positions as shown in the below diagram.
Figure 2 Boundary Conditions
Applying boundary conditions to these two key points acts the same as pinning the shaft as movement is restricted in a similar way balanced about the vertical centre of the shaft preventing rotation about the x & y axes and lateral movement in all 3 axes.
Since resistance to bending in the y–z plane is unknown simply supported conditions must be also be modelled. In the simply supported model, the shaft is unconstrained in z-axis & y-axis transitional movements. Is it however constrained in x-axis translations and all rotations except about the z-axis. This is representative of play in the bearings and roller rings permitting movement.
To obtain the stress and deflection measurements, the loadings were as per the below diagrams.
Figure 3 Loading
In these diagrams it can be seen that 3 loads were applied to the top surface of the shaft. Two of the loads had a magnitude of 400N and were applied at the ends, the central load had a magnitude of 380N. These loads are representative of the down force caused by the cutting wheels on the shaft as the wheels cut through the material.
In addition to the forces depicted in the diagrams, a rotational inertia was applied of 261.8 Radians /Second about the lateral axis.
In the Nodal analysis all loads, with the exception of the rotational inertia, were removed to demonstrate the free spinning of the shaft, as shown in the diagram below.
⦁ Simply Supported
The results of the simply supported model are shown in the following diagrams.
Figure 4 Vector Sum Displacement
Figure 5 X-displacement
Figure 6 y-Displacement
Figure 7 Z-displacement
Figure 8 Stress intensity
Figure 9 X-stress
Figure 10 y-stress
Figure 11 Z-stress
⦁ 5.2 Pinned At both Ends
The Results of the model which was pinned at both ends are contained in the following diagrams:
Figure 12 Vector sum
Figure 13 X displacement
Figure 14 Y-displacement
Figure 15 Z-displacement
Figure 16 Stress intensity
Figure 17 x-stress
Figure 18 y-stress
Figure 19 z-stress
⦁ 5.3 Nodal Analysis
The results of the Nodal Analysis are contained below:
Figure 20 Nodal Frequencies
The results of the simply supported model gave a maximum bending moment for the shaft of 0.589mm, this is only just outside the design tolerance for the mode at higher loads.
This when this was run for the pinned model the result was the same. This is due to the direction of force and bend on the shaft. The direction for both is towards the -Y. in these scenarios the restraints in the -Y direction are the same yielding similar results.
The Stress intensities for both models were also similar with the maximum stress being 241 GPa. This is primarily concentrated around the cooling holes which act as stress concentrators due to their geometry.
In both models, the stresses where higher in the Y-Axis than either X or Z again due to the direction of support and loading.
The differences between a simply supported model and fully constrained were not as stark as suspected. Potentially a finer meshing with improved licencing for the software may highlight more significant differences between the two models.
As it currently stands the model is not suitable for implementation at the higher loads as it does not fall within the design tolerances specified for either deflection or internal stress.
The nodal frequencies which were generated, do not fully match the calculated ones. This is after the model being run several times. This is believed to be due to the nature of modelling and how the software has interpreted the mesh and loads. Whilst some are close to the calculated ones there is not enough consistency in the dispersion to give confidence in the results.
The model does not accurately reflect the hand calculations or provide results close enough to them for the model to be fully trusted. IT is likely that there is an erroneous condition in the geometry or meshing of the model as a solid which is causing the variation. A further evaluation should therefore be conducted into which model and meshing type is most suitable for this evaluation.
In order for the model to meet the design requirements the rigidity of the shaft must be increased. This can be accomplished by using a more rigid material and maintain the same dimensions or by increasing the thickness of the steel.
In this scenario the current weight of the shaft is approx. 8kg increasing the thickness of the steel by 1mm will add 2 Kg to the weight. This will have implications for the motors and other equipment used to drive the shaft.
Alternatively, if the modulus of elasticity is increased by employing a steel with a different tempering process, the design criteria could be met without the requirement to alter the other components. This however could significantly increase the cost of the shaft.
Increasing the modulus of elasticity to E=320GP produces a design which is still outside of tolerance as can be seen in the models below achieving the design requirements by increasing the thickness of the material required a 2mm increase to the thickness. Adding approx. 4KG of weight to the material. Without access to the pricing and assuming the motors would have to be re-purchased, the recommendation would therefore be to make the shaft from the same steel but with a thicker wall.
Figure 21 Deflection at 220GPA
⦁ The Engineering toolbox, reference for Steel strengths, https://www.engineeringtoolbox.com/young-modulus-d_417.html