Calculations and Scaling
Calculations
We used a combination of equation based calculations and physical testing with Plasticine to determine the force requirements for our project.
Physical Testing
Physical testing was performed with Plasticine. We used our prototype die and a kitchen knife to get real world figures to validate and supplement our theoretical force calculations. By using a kitchen scale and carrying out the cutting and heading procedures on the Plasticine as it rested on the scale, the reactionary force could be roughly determined by reading the figures on the scale.
For the cutting operation, a 10mm round bar of Plasticine was swiftly cut (less than 1 second) with a kitchen knife and the peak force was recorded. For the heading operation, a 40mm disk of Plasticine was swiftly headed (less than 3 seconds) using a 20mm cylindrical extruder and the peak force was recorded.
Using a sample size of 10 test runs for both the cutting and heading operations, the following force requirements were derived:
- Cutting Operation: 18 lbs
- Heading Operation: 38 lbs
Calculation - Heading Force
For heading force calculations, we can apply F = K * Y * A
K = Impression Die Forging Constant (for flash-less simple shapes it ranges from 3-4, we assume 4 to overshoot required force)
Y = Flow Stress
A = Heading area
Flow Stress:
To find the flow stress of our process, we need to look up the material constants. We assume white Plasticine, and use constants from the course slide deck.
We get the constants:
k = 0.143
n = 0.167
ε = 0.2
Now we can calculate the flow stress:
Y = (0.143)(0.2)^0.162
Y = 0.1102 MPa
Area:
From SolidWorks model of manufactured bolt dimensions, we have that our heading area is 339.61 mm^2
Final Calculation:
Now we can calculate the force required for our process:
F = K*Y*A
F = (4)(0.1102)(339.61)
F = 149.7 N = 33.65 lbs
Calculation - Billet Feeding Conveyor Belt Torque
To calculate the torque needed from our DC-motor, the primary consideration is the force to eject the headed bolt out of the die chamber. To accurately calculate this force, we need to know the normal force in the die chamber, and the coefficient of friction between the ABS-M30, and the lubricated plasticine. This is not possible without testing, so we have assumed a force of 5N, which is 4* the gravitational force of the M20 bolt.
This force is being exerted by the billet on the conveyor belt. Therefore, we must ensure that the billet does not move on the conveyor belt. Our coefficient of friction at the conveyor belt needs to be 0.68, as the normal force from our billet is 7.3N, and the horizontal force is 5N. This is plausible to achieve with a high friction rubber against plasticine.
The 5N force must be set up by the torque from our motors. The total torque necessary is 0.08Nm, assuming a power transmission coefficient between the belt and motor of 0,7. Because the size difference between the motor-shaft and the conveyor belt-wheel is so big, we can exert a high resulting torque with a low-torque/high-rpm gearbox motor.
Calculation - Cutting Mechanism Force
Our blade has a volume of 98cm^3, and a weight of 790g. We are using a motor to elevate the blade, and cut the plasticine. The total gravitational force on the blade is 7.7N, so our motor should be able to exert a force of at-least 10N, to account for friction in the movement, and other unknown factors. This gives us a total minimum force of 5N per motor, as we are using two dual motors for this movement.
Calculation - Lead Screw Torque Requirement
Looking at the calculations shown to the side, we can see that the torque needed to lift the blade, using a force of 5N per motor is 54.7mNm. The torque needed to lower the blade using the same 5N force per motor, is 21.8mNm. The Stepper Motor that we have chosen has a Holding Torque of 90mNm which is more than what is required for the blade actuation. This motor has a safety factor of 1.644 which is acceptable in this application. Another thing to note is that the Angle of Friction is greater than the Lead Angle, this means that the screw will not back-drive (i.e. not lower on the weight of the load)
Calculation - Bottom Plate Deflection
The goal of this calculation is to approximate the maximum deflection and deformation behavior of the bottom plate during a heading operation. This is important because it leads to misalignment between the dies.
Our calculations show a maximum deflections of 0.6mm at the end of the ASTM A36 steel bottom plate. These locations make sense, since the end of the bottom plate has the largest moment arm, leading to max deflection.
Calculation - Die Misalignment with Bottom Plate Deflection
The goal of this calculation is to approximate the maximum deflection between the upper and lower dies during heading.
Our calculations show a maximum misalignment of 0.20mm. Since this is within our bolt tolerance specification of +/- 0.25mm, we chose to proceed with geometry and materials, however since the maximum deflection is relatively close to the boundary of the tolerance specification, we must monitor and proceed with caution.
Simulation - Full System Heading Deflection
The goal of this simulation is to model the maximum deflection and deformation behavior of our machine during a heading operation.
Our analysis concludes that there are maximum deflections of 1.3mm at the end of the ASTM A36 steel bottom plate. These locations make sense, since the end of the bottom plate has the largest moment arm, leading to max deflection. Based off of the following misalignment simulation, we chose to proceed with the geometry and materials since 1.3mm would be acceptable during operation. This is relatively larger than our calculations showed, however, the fixed support in the simulation better reflects the true deflection, since the bottom plate is not truly rigid.
Simulation - Heading Die Assembly Deflection Under Misalignment
The goal of this simulation is to model the deflection in the ABS dies and the steel alignment rods when the structural components of the machine are at maximum deflection, causing a misalignment in the linear actuator and dies.
Ideally, our force is purely in the horizontal component, however, when building the finite element model, we accounted for a maximum vertical deflection of 1.3mm in the bottom plate. As a result, our force vector was corrected, with a vertical component as well.
Our analysis concludes that there would be a maximum deflection of 0.41mm at the free end of the alignment pins. More importantly, from the plot, we see a maximum relative displacement of 0.05mm between the lower and upper dies. Adding the maximum gap that can occur between the alignment pin and upper die (0.15mm), there would be a maximum misalignment of 0.20mm, which is within our concentricity tolerance of +/-0.25mm. Based off of the maximum misalignment, we chose to proceed with the geometry and materials.
Simulation - Upper Die Strength Analysis
The goal of this model is to analyze the maximum compressive stresses in the upper die during a heading operation. The load that is applied comes from the plasticine that compresses against the die. The pressure that results from the press-fit of the steel sliding inserts are also included in this simulation to ensure that cracking will not occur in the die during insertion.
Our analysis concludes that there is a low probability of failure (Compressive Yield Stress for ABS-M30 is 90 MPa). While there is a maximum stress of -90MPa that is in the model's solution, further analysis points that the stress will not likely happen, since it does penetrate the thickness. Similarly, for the hoop stress analysis that results from the press fit interference, we see that the maximum hoop stress is at 25% of tensile yield stress. As a result, we chose to proceed with ABS M30 for the upper die material.
Stress Factor of Safety: 3.0
Simulation - Lower Die Deflection Analysis
The goal of this model is to analyze the maximum deflection that occurs in the lower die during a heading operation. The load that is applied comes from the upper die exerting a normal compressive force against the lower die.
Our analysis concludes that the maximum deflection will probably not exceed 0.16mm. Stress is also interpreted in the same solution, and the stress does not exceed 10% of ABS yield stress (90 MPa). As a result, we chose to proceed with ABS M30 for the lower die material.
Stress Factor of Safety: 9.0
Simulation - Acrylic Bottom Plate Strength Analysis
The goal of this model is to analyze the maximum stresses in the bottom acrylic plate. The load that is applied comes from the mount reacting the linear actuator heading force. This causes a bending moment on the acrylic plate, which causes stress in the plate.
Our analysis concludes that there is a high probability of failure (Yield Stress of Acrylic is 70 MPa). In our structural analysis, we expect a high stress around the bolt boundary condition, so that is not of concern necessarily. On the other hand, further away from the bolt regions, there is high stress in the acrylic plate (>100% yield stress) body, which points to high probability of failure. As a result, we chose not to proceed with acrylic as the bottom plate material.
Stress Factor of Safety: 1.0
Simulation - A36 Steel Bottom Plate Strength Analysis
The goal of this model is to analyze the maximum stresses in the bottom A36 Steel plate. The load that is applied comes from the mount reacting the linear actuator heading force. This causes a bending moment on the steel plate, which causes stress in the plate.
Our analysis concludes that there is a low probability of failure (Yield Stress for A36 is 250 MPa). In the regions outside of the bolt boundary conditions, the maximum stresses in the aluminum plate are at about 25% of the yield stress. Thus, we chose to proceed with A36 steel as the bottom plate material, rather than Acrylic.
Stress Factor of Safety: 4
Simulation - Blade Deflection Analysis
This model is the same model as the full system, with lower and upper dies hidden. The goal of this is to visualize blade deflection during heading.
This model shows that blade deflection will be low relative to other parts. As a result, we chose to proceed with the blade assembly geometry and components.
Forming Calculations
In order to validate our theoretical heading force, a simulation of the actual heading procedure is in order. Unfortunately, this simulation is very complex to set up and requires specialized metal forming software such as Deform or QForm so we will estimate the validity of our process using a forming limit diagram (FLD)
The Solidworks sketch on the left shows the undeformed billet (purple line) overlaid with the final bolt (black line). We considered what paths the billet could follow as it is deformed to its final state which would lead to the worst case strain states. The 3 paths considered are drawn with red arrows on the left. The tabulated major and minor axis strains for each path is tabulated below:
Using our calculated strains and a forming limit diagram for steel, the plot locations for each respective "worst case" path is drawn out to the left. As evident in the diagram, for each of the three paths considered, the strain state is still in the safe area of the FLD so we can expect the process to carry out successfully.
There are however limitations with this methodology. FLD diagrams are made for sheet metal operations so the material limitations for sheet metal may not be the same as the limitations for a heading procedure. Additionally, we can expect the FLD curves to change with temperature so the curves to the left may not accurately represent the forming limitations for steel at elevated temperatures.
Scaling Down
Initially we thought of basing our prototype machine on the Sutherland 315 metric ton machine. Since we were unable to ascertain the bolt size the Sutherland machine was used to produce, we estimated it to be at least a M50 if not bigger. Since we did not know the exact bolt size produced by the Sutherland machine, scaling down the force proved to be a challenge. We knew that utilizing such a scaling method would not result in an accurate heading force for a M10 bolt. Furthermore, bolt size and Heading Force are not linearly related. Taking all the factors into consideration, we realized that calculating the theoretical Heading Force required for a M10 bolt would give us a more accurate heading force. Thus, we chose to use the Impression Die Forging Equation to solve for a theoretical heading force, assuming White Plasticine.
Scaling Up
Our prototype machine was designed to head bolts made of plasticine. In real life, our machine would be able to make hot-worked steel bolts, and there are some additional considerations needed for our machine to make full scale bolts.
Prototype vs Full Scale Machine Forces
Our prototype machine would form a M10x40 with Allen Socket key. It's important to note how plasticine has similar material properties to steel when it is hot-worked. However, the biggest change between the two models is friction. The heading force required in the full scale model would be larger than that of our prototype machine because of the elevated temperature of steel and difference in die materials, which both add more friction that must be overcome by the machine.
Accounting for Elevated Temperature
When working with steel at elevated temperatures, there will be a relative increase in friction in the dies. Consequently, the tool life will shorten, the metal will flow slower, and more energy will be required to overcome the friction. As a result, we will need to use lubrication to decrease the friction force during the heading process. Other factors that we would also do would be decreasing the surface roughness of the dies through polishing, hardening the die surfaces through heat treatment, and regularly clean the dies with an acid and base.
Prototype Dimensions vs Full Scale Machine Dimensions
For the full scale machine, the stroke length and die geometry would remain the same. However, since steel is much denser, harder, and stronger than plasticine, we would need to change many dimensions and components of our prototype machine.
Stronger Heading Mechanism
Our machine in real life can produce up to 315 metric tons of heading force. As a result, the actuation mechanism we would use would need to be much larger in order to accommodate a larger mechanism with linkages and actuators. This would increase the overall length, width, and height of our machine.
Billet Heating Accommodation
The billet feeder assembly would need to increase in size to accommodate an induction heater to heat the steel up to elevated temperatures. This would also increase the overall length, width, and height of our machine.
Brackets, and Bolts
The real life machine would also be undergoing much more cycles of loading for structural parts throughout the lifespan of the machine. As a result, the frame walls would need to be thickened, brackets would need to be thickened, and bolts should be larger to increase the fatigue life of the machine.
Structural Frame Material
Our real life machine will have higher forces during the heading process. As a result, the frame material should be changed to 4130 steel. This is because we would need a high stiffness and strength material to minimize deformation in the frame and misalignment. We also cannot yield or break the material, which requires high strength. The steel also has a high endurance life, which will increase fatigue life.
Billet Feeding Rollers
Our prototype machine has a rubber conveyor belt that feeds the billet into the machine. However, if we were to feed high temperature steel into this machine, the rubber would melt and be unable to feed steel into the dies. As a result, steel rollers with a high melting temperature would need to be added, increasing the size width and length of our billet feeding mechanism.
Billet Cutting Mechanism
Our prototype machine cutting mechanism would not be adequate to cut heated steel, since our current design would effectively compress and shear the material apart. However, the heated steel would require a bandsaw or circular saw for cutting, because it would remove material with a smoother finish and it would take less time to cut the billet.