Tutor profile: Tomas C.
Subject: Mechanical Engineering
How do we dimension steel helical gears?
As with straight gears, we use a variation of the lewis equation and the contact equation in a iterative fashion to converge to a proper gear design. Helical gears have smoother contact between their teeth, and as such are better suited to high-rev applications, which is reflected in the different values of the lewis factors when compared to the values we get for straight gears. The procedure itself is quite simple, it basically boils down to taking educated guesses of the gear dimensions (Within the applicable norms and standards), then using the lewis and contact equations to check what sort of steel you would need to build the gear you selected out of to get the desired specs (Torque, life, etc). This is usually checked in a graph that compares the lewis equivalent stress (or contact stress) to the hardness-vs-stress profiles of the available steel grades. If the steel you would need is too expensive (or straight-up impossible), or if your gear is overdimensioned (You get a data point below the steels minimum values), then you use that information to adjust your design by making things bigger, smaller, bulkier or whatever is needed. Then you repeat until satisfied.
Why do we do Principal Component Analysis?
With most statistical data sets the main objective is to search for commonality (of lack thereof) between the data points so we can extract the trends. PCA addresses the problem of what to do when we have a data set where we suspect different components aren't completely unrelated. It is a tool that allows us to compress data into only the number of relevant degrees of variation, thus making the analysis easier. An example: Say we have a set of a 100 people and we take their height, weight, age and BMI. This 4-dimensional data set does not necessarily have 4 completely independent variables as there is a real possibility some variables will change together in some sense. PCA allows us to put this to the test and find exactly how many underlying variables are under the hood of that 4-D data set.
Subject: Aerospace Engineering
How does lift vary with the Mach number in the subsonic regime?
For a conventional airfoil, the growing effect of the compressibility as the Mach number increases in turn contributes to increasing the lift coefficient. It can be seen in experimental data that as the airspeed gets faster and faster, the sectional lift goes above the expected linear value of 0.5*Rho*V^2*cl, in a way inversely proportional to sqrt(1-M_inf^2), known as the Prandlt-Glauert rule after the original discoverers. However, Prandlt incorrecly assumed that this rule would hold for all values of Mach's number, coming to the conclusion that lift would be infinite at the speed of sound (And snap the wings off). In reality, as the regime becomes transonic, the interaction between the shockwaves and the boundary layer starts to affect the lifting properties of the airfoil, with the cl-vs-mach curve behaving in an oscillatory fashion until near Mach 1, where the shockwaves more or less dominate the airflow, making the flow around the relatively thin airfoil (compared to the shockwave profile) almost symettrical, thus making the lift plummet to near zero.
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