Originally Posted by
RChung
Ah. Well, I wish I could scan in my derivation but I can't. The "T" in your NP, and the basis for your claim that TSS varies with sqrt(T)? Um, that's not quite right. It's easier to see if you flip into the discrete formulation (since we have to use the vector norm anyway). You're taking the mean of N objects where, if T is time and dt is the time interval, N = T/dt. That extra dt is going to have units as time so N is a scalar, so NP/FTP = IF is a scalar.
It is no "claim". It is simply an observation. My calculation is correct. You are hung up on units. And I am not sure how you are seeing vectors in this. Power, time, ride time, etc.; all these variables are scalars. Nothing is a vector in this equation. Now then, IF is unitless, if that is what you a trying to say. That is true. Unitless variables can be functions of variables with units; that is certainly allowed. To take an example off the top of my head, Reynolds Number is clearly a function of fluid velocity, which might also be a function of time, and it is unitless by design.
Go over the math again. Regardless of units, and you can replace the integration with a summation if you like (the integral does not get modified in the derivation), it is pretty clear that you get a leftover T^1/2 in the final results. This is why TSS seems to change with the amount of rest you include in your session. It's because it does. The final equation actually has ride time, a given constant, as a term in the equation.
If you need experimental validation, try it yourself. Take a data set, add a bunch of zeros to the time period, and do the calculation using both the TSS and the IF^4 model. You will find that TSS increases proportionally to the sqrt(T), and the IF^4 formulation does not (obviously if you add a bunch of zeros to an integration, it does not change the value of the integral).