I am reverse engineering custom software for a stepper motor. The original software eases in and out of any motion, and the duration of the ramping up to speed is directly related to the speed that the motor is ramping up to.
In other words, when the motor is ramping up to a high speed, the duration of the ease-in is short / it eases in to the high speed quickly. When the motor is ramping up to a slow speed, the ease-in is long / it eases in to the slow speed slowly.
Here are some samples, with the caveat that there is potential for a .030 second margin of error in the ease-in durations...
Hz duration of ease-in
324 1.139
390 1.134
403 1.167
410 1.1
423.4 1.1
693.5 0.766
1040 0.567
1134 0.567
1480 0.434
How can I find the relationship between Hz and the ease-in duration, so that I can derive the proper ease-in duration from a given Hz?
As already said in comments, it could be good to have more data points.
It is not sure that a linear model is the most appropriate. You effectively obtained $$y=1.374 -0.000697369 x$$ to which correspond a sum of squares equal to $0.0381$ and $R^2=0.9951$.
Using this model, what would happen when $\text{Hz} > 2000$ ? Is a negative duration of the ease-in possible to consider ?
Looking at the scatter plot of the data, we could also think about an hyperbolic model (this is again a linear regression) $$y=0.268936 +\frac{328.735}{x}$$ to which correspond a sum of squares equal to $0.0362$ and $R^2=0.9954$.
Update
Using all the data you stored here : a scatter plot of $y$ as a function of $x$ clearly reveals that the model is nonlinear. At the opposite, plotting $xy$ as a function of $x$ clearly shows an almost linear trend. Then, the idea of an hyperbolic model as suggested earlier. Using all the data points, a least square fit leads to $$y=0.269704 +\frac{342.283}{x}$$ to which correspond a sum of squares equal to $0.1028$ and $R^2=0.9970$.
The following table reports your data and the fitted values $$\left( \begin{array}{ccc} 128.2 & 2.934 & 2.940 \\ 173.5 & 2.167 & 2.243 \\ 271.7 & 1.600 & 1.529 \\ 312.0 & 1.366 & 1.367 \\ 324.0 & 1.139 & 1.326 \\ 370.0 & 1.300 & 1.195 \\ 390.0 & 1.134 & 1.147 \\ 402.0 & 1.200 & 1.121 \\ 403.0 & 1.167 & 1.119 \\ 410.0 & 1.100 & 1.105 \\ 416.0 & 1.200 & 1.093 \\ 423.4 & 1.100 & 1.078 \\ 423.4 & 1.167 & 1.078 \\ 441.0 & 0.967 & 1.046 \\ 693.5 & 0.766 & 0.763 \\ 925.0 & 0.667 & 0.640 \\ 1040 & 0.567 & 0.599 \\ 1134 & 0.567 & 0.572 \\ 1190 & 0.566 & 0.557 \\ 1195 & 0.533 & 0.556 \\ 1200 & 0.566 & 0.555 \\ 1387 & 0.500 & 0.516 \\ 1480 & 0.434 & 0.501 \\ 1782 & 0.400 & 0.462 \end{array} \right)$$
which seems to be quite good.