In this post we discussed the use of the Taylor Series to evaluate the propagated uncertainties in measurements to a calculated result. We can also use a similar approach to determine the sensitivity of a transfer function to know perturbations in the independent variables.

In this case we will look at the volume of a right conical frustum or a truncated cone. The volume can be calculated from the three variables: , and shown in the figure. The volume is given as

The first step is to calculate the partial derivatives of with respect to the independent variables.

The total uncertainty in the volume, , is given in terms of the uncertainties in the independent variables, , and .

Each term of the form is the contribution to the total perturbation of the function by the perturbation of the independent variable, . Therefore we can calculate the sensitivities, , of the volume to small changes in each of the three variables , and , by calculating the percentage contribution of each term to the total perturbation. Typically, we wish to also define the direction that the pertubations in the independent variables will affect the total, so we remove the absolute values and evaluate the partial derivatives while maintaining their signs.

It is also evident that .

Example:

Suppose we have a design for a conical frustum with nominal dimensions of 0.500 in, 0.375 in and 1.250 in. Our manufacturing process can hold to ±.002 in, to ± .007 in and to ± .010 in. We then have the following inputs to our formulae.

From this we can calculate the nominal volume of our manufactured part.

Next we can calculate the partial derivatives.

And having the partial derivatives, we can calculate the total propagated uncertainty in the nominal volume.

From this we can obtain the predicted volume of the manufactured part subject to the manufacturing tolerances: 0.757 ± .021 in^{3}. We can then calculate the relative sensitivity of the total volume to the manufacturing tolerances specified.

It is important to note that these results are only valid for the initial conditions specified. If the nominal values for the dimensions or the tolerances change, both the uncertainties and sensitivities will change.

Caveat lector — All work and ideas presented here may not be accurate and should be verified before application.