1. Problem Statement: Volume of a Cylinder¶

This example follows the video tutorial on YouTube: Uncertainty Propagation Example: Volume of a Cylinder by Dr. Joshua Paul Steimel.

A tube of circular cross-section has a nominal length ($L$) of $50 \text{ m} \pm 0.5 \text{ m}$ and a radius ($R$) of $15 \text{ m} \pm 0.1 \text{ m}$. Determine the total uncertainty in the volume.

1.1 Given Nominal Values (Mean Values)¶

The nominal values (mean values, denoted $\bar{R}$ and $\bar{L}$ in standard notation, or $r$ and $l$ in the equations) are:

1.2 Given Uncertainties ($\mu$ or $u$)¶

The stated ± values are the Bias Uncertainties (or standard uncertainty $u$ if a coverage factor is not explicitly used, as is often the case in this type of problem, represented as $u_r$ and $u_l$):

1.3 Total Uncertainty Equation¶

The total uncertainty ($\omega_V$) is the root sum of the squares of the Bias Uncertainty ($B_V$) and the Precision Uncertainty ($P_V$):

As explained in the video (0:09, 2:35), since there is no information given about the number of samples or standard deviation, the Precision Uncertainty ($P_V$) is zero.

Therefore, the Total Uncertainty ($\omega_V$) is equal to the Bias Uncertainty ($B_V$):

2. Methodology: Uncertainty Propagation (Bias Error $B_V$)¶

The Volume ($V$) of the cylinder is a function of two independent variables, radius ($r$) and length ($l$).

The Bias Uncertainty ($B_V$) is calculated using the standard method for uncorrelated variables:

2.1 Calculate Partial Derivatives (Sensitivity Coefficients)¶

We first calculate the partial derivatives of the volume equation with respect to each independent variable. These are the sensitivity coefficients ($c_i$):

1. Sensitivity with respect to Radius ($c_r$):

   $$c_r = \frac{\partial V}{\partial r} = \frac{\partial}{\partial r}(\pi r^2 l) = 2 \pi r l$$

   (9)

2. Sensitivity with respect to Length ($c_l$):

   $$c_l = \frac{\partial V}{\partial l} = \frac{\partial}{\partial l}(\pi r^2 l) = \pi r^2$$

   (10)

2.2 Numerical Calculations for Total Uncertainty ($B_V$)¶

Now, we plug the nominal/mean values ($\bar{r}=15 \text{ m}$, $\bar{l}=50 \text{ m}$) into the partial derivatives to find the numerical sensitivity coefficients, and then calculate the total uncertainty $B_V$.

import numpy as np

# Nominal (Mean) Values
r = 15.0  # meters
l = 50.0  # meters

# Uncertainties (u_i or mu_i)
u_r = 0.1  # meters
u_l = 0.5  # meters
pi = np.pi

# 1. Calculate Nominal Volume (V)
V = pi * r**2 * l
print(f"Nominal Volume (V): {V:.3f} m^3")

# 2. Calculate Sensitivity Coefficients (c_r, c_l)
c_r = 2 * pi * r * l
c_l = pi * r**2

# 3. Calculate Squared Uncertainty Contributions (c_i * u_i)^2
term_r_squared = (c_r * u_r)**2
term_l_squared = (c_l * u_l)**2

# 4. Calculate Bias Uncertainty (B_V)
B_V = np.sqrt(term_r_squared + term_l_squared)

print(f"\nSensitivity Coefficient for Radius (c_r = 2πrl): {c_r:.3f}")
print(f"Sensitivity Coefficient for Length (c_l = πr²): {c_l:.3f}")
print(f"\nBias Uncertainty (B_V) in Volume: {B_V:.3f} m^3")

The calculated Bias Uncertainty ($B_V$) is approximately $589.845 \text{ m}^3$, which is the Total Uncertainty ($\omega_V$) for this problem.

3. Alternative Method: Fractional (Relative) Uncertainty¶

The video also suggests calculating the fractional uncertainty ($\frac{B_V}{V}$) first, as it helps with units consistency (7:11). The result is a unitless fraction (or percentage).

The fractional uncertainty equation is:

3.1 Simplified Fractional Terms¶

By dividing the partial derivatives by the volume formula $V = \pi r^2 l$, the terms simplify significantly:

1. Fractional Sensitivity for Radius:

   $$\frac{1}{V} \frac{\partial V}{\partial r} = \frac{2 \pi r l}{\pi r^2 l} = \frac{2}{r}$$

   (12)

2. Fractional Sensitivity for Length:

   $$\frac{1}{V} \frac{\partial V}{\partial l} = \frac{\pi r^2}{\pi r^2 l} = \frac{1}{l}$$

   (13)

3.2 Numerical Calculation of Fractional Uncertainty¶

The new unitless formula is:

# 1. Calculate Fractional Uncertainty (Unitless)
frac_r_squared = (2 * u_r / r)**2
frac_l_squared = (u_l / l)**2

frac_B_V = np.sqrt(frac_r_squared + frac_l_squared)

print(f"Fractional Uncertainty (B_V/V): {frac_B_V:.6f}")

# 2. Check: Reconvert to absolute uncertainty (B_V = V * (B_V/V))
B_V_check = V * frac_B_V

print(f"Absolute Uncertainty Check (V * (B_V/V)): {B_V_check:.3f} m^3")

The fractional uncertainty is approximately 0.016667, or $1.67\%$. Multiplying this by the Nominal Volume ($35342.9 \text{ m}^3$) yields $589.048 \text{ m}^3$, confirming the result from the direct calculation method.

4. Final Uncertainty Budget Summary¶

The total uncertainty in the volume of the cylinder is:

or in fractional terms: