How units of measurement in TechEditor helped to find an error in a scientific article

How units of measurement in TechEditor helped to find an error in a scientific article

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Today, I will share how the use of units of measurement in TechEditor helped me uncover a serious error in a scientific article.

Specifically, I am referring to the algorithm outlined in the publication [1] Arial Bombing Techniques (C. Pepper & C. Wilson, 2009, 11 pages), which was utilized to generate a calculation sheet for determining the coordinates of a target for a military drone.

As with other similar approaches, this algorithm relies on formulating differential equations of motion for a solid object, taking into account resistance forces. These differential equations are then integrated using analytical methods, which yield calculation formulas for determining the coordinates, velocities, and accelerations of the object. Let's delve deeper into this approach.

The Law of Projectile Motion

When a drone or any other unmanned aerial vehicle (UAV) moves at a speed of \( v \)  and an altitude of \( h \), it carries a concentrated explosive device, which can be referred to as a projectile.

The location where the drone releases the projectile is marked as point A, and the destination where the target is hit is marked as point B. By drawing a vertical plane that passes through these two points and placing a rectangular Cartesian coordinate system with the origin at point O, the plane motion of the system can be studied to determine the trajectory of the projectile:

Analytical model

Formulas for Coordinates

Upon release, the projectile begins to fall freely while the drone is still moving at a constant speed, resulting in an initial impulse that affects the projectile's trajectory. To determine the projectile's trajectory, we require both the vertical \( y \) and horizontal \( x \) coordinates, which can be computed using the following formulas [1]:

\( y(t) = y_0 + \frac{ g }{ k } t - \frac{ g }{ k^2 } \left( 1 - e^{ -k t} \right) \); (1)

\( x(t) = \frac{ v }{ k } \left( 1 - e^{ -k t} \right) \), (2)

where

  • \( y_0 \) — the height of the drone;
  • \( g \) — acceleration due to gravity (9.81 m/s2);
  • \( k \) — coefficient;
  • \( v \) — drone speed;
  • \( t \) — time point at which we determine the coordinates.

In the original publication [1], these analytical expressions (1) and (2) are presented without units. While this allows for the use of arbitrary units, it can also be a source of errors.

Implementation in TechEditor

Model 1 (Incorrect, No Units)

Initially, when we entered the algorithm into TechEditor, it lacked any units of measurement. This is a typical approach, where we assume SI units if no specific recommendations or explanations are provided for the parameters.

Here's how the algorithm appears in the TechEditor diagram:

Analytical model

Based on this model, the coordinates are calculated automatically, without any checks. This process involves pure mathematics, i.e., manipulating numbers, which implies that we will ALWAYS get some results. However, it's difficult to determine whether these numbers are accurate or not.

Model 2 (Incorrect, With Units of Measurement)

After the first iteration, we progressed to the next one, which entailed incorporating units of measurement into the calculation model:

Analytical model

It's worth noting that TechEditor will signal an error in the message bar:

Analytical model

How TechEditor works with physical quantities

TechEditor has a specific way of working with physical quantities. It follows standard mathematical rules for normal calculations but gives priority to physical quantities, which have units of measurement. In the case of an expression with physical quantities, the calculation revolves around those quantities. Any parameters without units are considered coefficients or prime numbers.

When TechEditor encountered the model with units of measurement, it signaled an error because of a mismatch between the units. It circled the problematic block with a red frame and displayed the message "Units must have equal dimensions. L<>L^2/T". This error message means that there is a mismatch between the expected and actual units of measurement. Here, "L" represents a length measurement in meters, and "T" represents a time measurement in seconds. The message indicates that the expression is trying to add a parameter to the length measurement (m), but the unit of measurement for that parameter is m2/s.

Upon further investigation, it was discovered that the problem was with the coefficient \( k \) in the formula.

Aerodynamics

The coefficient \( k \) used in the calculation model from [1] takes into account the aerodynamic drag forces acting on the projectile. It is calculated using the following formula:

\( k = \frac{ D \cdot \rho \cdot A }{ 2m } \), (3)

where

  • \( D \) — coefficient;
  • \( \rho \) — air density, kg/m3;
  • \( A \) — projectile contact surface area, m2;
  • \( m \) — body weight, kg.

However, for a more accurate calculation of aerodynamic drag, the drag coefficient \( \frac{ D \cdot \rho \cdot A }{ 2 } \) should be used. It is determined by the formula:

\( D = \frac{1}{2} C_d \cdot \rho \cdot v^2 \cdot A \), (4)

where

  • \( C_d \) — drag coefficient (dimensionless);
  • \( v \) — velocity of the body, m/s.

There are several sources that provide information on formula (4) and its parameters:

Now let's turn our attention to the fragments in publication [1] highlighted in yellow:

bug

As we can see, \( R \) in formula (7) is the drag force, which contains the square of the velocity \( v \). This formula is correct, it completely coincides with formula (4) from the above external sources. However, below, in the list of forces acting on the body, the authors of [1] introduce a separate resistance force \( F_d \), which should be identical to the drag force \( R \), but its expression contains the speed not in the second but in the first power. This is contrary to logic and indicates a likely error.

If you add the resistance force \( F_d \) with the square of the velocity, its expression will be:

\( F_d = \frac{1}{2} D \rho A v^2 \). (5)

Then the transformation (8) in publication [1] should look like this:

\( m a = m g - \frac{1}{2} D \rho A v^2 \);

\( a = g - \frac{ D \rho A v^2 }{ 2m } \), (6)

and the coefficient \( k \), respectively

\( k = \frac{ D \cdot \rho \cdot A \cdot v }{ 2m } \). (7)

Model 3 (correct, with units)

After making the necessary corrections to our calculation model by adding units of measurement, we were able to successfully calculate the coordinates of the projectile impact using TechEditor.

Analytical model

Conclusions and warnings

This case study highlights a systemic problem with physical calculations, where the lack of units of measurement can result in erroneous results, as demonstrated in Model 1. It is important to treat such results with a certain amount of distrust until everything is thoroughly checked.

Furthermore, when using spreadsheets or other similar solutions for physical calculations, it is crucial to ensure that all physical parameters in all formulas are consistent with one another. Using unverified solutions from the internet may result in incorrect results.

Therefore, it is recommended to use specialized software, such as TechEditor, for mathematical analysis and simulation of physical processes. In doing so, the software can provide additional assistance and alert users in cases where physical laws and correlations are violated.

Good luck!

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Vitalii Artomov

Dystlab (CEO & Co-founder), ex-associate professor of the Bridges Department (DNURT), TechEditor developer, PhD.

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