Experimental Investigation of Linear Drag Model

Author: Nikoloz Bitchiashvili

Abstract

This study investigates the conditions under which the linear drag model F ​= − bv becomes inadequate for describing the motion of a damped pendulum. Pendulum length of 2.28m was used with three objects of different shapes, and their motion was recording using Tracker. The damping behavior was determined by linear regressions of ln(A) vs t, while Reynolds numbers were estimated from the motion. A 20% change in the fitted damping coefficient relative to low-velocity baseline was used as the criterion for model failure. Under these experimental conditions, deviations from the linear damping model happened at Reynolds numbers between about 221 and 340 on different objects. The results suggest that nonlinear aerodynamic drag effects become significant within this range, although the exact transition depends on object geometry and experimental uncertainty. The results do not establish a universal Reynolds number for the failure of the linear drag model.

Introduction 

The accurate prediction of the motion of objects through air is essential in many areas of physics and engineering. From the design of aircraft and automobiles to the analysis of projectiles and sporting equipment, aerodynamic drag plays a crucial role in determining how objects move. The motion of objects through a fluid is influenced by drag force. In this research, I investigated objects moving through air under the influence of aerodynamic drag. Every single kid in this world has put their hands out of the car window on a family trip and experienced a force against the direction of the moving car. But some have realized that the faster the car goes, the harder it is to keep the hand out the window. Some also realized that how you position the hand against the air also determines how hard the air pushes you back. The research question is: under what conditions does the linear drag model fail to adequately describe the motion? How this question will be answered is not immediately obvious. What we know so far, which is quite intuitive, is that drag force is influenced by projected cross-sectional area, velocity, and shape of the object, which determines the external flow around it. This question will be answered in the context of projected cross-sectional area, velocity, shape, and Reynolds number, which will be explained in the next section.

Theory

Drag

A commonly used phenomenological model combining linear and quadratic drag is

F ​= − bv − cv∣v∣

where v is the signed velocity; the factor v∣v∣ ensures that the quadratic drag force always opposes the direction of motion. In the equation, b and c are proportionality coefficients that determine the strength of linear and quadratic term contributions. This formula combines linear term, −bv, which represents drag from viscous forces, and the quadratic term, −cv|v|, which represents pressure drag associated with the inertia of the fluid and the wake forming behind the moving object. Also, commonly used simplified models of aerodynamic drag are F = −bv and F = −cv|v| which are used to make the calculations easier rather than fighting both terms at the same time while deriving equations. Which model to use is determined by the relative contributions of the forces. Specifically, the F = −bv model would be used when viscous forces dominate, F = −cv|v| would be used when pressure drag dominates, and F ​= − bv − cv∣v∣ would be used when neither component's contribution can be neglected.

Reynolds number

Reynolds number Re is a dimensionless number that describes the flow regime of a fluid. This number describes how the fluid flows in or around an object. This dimensionless number represents the ratio of inertial forces to viscous forces. 

The equation

Re = ρvL / μ

defines the Reynolds number. Here, ρ is the density of the fluid, v is the velocity of the fluid, μ is the dynamic viscosity of the fluid and L the characteristic length. The characteristic length used in Reynolds number depends on the geometry and orientation of the object relative to the flow.

Damped pendulum

A damped pendulum is acted on by both a restoring gravitational force and a dissipative aerodynamic drag force. Starting with Newton's second law in the tangential direction,

ΣF = ma

the forces acting on the pendulum are the restoring gravitational force and the linear drag force. Therefore,

− mg sinθ − bv = mL (d²θ / dt²)

For small angles, sinθ ≈ θ, and since the tangential velocity is v = L (dθ / dt), the equation becomes

− mg θ − bL (dθ / dt) = mL (d²θ / dt²)

Rearranging gives

mL (d²θ / dt²) + bL (dθ / dt) + mg θ = 0

Dividing by mL gives the differential equation

d²θ / dt² + (b / m) dθ / dt + (g / L) θ = 0

Since x = Lθ, the same equation can be written in terms of linear displacement as

d²x / dt² + (b / m) dx / dt + (g / L) x = 0

Solving this differential equation for the underdamped case gives

x(t) = A₀ exp[− b / (2m) t] cos(√(g / L − b² / (4m²)) t)

In our investigation we don't really care about the cosine term, since damping will be really weak (as observed in the experiment) and we are only going to observe amplitudes. Since damping is weak, A(t) = A₀ exp[− b / (2m) t], and subsequently ln(A) = ln(A₀) − (b / 2m) t. This is a linear equation; if we plot ln(A) vs t we can easily find the slope, which can be used to calculate b, the damping coefficient. 

Experimental methodology

For the experiment, I have used three pendulum bobs of different shapes and cross areas which are described below.

Bob	Dimensions	Cross area	Mass	Characteristic length
Sphere	diameter = 1.422 cm	1.561 cm²	15.32g	1.422cm
Cylinder	length = 4.853 cm; diameter = 0.734 cm	3.676 cm²	24.73g	0.724 cm
Weird-shaped object	length = 4.186 cm; max width = 1.358 cm; min width = 0.728 cm	4.275 cm²	37.77g	1.043 cm

The three pendulum bobs used in the experiment: spherical, cylindrical, and weird-shaped objects, shown alongside a measuring tape for scale:

Experimental setup and methodology

My goal in this experiment was to essentially make the F = −bv model fail. This would happen when the ln(A) vs t graph would not be linear (why this happens is explained later on). My approach was therefore to reach sufficiently high speeds that departures from the linear-drag approximation could become detectable.

The first setup was smaller. I had put white cloth in the background to obtain better frames of the bob but after having multiple trials with a 1m pendulum, releasing different bobs from about 10 degrees, and analyzing data, it showed no indication of the quadratic term ever being significant. Finally, I used all the remaining fishing line, and moved the setup away from the wall so I could have higher release angle. The final setup turned out to be a pendulum with a length of 2.28m and release angle about 38 degrees. I'll state here that release angle was 38 degrees and the sinθ ≈ θ assumption has quite a large error at this value. However, this did not matter much since the significance of the quadratic term appeared at smaller angles. The camera was placed about 180cm away, but distance from the camera to the plane of motion often changed and was recalculated. Since the pendulum moved at high speeds and camera couldn't capture the bob every time. I therefore analysed every second turning point, corresponding to successive peaks on the same side of the pendulum, using Tracker. For example, if the coordinates of the bob were 1.3, −1.25, 1.2, and −1.15 than only −1.25 and −1.15 were written down since the entire setup couldn't be fit in the frame (even if it did, the bob would be very tiny in the frame and probably invisible). I had two variables from Tracker: coordinate and time. The amplitude used in the analysis was the magnitude of the measured displacement, A = ∣x∣. According to the formula ln(A) = ln(A₀) − (b / 2m) t, all I had to do was plot ln(A) vs t and see where the slope changed by about 20% from the baseline (the baseline was at lower amplitudes, I chose 20% as failure of the model which will be explained later). The percentage of deviation was calculated using the formula S = (b − b(baseline)) / b(baseline) * 100.

Results 

RMS velocity​ = root-mean-square velocity within each analysis interval

MRN​ = Reynolds number calculated using RMS velocity (v)​ for that interval

Sphere

Interval	Slope	R²	Damping Coefficient (kg/s)	RMS Velocity (m/s)	Status	MRN / Deviation from baseline
1 - 10	-0.01517680983	0.9987	0.00046461745	1.747	FAILED	1681 / 215.9%
11 - 20	-0.01193602291	0.9994	0.00036571974	1.129	FAILED	1087 / 148.7%
21 - 30	-0.00998003992	0.9997	0.00030578842	0.805	FAILED	775 / 107.9%
31 - 40	-0.00866551126	0.9998	0.00026551006	0.604	FAILED	581 / 80.5%
41 - 50	-0.00763358778	0.9997	0.00023348913	0.472	FAILED	455 / 58.7%
51 - 60	-0.00662690523	0.9944	0.00020304758	0.376	FAILED	362 / 38.1%
61 - 70	-0.00638162093	0.9928	0.00019553284	0.310	FAILED	298 / 32.9%
71 - 80	-0.00566893424	0.9988	0.00017369735	0.258	FAILED	248 / 18.1%
81 - 90	-0.00534188034	0.9986	0.00016327442	0.218	ACTIVE	210 / 11.0%
91 - 100	-0.00481463649	0.9956	0.00014708046	0.188	ACTIVE	181 / 0.0%

Cylinder

Interval	Slope	R²	Damping Coefficient (kg/s)	RMS Velocity (m/s)	Status	MRN / Deviation from baseline
1 - 10	-0.01374192661	0.9980	0.00067967569013	1.683	FAILED	824 / 187.2%
11 - 20	-0.01032204789	0.9992	0.00051052848863	1.144	FAILED	561 / 115.7%
21 - 30	-0.00841042893	0.9995	0.00041597981	0.853	FAILED	418 / 75.8%
31 - 40	-0.0071942446	0.9999	0.00035582733	0.671	FAILED	329 / 50.4%
41 - 50	-0.00636942675	0.9984	0.00031503184	0.543	FAILED	266 / 33.1%
51 - 60	-0.00589970501	0.9999	0.0002917994	0.451	FAILED	221 / 23.3%
61 - 70	-0.00516262261	0.9976	0.00025534331	0.380	ACTIVE	186 / 7.9%
71 - 80	-0.00486854917	0.9995	0.00024079844	0.327	ACTIVE	161 / 1.8%
81 - 87	-0.00478468899	0.9991	0.00023665071	0.291	ACTIVE	143 / 0.0%

Weird shape

Interval	Slope	R²	Damping Coefficient (kg/s)	RMS Velocity (m/s)	Status	MRN / Deviation from baseline
1 - 10	-0.01012760785	0.9982	0.00076504	1.780	FAILED	1256 / 230.6%
11 - 20	-0.00793021411	0.9991	0.00059905	1.329	FAILED	937 / 158.9%
21 - 30	-0.00670690811	0.9996	0.00050664	1.054	FAILED	744 / 119.0%
31 - 40	-0.00564334085	0.9986	0.00042630	0.867	FAILED	612 / 84.2%
41 - 50	-0.00515463917	0.9974	0.00038938	0.732	FAILED	516 / 68.3%
51 - 60	-0.00474833808	0.9998	0.00035829	0.628	FAILED	443 / 54.8%
61 - 70	-0.00428816466	0.9973	0.00032352	0.547	FAILED	386 / 39.8%
71 - 80	-0.00389711613	0.9989	0.00029479	0.482	FAILED	340 / 27.4%
81 - 90	-0.00344352617	0.9986	0.00026072	0.429	ACTIVE	303 / 12.7%
91 - 100	-0.00353857041	0.9979	0.00026790	0.383	ACTIVE	270 / 15.8%
101 - 110	-0.00333444481	0.9981	0.00025198	0.346	ACTIVE	245 / 8.9%
110 - 129	-0.00306184935	0.9985	0.00023139	0.307	ACTIVE	216 / 0.0%

Where the linear model failed

The linear drag model was considered to have failed when the slope of the ln(A) versus t graph changed by about 20% from the initial slope. For the sphere, the first significant change occurred around the interval corresponding to a Reynolds number of approximately 298. For the cylinder, the corresponding Reynolds number was approximately 221, while for the weird-shaped object it was approximately 340.

These results show that the linear drag model did not fail at exactly the same Reynolds number for all three objects. Under the conditions of this experiment, failure was observed at Reynolds numbers between approximately 221 and 340. 

Discussion / Conclusion

Limitations / sources of uncertainty

First I want to say why the failure of the linearly-damped model of the pendulum means failure of the linear drag model itself. The damped pendulum model is essentially the combination of two models: the ideal pendulum where the theory holds for small angles and the linear damp model where we neglect the quadratic term. In the investigation we can see that the combined model failed at about 5-9 degree angles where the error of the sinθ ≈ θ approximation is not even 0.5%. So by the 20% criterion, failure of the combined model means failure of the linear model (theoretically, not taking the experimental uncertainties into account yet).

The 20% criterion was selected as a conservative threshold for identifying a meaningful change in the fitted damping coefficient. The maximum relative standard error of the fitted slopes was approximately 3.1%, while additional uncertainty was introduced by manual tracking of the pendulum bob. Therefore, a 20% change was chosen to provide a substantially larger margin than the observed regression uncertainty and to reduce the likelihood of interpreting small measurement fluctuations as evidence of a significant quadratic-drag contribution.

Also, it is important to emphasize the uncertainty of the manual tracking. Since the bob was hardly visible to the program for it to automatically track it, I chose to manually track the center of each bob. Of course, after tracking over 500 points, I made some mistakes, including failed trials, sometimes because of non-planar motion and sometimes because the phone kept skipping frames due to low battery. Even when zooming in on the bob, there was still some uncertainty. I would estimate this uncertainty as ±2 pixels, which translates to ±1.81mm. This uncertainty, even for the smallest x-coordinate observed, corresponds to only about a 1% error. The fishing line also introduced a small additional source of uncertainty. The line had a diameter of approximately 0.5mm and therefore contributed some aerodynamic drag of its own. These effects were not independently quantified.

A 10-point window was selected for the linear regressions as a compromise between fitting reliability and temporal resolution. Using too few points would make the fitted slope more sensitive to individual tracking errors, while using substantially more points would combine data from a wider range of velocities and potentially obscure changes in the damping behavior.

I chose RMS velocity of the motion to calculate the Reynolds numbers. Calculating Reynolds number in this manner is a source of error itself. Where I interpreted the model failed, the exact deviation from the baseline was not exactly 20%, but it ranged from 18.1% to 27.4%.

Lastly, even though it doesn't matter for the investigation, I want to point out that high Reynolds numbers in the tables are not completely reliable since the sinθ ≈ θ assumption can contribute significant error at the higher angles. Another source of error is my own measurements. I tried to keep the camera parallel to the plane of motion; I measured all the distances from the camera to the plane of motion, and the length of the rope, but of course I probably had made some small mistakes there which was shown even in the calibration of the Tracker. Since the camera had to be moved sometimes, the distance from the camera to the plane of motion had to be recalculated and Tracker had to be re-calibrated. For the sphere, Tracker identified the release angle as 38 degrees whereas for the weird-shaped object it identified it as 36 degrees even though the release angle was the same. The damping effect of the fishing line was not considered since it was extremely thin and its effect was not calculated since it operated at very low Reynolds numbers, where drag is assumed linear. Of course, this introduces another error. Cross-sectional areas and characteristic lengths were calculated using ImageJ in which, again, I probably made mistakes while marking the boundaries of the objects by hand to calculate their cross areas. 

Characteristic length and projected area

A comparison of the objects suggests that characteristic length may have a stronger influence on the Reynolds number at which the linear model fails than projected cross-sectional area. When comparing the cylinder and sphere, characteristic length increased by approximately 1.97x, while cross-sectional area decreased by approximately 2.35x, and the failure Reynolds number increased by approximately 1.12x. When comparing the sphere and the weird-shaped object, characteristic length decreased by approximately 1.36x, while cross-sectional area increased by approximately 2.74x, yet the failure Reynolds number increased by only 1.37x.  This observation is suggestive rather than conclusive, since characteristic length and cross-sectional area were not independently controlled in this experiment.

Implications / Interpretation

The results of this investigation show that, under the experimental conditions tested, the linear drag model F ​= − bv reached the predefined failure criterion at Reynolds numbers between 221 and 340. This does not establish a universal Reynolds Number above which the linear drag model is always inadequate. Rather, it identifies a range in which failure of the model was experimentally observed for the objects investigated. The experiment therefore provides evidence that the quadratic component of aerodynamic drag can become significant within this Reynolds-number range. At higher Reynolds numbers, including values above 1000, the linear model continued to exhibit much larger deviations (but with bigger error) from the expected behavior according to the same failure criterion.

Software and Resources Used

• Website used for linear regression was graphpad.com.

• Videos were analyzed using the program Tracker.

• Objects were measured using the program ImageJ.

Appendices

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