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Calories and Power in Exercise Equipment

Determine the thickness of resistance bands

Resistance bands have been used since the early 1900’s as an exercise tool, and are now also widely used as a rehabilitation tool to strengthen (injured) muscles and joints. They became popular due to their affordability and convenience. Just like weight training, they can increase muscle strength, endurance and flexibility. When designing a resistance band, it is important to determine several aspects, like what material to use, and how long, thick and wide to make the bands.

Image result for resistance band with handles

Figure 1. Commercially available resistance bands.


Image result for chest exercise resistance band

Figure 2. Chest exercise performed with resistance band.








The question we will consider here is what the appropriate dimensions are of resistance bands so that they are able to provide the appropriate resistance forces. It is important to be able to produce several bands that gradually increase the force needed to stretch them for several reasons. For example, strength varies between individuals and between different muscle groups within an individual so need different resistance levels are needed, and people (re)building strength after an injury need to be able to gradually rebuild with different resistance levels. Here, we will focus on circular resistance bands, and therefore the specific variable that will be determined is the radius of the resistance band.

A resistance band is usually made of natural rubber, as this provides enough strength while still stretching with the user’s movements.  Natural rubber behaves as a linear elastic material, with a Young’s Modulus (E) between 1 and 5 MPa. As you can see in Figure 3, the Young’s Modulus is a measure of stiffness for a material’s elastic regime. It represents the ratio of tensile stress over strain, where tensile stress is the tension force applied on a surface area and strain is the amount of deformation (amount of stretch of the resistance band).

Figure 3. Tensile stress-strain curve.

We will simplify the problem by assuming this is a simple an exercise as shown in Figure 2, and the resistance felt is only coming from the band itself. We also simplify the situation by assuming a constant position of the stretching band in front of the user’s body, so that the increase in resistance felt is for the same muscle group, and coming from the band properties only. See Figure 4 for a free body diagram of the resistance band.

Figure 4. Free body diagram showing the forces acting upon the resistance band when performing the exercise shown in Figure 2.


  • Natural rubber behaves as a linear elastic material, such that E = σ/ε
  • Tensile stress,  σ = F/A
  • Strain, ε = ΔL/L
  • Circular resistance band with
    • Length, L
    • surface area, A=π*R2 (where R=radius)
  • Stress applied is below yield strength

We will estimate the following values:

  • E = 2.5 MPa = 0.0025 N/mm2  (average value (3))
  • ε=1.5 and ε=3 (we determine the force needed to stretch the band 150% and 300%)
  • Resistance force/force applied to band: 1lb, 3lb, 7lb, 9lb, 10lb, which equals 4.4N, 13.3 N, 22.2N, 31.1N, 40N, 44.5N.

We can now solve for the radius (so the thickness) of the band, using the following steps:

  1. E = σ/ε
  2. Rewrite σ = E*ε
  3. Plug in the values for E and ε=1.5 you get σ = 0.00375 N/mm2
  4. Since σ=F/A, we can rewrite this as A = F/0.00375
  5. We can now plug in the different values for F and find the surface area, A
  6. Since A=π*R2, we can solve for R using R=sqrt(A/π)
  7. Similarly, for ε=3 you obtain values for R by using A = F/0.0075

Plugging in these values in the equations gives the following results:

F (lb) F(N) Equation [ε=1.5] A (mm2) [ε=1.5]

R (mm) [ε=1.5]

to stretch 150%

R(mm) [ε=3]

to stretch 300%

1 4.4 A = 4.4/0.00375 1173 25 14
3 13.3 A = 13.3/0.00375 3547 34 24
5 22.2 A = 22.2/0.00375 5920 43 31
7 31.1 A = 31.1/0.00375 8293 51 36
9 40 A = 40/0.00375 10666 58 41
10 44.5 A = 44.5/0.00375 11867 61 43

According to the above calculations, the radius of a resistance band made of natural rubber could range from 25 mm to 61 mm in order to create resistance forces ranging from 1 to 10 lb for stretching 150%, and 14 mm to 43 mm for stretching 300%. Intuitively these values seem high. I have personally never seen tubular resistance bands with a radius much larger than 7mm, but according to the calculations above twice as large diameter (14mm) is needed only to create a 1lb force to stretch the band 300%. If we compare these values to the forces created by the commercially available Thera-Band®, a resistance band made of natural rubber with a radius around 40 mm would create similar forces as the red band (which is the second strongest of the 5 colored bands).

It seems fair to say that the solved values are not reasonable, considering the forces produced by existing Thera-Band® and the dimensions of commercially available tubular resistance bands. I think the solution here is limited due to several simplifications. We assumed that the force to stretch the band is applied to the end of the tubular stretching band. However, if a band is directly grabbed by a person, it is a more complex situation as he/she will probably use their hands to grab the band over a larger surface than estimated above. It seems like the resistance band cannot be modeled as a tubular piece of solid material on which a stress is applied to the end surfaces, as shown in Figure 4. Another aspect that is not taken into account here, is that the different resistance bands can also be altered in design in “how hollow vs. solid” they are, in other words how dense the natural rubber is packed. The more dense it is packed, the more resistance it will provide as more fibers in the band need to be stretched.

In conclusion, the approach provided above does not seem to provide reasonable answers to the question what the radius of tubular resistance bands should have in order to provide certain resistance levels. The model seems to simplified, and might therefore not be appropriate to apply when designing resistance bands.


Additional Resources:


The Unofficial Soundtrack of Basketball

If you are a sports fan, like me, March is not just another month in your calendar. No, March means March Madness. The month to watch the most fun basketball games. Last year, in 2016, a total of 74,340 fans attended the final game, while 3.4 million people watched live streams through the app, and 22.3 million people via Turner Sports’ at the end of the game. An important feature of the game that I, like many others probably, have never really thought about is the squeaky noise of the shoes on the court, “the unofficial soundtrack of basketball”. It is an important feature to both the players and the fans. The New York Times wrote an article about how this phenomenon can be explained by shoe designers, rubber scientists, mechanical engineers and a biologist.

A detailed look at the Under Armour basketball shoe worn by Maryland
in a game at Michigan State in 2014.

Credit Leon Halip/Getty Images
From: New York Times, March 17, 2017              

Sheila Patek, a biologist at Duke University, found that her discovery on the defensive mechanism of spiny lobsters can be related to the squeaky noise from basketball shoes. These lobsters rub a smooth, rubbery part of their antenna against a smooth, hard part of their head, creating a squawk. Contact between these types of surfaces is similar with the basketball sole and court, as Martyn Shorten, owner of a biomechanics consulting firm in Portland, Ore., found. He is one of the few people that has researched the squawk specific to basketball shoes. With his research partner Xia Xi, he concluded that the herringbone structures of the shoe outsole are induced to vibrate at their low-order natural frequencies by stick-slip contact with the surface. These vibrations turn into the well-known high-pitched squeaks. Leo Chang, senior design director at Nike, says that “the squeak is reassurance to a lot of players. They listen for it. It gives them that audio sense of reassurance that they’re sticking.” Judit Puskas, a chemical engineering professor at the University of Akron, explains the rubber technology, as that is what the sole is made of. In designing, for example a basketball shoe, you need to find the right balance between the ability of the rubber material to “stick”, “slip”, and wear. The rubber sole needs to allow a player to stop and turn, a too sticky sole creates a too high impact on the body. Greg McDaniel, an assistant professor of mechanical engineering at Boston University, explains how the squeaky noise is created under a rubber sole. Air gets compressed in the tiny, vibrating spaces in the rubber sole, which sucks in neighboring air. This causes the air to expand, which leads to compression of neighboring air. As rubber moves, it compresses air at a frequency that is the same as the vibrations. He found that different herringbone designs lead to high-pitch squeaks of frequencies between 5-6 kilohertz, which can be carried through an arena very well. That is why a basketball game will never be silenced.

This article immediately made me think of one of the discussions from The Sports Gene, when they talk about how technology has impacted performance. I really wonder how much the design of the basketball shoe has impacted the performance of the players. The chapter of The Vitruvian NBA Player mentions how the height of players has influenced the sport so much in terms of recruiting, but maybe the development of the basketball shoe has also contributed to how quickly players can turn, and how high they can jump? In general I thought this article related to the course as it discussed the design of a piece of equipment, the shoe, that provides exercise.

I found it very interesting that people took the time and effort to research the squeaky noises on the basketball court. The findings from Martyn Shorten and Xia Xi on the different frequencies of squeaky noise caused by different shoes were pretty specific. I also found it interesting that people from so many different disciplines contributed to researching this topic. Like mentioned earlier, it was shoe designers, rubber scientists, mechanical engineers and a biologist whose knowledge was useful. Who would ever think to relate a spiny lobster to a basketball shoe? I also think it is interesting how the design of the basketball shoe sole has an impact in many different aspects. Not only does the rubber design of the sole matter for friction between the shoe and the court which impact the forces felt on their bodies when they are turning, it also plays a role in the mental game. It gives the players confidence in their moves, as they said it re-insures that sticking feeling for them, and I think that potentially even influences risk of injury. I wonder if a basketball player would move differently when you would give him a non-squeaky shoe, and if that would put him at higher risk for injury.


Direct Measurement of Sweat Activity

Method and Kit for Sweat Activity Measurement
Patent Number: US 8,565,850 B2
Filed July 1, 2008 – Issued December 22, 2009
Inventors: Orjan Grottem Martinsen, Stabekk (NO); Sverre Joran Grimnes, Oslo (NO); Erik Fosse, Oslo (NO)
U.S. Classification: USPC 600/346; 600/306
15 claims

The method and kit for sweat activity measurement is an invention that is aimed at directly measuring sweat activity. It does so by obtaining a measure for the degree of sweat duct filling, which is found to be a reliable estimate of sweat activity. Sweat ducts are the way through which sweat, produced in the sweat glands, is secreted onto the skin surface. The invention claimed is a kit for monitoring sweat activity, consisting of three electrodes to be placed on the subject’s skin and an electronic processing unit. This electronic processing unit applies a periodic signal to one of the electrodes, and a circuit measures the conductance signal received from another electrode. It can use identification of fluctuations in this signal to measure sweat activity. Additionally, it is able to determine a coefficient of change in sweat activity over time and activity level of the subject, using a signal analysis and frequency analysis module respectively. Additional claims include the ability to determine a quantitative expression for the sweat activity. Also, information obtained on the physiological state of the subject is claimed to relate to blood sugar.

Based on the claims of this invention, this technology might be interested to several groups. It could be useful for physical training and exercise for several reasons. It can give an early warning of thermal imbalance of the body, it can be used to advice athletes on how much they should hydrate (especially given the variation of sweat rates), and it could potentially be useful for athletic trainers present at athlete’s practices and games to track the athlete’s hydration status. Medically this technology could be relevant for any scenario in which sweat levels are relevant, such as hidrosis (excessive sweat activity), illnesses and fever, such as malaria, diabetes and in general diagnostic activities correlating fever and sweating.

The presented technology takes advantage of the conducting characteristics of the skin. The skin may be electrically modeled as a poorly conducting stratum corneum layer (the outer layer of the skin) shunted by sweat ducts containing variable amount of well conducting sweat. As sweat ducts fill, conductance increases. This is followed by a refractory period, during which sweat ducts empty themselves by a re-absorption process through the duct walls into surrounding epidermis layers. Hence, sweat activity is most strongly correlated with increase in conductance.

The basic kit consists of several components, as shown in Figure 1. An electronic processing unit comprises operational amplifiers A and B, a signal multiplier X, a low pass filter LPF, and a conductance output G. V denotes a constant voltage generator. M, R, and C, respectively the measuring electrode, reference electrode, and current carrying electrode are placed on the skin, with an electrolyte providing electric contact between these electrodes (the lines from C to M indicate the electric current paths through the skin). Zsc and Zt denote the impedance of the stratum corneum and deeper layers of the skin respectively (note, impedance is a measurement of resistance to alternating currents, which is the same as resistance in direct currents).

Figure 1. A schematic of the components of the method kit for sweat activity measurement. It shows circuit in the electronic processing unit, and the 3 electrodes (M, R, C) placed on the human skin.

The circuit functions as follows. The voltage generator V, an oscillator, supplies a periodic signal with a predetermined frequency to the electrodes C via operational amplifier A. The current travels to electrodes M, after which the second operational amplifier B with a resistor R in the negative feedback loop serves as a current to voltage converter. Then, through multiplier X, the voltage is multiplied with the excitation sine wave signal from signal generator V by multiplier X. This allows to extract the real part, the conductance, from the complex admittance (a measure of the allowance of current). Remember, this conductance is the desired signal as it relates to sweat activity. Lastly, this multiplied signal is low pass filtered by LPF to obtain a direct current value proportional to the conductance in the skin. Remember, this is relevant since increased conductance is directly related to increased sweat activity. Some potential aspects of the design include incorporation of a micro controller to perform certain functions of the processing unit, reducing the number of separate components. Extension to a multi-channel system, for example by connecting a radio frequency transmitter, can potentially allow for data storage as well.

In contrast to prior art, this technology provides a direct measurement of sweat activity. Here, a measure for the degree of sweat duct filling is obtained, which is directly related to sweat activity, as opposed to the indirect measurement of moisture content. Additionally, the inventors have taken into account that it is important that the measurement itself does not interfere too much with the filling of the sweat ducts, and they have discovered that prior art measurements did not take this into account. A previous intention also aimed at estimating sweat activity based on conductivity is inconvenient in use due to its complexity.

I choose to discuss this particular patent based on the blog about electrolytes that I wrote earlier. Many studies that I read involved a way of measuring sweat, which is what got me thinking about the efficiency and accuracy of current methods. I think it is interesting that this invention allows for a convenient way to directly measure sweat activity. In addition to its relevance to exercise, sweat is also an indicator of for example lying, so I believe the invention has great potential for several applications.

Additional information on this patent can be found here.

Seawater, The Ideal Rehydration Beverage?

You have probably seen the cool commercials showing sweating professional athletes making a legendary play, or training to their maximum level, while drinking their Gatorade or Powerade bottle. Many people, including the brands of these commercially available sport drinks, believe these sport drinks are good for you because they replenish the electrolytes you lose when sweating. However, certain nutritionists and other health and fitness guru’s tell you to stay away from them because they do more harm than good. How does this really work? Do we need to consume these sport drinks when exercising in order to replenish electrolytes?

It is well-known that we sweat during exercise, and that our sweat mainly consist of water and electrolytes. These electrolytes include sodium, potassium, calcium, and magnesium, sodium being the most abundant in our sweat. Sodium is the main cation in extracellular fluid, and its presence has a large effect on plasma osmolality. The amount of sweat and electrolytes lost during exercise varies greatly between individuals, showed by a study done on soccer players. They showed it is affected by factors such as fitness level, sweating rate, and prior diet. For example, sodium concentration in sweat can vary from 20-80 mmol/L. As sodium is the most important electrolyte in our body, it will be the main focus of this review.

Despite the variability within individuals as described above, several studies have researched the effect of sodium concentration in drinks on rehydration during and post exercise. A study published in the European Journal of Applied Physiology found that the addition of sodium to fluids consumed after exercise-induced dehydration has an effect on the rehydration process. More fluid was retained if the sodium concentrations were higher. Subjects did not remain in a positive fluid balance for more than 2 hours when the sodium concentration was low (20 mmol/L). However, drinking fluids with a volume of 1.5 times their sweat loss and a sodium concentration of 60 mmol/L did allow them to remain in a positive fluid balance (meaning that the fluid intake is greater than the fluid output). Another study from University Medical School in Scotland got similar results. They found that in order to sufficiently rehydrate after exercise, both the sodium concentration and volume of the beverage need to be high enough. It is important to note that both of these studies only looked at males, and included a minimum number of subjects. It should also be mentioned that these studies did not take effects of any other components into consideration that are present in many commercially available sports drinks, such as carbohydrates. According to a study about the maintenance of fluid and electrolyte balance during exercise, the total carbohydrate concentrations in drinks consume during exercise should be 5-10%, to avoid delay of fluid and electrolyte absorption. Therefore, sodium concentrations in sport drinks cannot simply be compared to these data in order to determine its efficiency on rehydration.

In addition to drinks, food is also a source of electrolytes. Maughan et al found that urine production was significantly less in subjects that consumed a meal with water after exercise, than in those that had a carbohydrate-electrolyte beverage (sport drink) after exercise. However, they did not give any data on the electrolyte content of the meal and sport drink. Melinda L. et al compared post heat- and exercise-rehydration when consuming chicken broth, soup, sport drinks, and water. They found that plasma volume was not significantly different from predehydration values in the chicken broth and soup trials, but remained significantly below predehydration values for the water and sport drinks trails. Considering the fact that chicken broth and soup had the highest sodium concentrations (109.5 and 333.8 mmol/L respectively versus 0.0 and 16.0 mmol/L in water and sport drinks), this might indicate that higher sodium concentrations have a positive effect on plasma volume recovery. However, it has to be taken into account these four products have large variation in substance (solid/fluid) and electrolytes and carbohydrates composition, so conclusions have to be drawn very carefully.

Analyzing various studies that have been performed on electrolyte and fluid replenishment after exercise, I think it is fair to say that electrolyte composition of fluids consumed after exercise does matter for a successful rehydration process. Looking at the results from University Medical School in Scotland and the article from the European Journal of Applied Physiology, we might be able to say that after intense exercise sport drinks might positively influence the rehydration process, especially compared to water. However, considering the study from Maughan et al and Melina L. et al, I do not think we should say that sport drinks are the ultimate way to rehydrate, as some commercials might suggest. According to this study, subjects remained in positive fluid balance when consuming beverages containing sodium concentrations of 60 mmol/L. This is equivalent to 1379 mg/L, which is not close to the 423 mg/L that for example Gatorade contains. The sodium content of sport drinks is relatively low, while the carbohydrate concentration is relatively high. This makes sense, as a very salty drink most likely does not have a preferable taste. However, if you were to replenish your water and electrolytes purely with sport drinks, you would consume a lot of (unnecessary) sugars. Therefore, I think that consuming sport drinks in order to rehydrate is mostly beneficial during longer training sessions, as the carbohydrates function as a fuel source. However, after exercising I think the calories from the sport drink might counteract the purpose of exercising, and should therefore mostly be consumed after exercise if you don’t have access to food containing high concentrations of sodium. Otherwise, the combination of consuming water and a “salty meal” might be the most efficient way to rehydrate.


Recommended Further Reading:

Rachael Rettner, Are Sports Drinks Better of Worse Than Water? April 25, 2016. Available from:

Christie Wilcox, Sport Drinks: More Harm Than Good? 2009 [cited 2017Feb19]. Available from:

Gatorade [Internet]. c2016 [cited 2017Feb19]. Available from:

Maughan R.J., Shirreffs S.M., Rehydration and Recovery After Exercise. Science & Sports, 2004, 19;234-238. Available from:

Maughan R.J., Leiper J.B., Sodium intake and post-exercise rehydration in man. European Journal of Applied Physiology and Occupational Physiology, 1995, 71;4:311. Available from:

Gethin H. Evans, Susan M. Shirreffs, Ronald J. Maughan, Postexercise rehydration in man: The effects of osmolality and carbohydrate content of ingested drinks. Nutrition, 2009:905-913. Available from:

Eric T. Wittbrodt, Maintaining Fluid and Electrolyte Balance During Exercise. Journal of Pharmacy Practice, 2003, 16;1:45–50. Available from:

Melinda L. et al, Effect of sodium in a rehydration beverage when consumed as a fluid or meal. Journal of Applied Physiology, 1998, 85;4:1329–1336. Available from: