Exploring the Physics of Badminton Rackets Part 2: Oscillatory model
In Part 1 of exploring the physics of badminton rackets, we used an abstract static model of a cantilever beam to gain some insight into what factors affect a racket’s stiffness. However, the static model only serves to show us part of a far more complex system, one that in practice is always in motion. This time we’re going to look at one component of motion in the racket: oscillation.
Anyone who plays badminton should intuitively know that the racket bends when hitting the shuttle, or at least you now should know that after having read the previous article. The static model captured a moment in time when the shaft of the racket was deflected but none of the motion involved. We know that after experiencing deflection, a badminton racket always returns to it’s resting form (if it doesn’t you might want to get that checked out). In this article we’ll explore the motion of the shaft deflecting then returning to rest by again. We’ll accomplish this by using an abstract model of oscillation to theoretically represent the racket shaft so we can gain insight into how different racket shaft specifications might change the way it oscillates.
So what is oscillation exactly?
Well anyone reading this probably has an intuitive understanding of a few oscillatory models. A spring is the simplest example that is used in most physics textbooks, but there are many everyday examples of oscillation that can be found around the house. Anything resembling a pendulum, that is, if you’ve ever left something dangling off a table that was then left to swing back and forth until coming to a rest, is an example of oscillation. If you have a spring door stop behind any of your doors, yes the thing that makes a loud sound if you accidentally kick it when sneaking to the kitchen to eat potato chips at night, that is an example of an oscillating object. If you play an instrument and have an analog metronome, that is also an example of oscillation. Oscillation is defined as “The process of repeating variations of any quantity or measure about its equilibrium value in time.” [5]. In all the examples I’ve brought up, the “variation of quantity or measure” refers to a physical displacement. In the case of a racket shaft, it would refer to the action of bending the shaft to displace it, similar to the model we analyzed in the previous article. The wording is a bit vague on purpose because oscillation can refer to any measurable quantity such as temperature, light, sound, etc.
Now obviously a good badminton racket should not be oscillating continuously, but rackets will generally experience one cycle of oscillation during an overhead swing. The racket shaft will first bend backwards as the arm accelerates and the racket head lags behind, and then the racket shaft will bend forward as the arm decelerates the racket. How much the racket shaft bends in this acceleration or deceleration phase of swinging can vary depending on the method of swinging, but that’s not what we’re going to focus on in this article (that will be covered in the next part). In this article we’re going to focus mainly on the frequency of oscillation and how that is affected by the specifications of the racket shaft.
Defining the Abstract Oscillatory Model
Because we are now analyzing a system in motion, mass in the system will play a significant role where it hadn’t before. In the previous article we used a model of a cantilever beam where we assumed the cross-section of the beam was constant and made of a homogeneous material, in other words the mass per length of the beam would have been the same throughout the whole beam. Literature on the oscillation of cantilever beams does exist but the math gets extremely complicated extremely quickly. You can skim some articles I found on the topic in the citations, particularly [1] and [2].
Luckily we won’t have to deal with these equations because the model we used previously of a cantilever beam with uniform mass throughout will actually not be accurate in analyzing the oscillation tendencies of a racket, because in reality the racket is not of uniform mass above the handle. The head of the racket accounts for a large portion of a racket’s mass, and from a physics standpoint the mass of the shaft itself is trivial compared to the mass of the racket head. For the following analysis we’ll consider the racket head as a single massive object with no systemic properties of its own. In practice, we know this to absolutely not be true, as the racket head accounts for quite a lot of the racket’s total length, and that a racket’s head can and will flex during hard impacts, often by design, but again we’re simplifying the system here to focus on the physical properties of the racket shaft. Now our model consists of an end-loaded massless cantilever beam.
Racket shafts can be made of a number of materials but over time they have shown an emphasis on maintaining some level of pliability yet resisting deformation. In other words, a good racket shaft is one that is elastic always returns to its original shape even if it’s bent. In being elastic, the racket shaft will store potential energy when displaced, and release it as it returns to rest. If you’ve taken any physics classes this might begin to sound familiar, because I have basically described the model for a spring system.
This makes sense because most of us intuitively know that the racket shaft does in fact act like a spring. “Well it doesn’t look like a spring though” you might say, picturing a coil spring in your mind’s eye. Turns out there are many types of springs (helical compression/extension, leaf, torsion, etc). It is a spring as long as it matches the following definition: “Springs are objects that produce a restoring force when stretched, compressed, or deflected by an applied force and will return to their natural shape after the applied force is taken away.” And a racket shaft is exactly that.
Equation for a Spring System
Like in the previous article, we won’t go too deeply into the math because that’s not what you’re here for. Instead I’m just going to show you this equation and we’ll talk about what it means.
We’ve established that in a dynamic oscillatory environment, a racket is better represented by a cantilever beam that is end-loaded rather than of uniform mass, and that the racket shaft ought to represented by some sort of spring system.
The equation for a spring is as follows:
Where:
- f is the frequency
- m is the mass at the end of the beam (racket head weight)
- k is the spring constant expressed in the unit of force over displacement, typically N/m in intro physics courses
Before we go any further, let’s talk a bit about what these variables actually mean in context.
Let’s also bring back the equation for static beam deflection from the previous article:
This time I’m using d to represent displacement since we already have f representing frequency. Notice anything interesting or familiar? That’s right, this equation also contains a force and a displacement variable. First lets rearrange the equation to get the proper unit of force F over displacement d:
Since the left side of the equation is now the same unit type as spring constant k, we can substitute the right side of the equation for k in the first equation like so:
To recap the variables in our final equation:
- f is the oscillation frequency
- L is the length of the beam along its longitudinal axis
- E is the elastic modulus (how rigid the beam’s material is)
- I is the area moment of inertia of the beam (simply put, how thick the beam is)
- m is the beam’s end-loaded mass
Here we can see that all of the variables are inside a square root, with E and I being on top of the fraction, meaning that these two variables have a square root relationship with f. On the bottom, m has an inverse square root relationship with f, and L has a weird inverse square root cube relationship with f. In the next few sections, we’ll discuss how to interpret these relationships and what kind of implications each variable has on the actually real world performance on the racket.
Frequency of racket shaft oscillation
Before we discuss variable relationships, let’s remind ourselves what frequency f means in context. f represents our frequency, which is defined as “The reciprocal of the time period of an oscillation.” [4]. In other words, frequency is the number of oscillations which happen in a constant period of time. A higher frequency value means the object is oscillating more quickly, or in the context of the badminton, means the racket will load and rebound more quickly. A lower frequency value means the object oscillates more slowly, or that the racket takes longer to load up and unload through the swing.
Relationship between frequency and elastic modulus/moment of inertia
I’m not going to spend too long on these two variables since we already covered them in the last article even though the context is slightly different. Both of these variables have a square root relationship with frequency f, meaning that as E or I increases, f also increases, but the amount that f increases kinda begins to taper as E or I reach higher values. We can see this reflected in the graph above. As the shaft of the racket becomes thicker or made of a stiffer composite, the frequency of oscillation will increase, but not linearly. There is no theoretical upper bound to this relationship, so frequency will continue to increase as either E or I approaches infinity, but the frequency will increase more slowly as E or I reaches higher values.
However you might ask be thinking, “but it does matter where on the graph we are, since the relationship is non-linear and so depending where we are on this graph, we don’t know if a slight increase in either E or I will result in a low increase in frequency (for example, increasing the x-axis value from 15 to 20 on the graph above) or if a slight increase in E or I will result in a larger increase in frequency (for example, increasing the x-axis value from 0.5 to 1)” and you’d be right. However if we only want to do relative calculations we can easily accomplish that with some basic algebra. In the next section I’ll explain and prove the relative calculations, but if you don’t care to read and just trust that I did the math properly, just skip this section.
Math interlude: relative calculations
For example, maybe we don’t know every variable in this equation, so we can’t calculate the exact frequency value, but if theoretically we had two rackets A and B, each completely identical, but the elastic modulus value E in racket B is 20% higher than in racket A, we can calculate what affect this 20% higher E value will have on the frequency.
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First we can isolate variables f and E from the rest like so:
And since we know that between rackets A and B, every other variable must remain constant, we can just assign the rest of the equation to some arbitrary constant C:
Then substitute it back into the original equation to give us something much simpler:
Now figuring out the relative value is quite straightforward. All we have to do is come up with a second equation where E is replaced by 1.2E (representing a 20% increase), and f is replaced by xf to represent the scaling factor that we need to solve for.
Which, using the original equation to cancel out every other variable, we find that x = 1.0954, meaning that a 20% increase in elastic modulus E results in an approximately 9.54% increase in frequency f.
You might notice that all we ended up doing was taking the percentage increase or decrease as a decimal (20% increase is 1.2) and take the square root of that. If rackets A and B were exactly the same, the scaling value would just be 1, of which the square root is just itself, 1. From this we can deduce that the important part of the graph is actually the shape of the graph around where the x = 1, since 1 represents 100% of the objective value, and we’re simply playing around with scaling that value up to 20% higher or lower, no more.
End math interlude
So if you didn’t care to read about the math here are the numbers:
- 20% increase in E or I => 9.54% increase in f
- 10% increase in E or I => 4.88% increase in f
- 10% decrease in E or I => 5.13% decrease in f
- 20% decrease in E or I => 10.56% decrease in f
Since manufacturers generally don’t want to make badminton racket shafts too thick, there is a limit to how much we would realistically use these variables to affect oscillation frequency, that is to say most likely these aren’t the things manufacturers would change if they want to modify the oscillation frequency of a racket, but it’s interesting to see the affects nonetheless.
Relationship between frequency and end-loaded mass
Frequency has an inverse square root relationship with end-loaded mass m. As we can see from the graph, there are significant changes in f when m is small, but very little change as m approaches infinity, as f approaches the limit of 0. In other words, changes in head weight for head light rackets result in more significant changes to oscillation frequency than changes in head weight for head heavy rackets. This graph appears to have little affect on frequency f since it tapers extremely slowly as m approaches infinity, but remember we discussed in the previous section that we only care about the values near where the x-value = 1. Performing similar math as before, we find that various modifications to the head weight are as follows:
- 20% increase in m => 8.71% decrease in f
- 10% increase in m => 4.65% decrease in f
- 10% decrease in m => 5.41% increase in f
- 20% decrease in m => 11.80% increase in f
We can see in the math that indeed there are more significant frequency changes as the head weight m is decreased rather than if it was increased, since a 20% decrease resulted in a 11.80% change in f, while a 20% increase only resulted in a 8.71% change in f. Still, these resulting changes in f are relatively small considering the significant head weight changes that would need to occur. Put in perspective, a typical even balance 4U racket might have about 40 grams of weight in its head, so a 20% change would mean an 8 gram increase or decrease in the head weight, which any badminton player could tell you is a ludicrous amount. Such a change in a racket would have far more significant implications in racket feeling aside from oscillation frequency.
Relationship between length and frequency
Frequency has an inverse square root cube relationship with the length of the shaft L. At first glance this graph looks similar to the relationship with m due to its square root nature, where f approaches the limit of 0 as L increases to infinity. However if we remind ourselves that only values around x-value = 1 matter, we can see that the curve at this point is indeed steeper than it was for the relationship between f and m. Doing the math reveals:
- 20% increase in L => 23.93% decrease in f
- 10% increase in L => 13.32% decrease in f
- 10% decrease in L => 17.12% increase in f
- 20% decrease in L => 39.75% increase in f
You’ll immediately notice that the corresponding changes in frequency are far greater than what we saw when modifying any of the other variables. Merely a 10% change in L yields a greater change in f than even a 20% change in any of the other variables. Furthermore we can see that the change in f is extremely significant as L is reduced, as a 20% decrease in L results in more than a 1/3 reduction in f. You likely could have predicted this looking at the equation. Despite being inside the square root, L is cubed which means that the L value would have significantly more impact on the f value than any of the other variables did. For reference, Arcsaber 11 Pro and Arcsaber 7 Prohave a shaft length of 224 mm, which seems to be relatively median. The Astrox 100ZZ, which is known to have one of the shortest shafts on the market, measures in at 205mm, a difference of about 19mm and a 8.5% decrease in length. If every other specification of the racket was the same, that would lead to a 14.25% increase in frequency. We know that’s not the case because these rackets have vastly different designs, but this just illustrates what a difference in racket behavior the shaft length can cause.
As with in the previous article, the analysis performed in this article only serves to cover one facet of racket shaft behavior, the oscillation frequency of the racket. While we did model dynamic motion in the racket which is a step forward from the static model in the previous article, we still haven’t accounted for dynamic forces, especially the interaction of actually striking the shuttle, and how racket specifications or contact timing can change the behavior of the racket on contact. That’s something I intend to explore in a future article.
Now armed with the knowledge in this article, we can combine our findings with the takeaways of the previous article. We know that shaft length, thickness, and material have an effect on the objective stiffness of the shaft, but also that these factors in addition to head weight have an effect on oscillation frequency. By tuning specific parameters, it should theoretically be possible to maintain a certain objective shaft stiffness, but raise or lower the oscillation frequency, creating a racket that has the same stiffness but rewards long swings or short and snappy swings.
Takeaways
The oscillation frequency of the racket depends on a variety of factors including shaft thickness, shaft material, head weight, but most importantly shaft length. Changing any of these factors can cause a racket shaft to oscillate with differing frequencies, but we know that shaft length by far has the most impact on the oscillation frequency of a racket. Now that we’ve discussed some of the physics of a badminton racket at rest and in motion, in the next part we’ll discuss how racket design plays a role in making contact with the shuttle.
Citations
- http://emweb.unl.edu/Mechanics-Pages/Scott-Whitney/325hweb/Beams.htm
- https://vlab.amrita.edu/?sub=3&brch=175&sim=1080&cnt=1
- https://www.ntmdt-si.com/resources/spm-theory/theoretical-background-of-spm/2-scanning-force-microscopy-(sfm)/23-linear-oscillations-of-cantilever
- https://byjus.com/question-answer/what-is-oscillation-frequency/
- https://byjus.com/jee/oscillation/
- https://www.ucc.ie/en/physics/study/undergraduate/thelaboratories/firstyearphysicslabs/pendulum/
