Physical principles for the golf swing

Biology

Dave Tutelman  --  October 3, 2022

Muscles

So far, we have talked about what forces and torques do; we have not mentioned how they happen in the first place. That part is biology, specifically the muscles and joints. The muscles are the engine of the golf swing. That is quite literal. Like the engine of an automobile, the muscles take other forms of energy and convert them into force. Similarly, the joints may be likened to the crankshaft of an automobile; they take the force produced by the muscle and convert it into torque to be conveyed and used elsewhere.

I'm going to try to keep the biology section simple. We'll limit the discussion to skeletal muscles and the joints most important to analyzing the golf swing. I'm not going to even try to get into non-skeletal muscles, finger joints, and other things that only occasionally arise in golf biomechanics discussions. If these turn out to be critical to a point you want to understand, there are lots of resources you can use for help. However, this tutorial is not one of them.

In fact, I'm not even going to write this section as a biologist might. Instead, I will treat muscles and joints as an engineer might look at mechanical components in a catalog, describing them in terms of their mechanical specifications and not worrying about the details of what makes them work.

Skeletal muscles

Muscles are soft tissue that can change shape and exert a force by doing so. Skeletal muscles are those muscles that connect bones to bones. Let's look at what muscles do. (We can't look at "how they move our body" without talking about joints, so that will wait until the next chapter.)

A muscle consists of a bundle of generally parallel fibers. When you exert an effort with a muscle, it contracts: it shortens and thickens. It shortens forcefully enough to pull along anything attached to its ends, with considerable force. For a skeletal muscle, that means the bones attached to its ends are pulled closer together.

How much force? The force is proportional to the cross-sectional area of the muscle. When you exercise enough to add muscle, you are increasing the thickness (cross-sectional area) of the muscle, and that is what makes it stronger.

I'd like to emphasize that the force is a pull, never a push. Think of a muscle more as a string than a rod. The ends of the muscle pull the bones it is attached to closer together. There is nothing it can do to move its bones farther apart; just not going to happen!

I have heard some golf instructors claim the muscles act as if they were rubber bands. They do not! You can't consider the backswing to be a stretching of springs or rubber bands that will make the downswing happen by releasing that stored stretch. The downswing (like the backswing) is motivated by muscles contracting energetically, pulling on bones in such a way as to move the club the way it should be moved. There may be a small dose of potential energy helping to power the downswing (gravity, and the Stretch-Shorten Cycle described below), but you would be ill-advised to focus on developing it in preference to driving the swing with the muscles.

Muscle energy

Let's see if we can put a number on the work that muscles can do. Later, we will compare this with the energy required for a golf swing, but first we should understand how joints work. Let me encourage you to take a look at the references linked; as I said, I am no expert on the biology of sports, and these were resources I found by poking around and searching. I tried to keep my searches on the internet if I could, so you can follow them as well.

Let's start with what we know about skeletal muscle tissue:
  • It is made of parallel fibers that individually contract and relax, but in synchronism with other fibers in the tissue. Each parallel fiber adds force to the contraction of the muscle itself. Therefore...


  • The force available from the muscle is proportional to the cross-sectional area of the bundle of fibers. I have found a few estimates of the force per square centimater of muscle tissue. The number I have most confidence in is 9.5 N/cm2. The diagram shows a cross section of muscle tissue with a square in it. If the square is 1cm on a side, then contraction of a muscle in just that square would be capable of exerting one Newton of force.

    That's for men. The number is slightly lower for women, 8.9 n/cm2. It isn't much lower; the standard deviation of either number will cover the other number. This says there was a considerably larger variation from individual to individual than between genders.
  • If we extend that square in the direction of the fibers themselves, we can get a cube 1cm on a side -- therefore 1cc (cubic centimeter) in volume. Here are two interesting things about the cube:
    • Let's assume that all skeletal muscle in an individual is pretty similar, whether in the arm, the leg, the glute, or elsewhere. That's actually a fairly safe assumption. And research shows that skeletal muscle has a density of 1.06 g/cc, so the muscle tissue in the cube is a mass of 1.06g. (Actually, the reference says 1.06kg/L. But a liter is 1000cc, so a little playing with units gives the number I provided above.)
    • If the muscle contracts by X%, then the cube will shrink by x% along the length of the muscle fibers. This contraction is motion in the direction of the force the muscle is exerting. Therefore, we can use it -- and the force -- to compute the work the muscle is doing.
That brings us to the question of how much work the muscle can do. (I.e.- how much energy the muscle can create from the chemicals that nourish it.) To find that, we need to know how much the muscle contracts, and how the force varies as the muscle contracts. Let's start with the simplest possible set of assumptions, and then figure out the real-world conditions.

Naive model


Assume the muscle is capable of contracting by a fraction x, and it does this at a constant force, the nominal 9.5 N/cm2. How much work can each gram of muscle perform?

First, let's see about the volume of a cube of muscle. Since the density of the muscle fiber is 1.06g/cc, the volume is:
V  =  1 / 1.06 = 0.94cc
We can show that only the volume will matter in these calculations; it doesn't matter whether the rectangular solid stays a cube or shrinks in just one or two dimensions. So let's keep the cross-sectional area at 1 by 1 cm, and reduce the dimension in which contraction occurs. We can then go through the calculation for work per gram of muscle.

Force  =  9.5N

Distance  =  .94 * x  cm

Work  =  Force * Distance  =  9.5 * .94x   N-cm  =  9.0x N-cm

In order to get to Nm or Joules, we have to divide by 100, giving us
Work  =  .09x  Joules per gram of muscle

To complete the units conversion, find work per Kg of muscle by multiplying y 1000:
Work  =  90x   Joules per kilogram of muscle

More detailed model


Now let's go back and repeat the process, but with more reality built into the characterization of muscular contraction. Our prior attempt assumed constant force and a percentage-limited contraction. I have found a fairly detailed description, with a graph of the force vs contraction function.

Here is a reproduction of the graph, edited by me:
  • I have cropped it down to the details we will need to compute work. I have cropped out one of the three pictures of how muscle cells work internally. The other two are still there, but play no part in the analysis below. (We will not concern ourselves with the internal mechanisms of muscles; we will treat them here a "black boxes" that can apply a force as they contract. If you want to know more about the biology, there are sources for that -- starting with the one where I got this diagram.)
  • I have highlighted part of the graph in yellow. This is the part where the muscle contracts from its neutral (no contraction) state.
The highlighted area shows the ability of the muscle to apply a force (the vertical axis) to be a maximum at the muscle's neutral, uncontracted length of Lo. I digitized the graph into a spreadsheet, so we can do calculations using the force vs length graph. (If you are interested in the digitizing, it is a tedious process I do by hand in photo-editing software. I described it in detail in another article.)


The blue curve is the graph of muscle force vs muscle length. It should look like the part of the curve above that we highlighted in yellow.

As part of digitizing the graph, I normalized the peak force to 1.0. Setting it to 1.0 allows us to multiply by the area of the muscle and the strength 9.5N/cm2 of skeletal muscle to get the force exerted by the whole muscle, not just a single strand of muscle fiber. We also normalize length to 1.0, because we are going to describe things in terms of percentage contraction, not just the contraction in mm of a strand of muscle fiber of length Lo.

Let's think about what it means for a muscle to contract. It's very simple; the length shortens. For instance, as the length goes from its neutral, relaxed length (which is 1.0 on the graph) to a length of 0.8, it contracts an amount of 0.2. More generally, 1.0 - length is the amount of the contraction.

The graph shows a green shaded area representing a contraction from 1.0 to 0.8, a contraction of 0.2. The shaded area under the curve represents work! It is a force times the distance that force acts while the muscle is contracting. This time, the force varies as it moves, so it isn't a simple multiplication to find the work. But if we divide the are into little vertical slivers and add them all up, we get all the force-times-distance slivers all collected into the green shaded area. That is called "numerical integration", and we need it because we have an empirical curve, not one described by an algebraic function. Integration is usually the province of integral calculus, but it doesn't quite work here because the curve is completely empirical. So instead of calculus, I used a spreadsheet to add up all the force-times-distance slivers.

When we add up the force-times-distance slivers for a muscle contraction of amount x, here is the graph of work for a contraction x (the blue curve).

The red curve is the naive model we used earlier, where we assumed the force was constant for any contraction. It should not be surprising that the two answers are close for small values of contraction. If there is not much contraction, the force is close to the force at Lo, a constant, so the work done by that force should be close to the same thing. As the contraction increases, the muscle can't exert as much force and the curves separate from one another.

Interesting note on this. Many joints limit the amount their driving muscle can contract. For instance, consider the experiment that determined the force a sq-cm of muscle can exert. The muscle being used was the quadriceps (the front of the thigh), to straighten a leg at the knee. When you look in more detail at the knee joint, the quadriceps can only contract by about 10% of its length before the knee joint reaches full extension and won't go any further; it is pinned. So any work done by the quads involves a contraction of 0.1 or less, and the naive model is pretty good for a first estimate. We will look at this in more detail in an example at the end of the Biology chapter.

Moral of the story: know how much the muscle you are considering can contract; most skeletal muscles are limited not by muscle characteristics, but by the range of motion of the joint that the muscle activates. If that is a small contraction (say 10% or less of the muscle length), then the naive model works pretty well.

Fascia, tendons, and the Stretch-Shorten Cycle

OK, I promised to mention the Stretch-Shorten Cycle (SSC). I'm far from an expert here, but this is my understanding of it.

As noted earlier on this page, the muscles are not springs; muscles themselves are not elastic. They operate by turning chemical energy into mechanical energy (contraction with force). But there are two types of tissue that enjoy the company of skeletal muscles, tendons and fascia, that have some elastic properties.
  • Tendons connect the muscle to the bone that the muscle intends to move. They are the white tissue in the photo; the muscles are red and the bones brown. If the tendons are able to exert a force by lengthening or contracting, that force is added to what the muscle produces.


  • Fascia surrounds, protects, encases, and "organizes" the muscle fibers. (Image courtesy of Marquette Physical Therapy.) Its composition has some commonality with tendons, and it has some similar elastic properties.

The elastic properties of tendons and fascia are not pure elastic. They are a form of kinetic-kinematic behavior known as viscoelasticity. Before defining viscoelasticity, let's think about an example. How about the most extreme example I can imagine: Silly Putty. Its properties push beyond the limits of ordinary viscoelastic materials, but let's start with it anyway.

  • Squeeze it or stretch it and it behaves like clay or putty; it deforms to the shape you give it. Viscous behavior.
  • Roll it into a ball and it bounces. Elastic behavior.
The difference between the viscous and elastic behaviors is time-dependent. Here is a simplified version of what happens when you suddenly stretch a viscoelastic material.
  • At time To, the material is suddenly stretched, then held in that position.
  • At that moment, the elongation is elastic; if you released the material, it would snap all the way back. It is, in essence, a spring.
  • If you simply hold it stretched, the elastic force wanting to return the material to its original shape shrinks. (That's the red part of the graph.) After a while, there is no elasticity left to make the material spring back. It has been 100% deformed.
  • Depending on the material, it might gradually assume its former shape if you release it after a while. Or it might just stay elongated. Either way, it no longer exerts a strong force; the spring action is gone.
Silly putty is at one extreme of viscoelastic performance. It has a lot of initial elasticity, but it goes away in milliseconds. That's why it bounces (contact is made and rejected in a millisecond or two, while elasticity is still strong), but also can be molded (no elasticity left in a tenth of a second or so). There are viscoelastic materials at the other end of the spectrum as well. They may take minutes to hours, perhaps even calendar-measured time, to fully assume the new shape. (I encountered that with plastics in the design of the NeuFinder-4, an instrument to measure golf shaft stiffness. The plastic was strong, but it had a viscoelastic "creep" of more than 1% of its length. So you have to note your reading quickly; in 20-30 seconds, the shaft may register as more than 1% less stiff.)

So where in this spectrum are tendons and fascia, the biological structures that would make possible an elastic rebound from a stretch? BTW, that rebound is referred to as the Stretch-Shortening Cycle (SSC). I have seen articles and papers that estimate the time for elastic behavior from a few hundredths of a second to a quarter of a second. That's quite a range. The shorter estimate would be of very little use in the golf swing, while the longer estimate might make it worthwhile to cultivate a stretch-shorten cycle. But it does seem clear that a discernible pause at transition will blow any chance you have of using it. Other questions I have seldom seen answers for is quantification of how much energy there is to be had from the SSC, assuming we can do things fast enough so the elasticity doesn't expire. I doubt it is a major fraction of what the muscles give us, but perhaps there is enough there to be of use, enough that it isn't negligible compared to muscle power. There must be something significant somewhere, because the SSC is deemed important in track and field sports like running and jumping.

So why track and field -- and, by the same token, why not golf? The shorten must follow the stretch very closely, or the advantage will dissipate; most sources deem it as effectively gone by, at the absolute outside, a half second after the stretch. That works fine for running and jumping -- track and field stuff -- where the stretch-shorten cycle takes place so fast it is almost a bounce. The golf swing is slower. It typically takes about a second -- perhaps a little less for elite players and up to a second and a half for handicap players. That is marginal for an advantage based on the counter-move less than a half second before the move.

As a result, much of the research on SSC has been on track and field sports, running and jumping. But there is a paper in a sport whose time scale is closer to golf -- rowing. In fact, with a stroke rate between 30 and 40 strokes per minute, its complete cycle is 1.5-2.0 seconds, even slower than a golf swing. The paper shows a 10% increase in power if the stroke includes a move that recruits a stretch-shorten cycle.

Here's how the study was conducted. A single, isolated rowing stroke (phases C-D-A in the diagram) was done on a rowing machine that measured the energy in the stroke. The difference between an SSC stroke and non-SSC was the slide phase (B in the diagram). To measure without an SSC, the subject would get to position C statically. For an SSC stroke, the subject would use a slide-drive combination in rapid succession. (The paper refers to the cycle as "FEC", flexion-extension cycle. It means the same thing as SSC.)

We can compare slide-drive to backswing-downswing. If the backswing is dynamically linked to the downswing in rapid succession, then perhaps the golf swing can attain a similar gain of 10% in work performed in the downswing. The muscles used are probably different, as is the timing, but there are certainly similarities.

Provisionally, I am going to assume 10% or less improvement in golf power due to the stretch-shorten cycle. If and when there is more conclusive data, I am ready to adopt the new conclusion. In the meantime, it seems to me the jury is still out on this one. (Written in April, 2023.) There is ongoing research, and it will be needed to to tell us what, if anything, we can do to improve the golf swing using fascia and tendons. Here are a few additional resources you can read if you are interested in the topic.
  • A very elementary introduction to fascia.
  • A summary of tendon biomechanics, including graphs showing creep, stress relaxation, and hysteresis, plus a good bibliography of further references.
  • A paper that consists of a quick summary of a lot of research papers in the field of SSC -- with links to those papers. The links make this an excellent jumping-off point to a more serious investigation of the SSC. The author of this paper appears not to distinguish between the effect in fascia and in muscles, and includes research attributing the effect to both.
  • A blog that is as much advocacy as information about myofascial meridians, chains of fascia that may have importance in certain motions like running or pitching a baseball.


Last modified -- Apr 3, 2023