From complex to simple 1: Exploring elasticity

In the last post, it was proposed that when two advanced dancers dance together, they control their muscles in such a way that their centres of mass (COMs) behave as though they were simply connected by a spring. We have called this system elastic connection. This post will start at the other end and meet the previous post in the middle; here we shall begin by discussing elasticity in a general sense and then describe dancing as a specific example.

The whole point: Learning how to think simply about a complex thing
Something to keep in mind as we proceed: the whole point of attempting a scientific analysis of good dancing is to make it simpler to think about and easier to learn. When we first come to dancing, either as observers or participants, it's an impressive, complex-looking thing. We sit and stare, awestruck, at world-class dancers and think, "What is that magic that they've got, that exquisite quality of movement? How can I possibly take that magic and make it my own?" Trying to learn to do something that's complex is difficult. We fumble around enthusiastically, learning lots of 'moves' and 'variations' and feeling spiritual about a few vague concepts of connection. We imitate the eye-catching shapes and stylings of our favourite dancers, and try to hold a thousand little snippets of information together in our minds as we dance, hoping that it will also somehow click together and voila, we too will make the magic happen. My experience with trying to learn dancing this way has often been frustrating. The whole process can be made easier and less dramatic, I believe, if we are able to dig deep enough down into the fundamentals of what's going on in good dancing, to show that it's actually not that complicated after all, but rather is just a small collection of simple things added together. Those simple things can then be learned one at a time and put together piece by piece, to build solid dancing from the ground up.

Luckily, this approach is applied all the time in physics, with great success. The process for doing so was described earlier in this blog, in the post on simple models of the human body and the magic of its centre. There is an important point to add here: physics typically proceeds when it is recognised that a newly-studied system in nature, which isn't yet understood, looks like some other system which has been studied in the past and is understood. One can then take the model which was developed for the old system and see if it works for the new system too. Sometimes it works brilliantly, right off the mark; sometimes a bit of tweaking is required. In many cases, the new thing quickly becomes just as well understood as the old thing; suddenly, it becomes possible to think about a complex, mysterious looking thing in simple ways.

I have spent some time trying to figure out what well-understood model might be applied to a dance partnership in order to allow us to think about dancing more simply. In the last post we introduced a key feature of that model: elastic connection. In this post, we will explore more deeply, the consequences of having dance partners connected elastically. We will begin by talking about some abstract things but see if you can spot the connections to dancing along the way. I promise that if you stick with me through the abstract ideas and get a feel for them, new vistas of simplicity will open up in how we are able to talk about dancing down the line. Here we go.

Something well understood: A mass on a spring

Let's consider the above picture, which shows a mass M connected to a solid wall by a spring. We assume that the mass is sitting on a frictionless surface. In the top frame, the mass is sitting such that the spring is at its rest length. ie. its natural length when it's not being stretched or compressed - all springs have one of these. At this length, there is no elastic potential energy stored in the spring. If we also assume that the mass is sitting at rest, there is no kinetic energy in the system either. We will call this the zero energy system. In the lower frame, the mass has been moved by a distance x, such that the spring is now stretched. Now there is some elastic potential energy stored in the spring. If we assume that the mass is again at rest in this new position, there is still no kinetic energy in the system. We will call this a pure potential system.What will happen if the mass is released from this position?

The answer, of course, is: The mass will be pulled towards by the wall by the spring as it shortens back to its rest length. As this happens, the potential energy stored in the spring is gradually converted into the mass' kinetic energy as it speeds up. By the time the mass has returned to the position shown in the top frame (the spring is again 'relaxed' at its rest length), the system has become a pure kinetic system, with all of the spring's original potential energy now having been converted into the mass' kinetic energy. The mass does not stop here. It keeps moving and the spring is gradually compressed until all of the mass' kinetic energy has been returned to potential energy in the spring (at which point, the mass momentarily stops again), which now stores it as a compression, rather than a stretch. And so on, the mass will oscillate back and forth as energy is traded between potential and kinetic states. If the system is genuinely frictionless and no energy is either given to or lost from the system by any other mechanism, the mass will bounce back and forth forever. The technical term for this kind of system, which shows up everywhere in physics, is a simple harmonic oscillator (SHO). Readers who would really like to get a feel for how SHOs behave are encouraged to check out the wikipedia page on SHOs, which is excellent and includes some great animations, which help to make things more intuitive.

There is an important property of this system, which must be mentioned because it is essential to understanding the elastic model of dance connection. Let's introduce it with a question: How quickly does the mass oscillate? Are the vibrations quick like a plucked guitar string or slow like a kid on a swing? At this stage, we can't say either way; the model as we've introduced it is entirely general so we need to specify a couple of things in order to answer this question. There are two properties of the mass-spring system, which determine how quickly it oscillates: The size of the mass M and the elastic constant of the spring k (ie. whether it's a stiff spring like in car suspension or a soft spring like a slinky). If those two quantities are known, we can calcuate exactly, the resonant frequency of the system. Every mechanical system has a resonant frequency. It's the system's natural frequency of vibration - how fast it will vibrate  (the number of times that its mass moves back and forth per second) if it's struck or plucked. Two familiar examples are a crystal wine glass and a guitar string, both of which will vibrate fast enough to make a musical note.

A quick summary, which is important to remember: When a mass on a spring is given some energy (ie. the mass is moved by some external force - like a hand pulling it - and the spring is stretched/compressed accordingly, then the external force is removed and the system is allowed to freely move), it will naturally oscillate at a particular frequency, which is known as its resonant frequency. Or, stated in reverse, if a mass of certain size, attached to a spring, is required to resonate at a particular frequency, the property of the system that must be carefully chosen is the spring constant (or elastic constant) k.

Losing energy: Damping
In real mechanical systems, there is always some loss of energy through friction, so there is no such thing as a mechanical SHO that oscillates forever. The vibrations of a SHO under friction will gradually get smaller and smaller (though they will keep the same frequency) until the mass stops moving altogether (the SHO has returned to a zero energy system). But unwanted friction isn't the only thing that can steal energy from a SHO; the general name for a force that does this is a damping force. Sometimes mechanical systems are deliberately designed to include significant damping forces because ongoing vibrations are not desired. The classic example is a suspension system in a car, which is composed of a spring and a shock absorber. Here, 'shock absorber' is just a fancy way of saying 'damper'. When a car hits a bump, the suspension springs compress and energy is stored in them. Without shock absorbers, the car will bounce up and down at its resonant frequency until its vibrations are gradually damped by friction with the environment (air resistance, etc). This would make for an awful ride. So, shock absorbers are added and the oscillations of the car are very quickly damped, making the ride more comfortable.

Adding energy: Driving
No, not that kind of driving. We're not talking about cars now. A driving force is the technical name given to any force which gives energy to the oscillations of a SHO. The classic example for this is a parent pushing his/her child on a swing. Without repeated pushing, the swing's oscillations will be damped by air resistance and the ride will soon be over. So, a periodic driving force is required in order to keep the swing going. If the swing is to be kept going at the same height, the driving force and damping force must cancel each other exactly, and the swing will move simply as if there were no driving or damping at all.

Muscles: drivers, dampers
Muscles work in groups to achieve complex tasks. Sometimes they contract, sometimes they stretch. Sometimes they give energy to limbs, sometimes they take energy away from limbs. How a given muscle behaves is dictated by the role it is required to play in making an overall body movement happen. Consider the act of throwing a ball in the usual way (like pitching a baseball). Early in the movement, the muscles of the chest and shoulder contract quickly, giving the arm great speed and energy (ie. acting as drivers of the arm's movement). But not all of that energy can be given to the ball, so after its release, there will be an arm left behind, moving at dangerous speeds. In order to prevent the injury that might result from this speeding arm crashing into the body after the ball is released, the shoulder and chest muscles switch into 'active stretch' (as we called it in the last post) and damp the arm's motion, slowing it down to a safe stop.

Stripping the dancers
Ok, great. We've talked some physics and some physiology, and now we can get down to talking about dancing. Let's begin here, by describing the step-by-step process of reducing the complex interactions of a dance partnership to a simple, elastic connection model. Consider the following figure:

In frame A, we see a connected dance partnership in all their silhouetted glory. If we could see them moving, it might look quite complex and at this stage, we think of them with all the complexity with which we might usually imagine two connected human bodies, each with the potential to move in so many amazing ways.

A few essential mechanical features have been added in frame B. We see both partners' centres of mass (COMs) as blue blobs around the heights of their belly buttons. The basic structure of the skeleton is shown by the white 'wireframe' and overlaid on it are grey springs and some red and green lines, which are all together intended represent the major muscle groups. Green represents the potential to apply a driving force, which adds energy, and red the potential to apply a damping force, which subtracts energy. The presence of the springs represents the potential of each muscle to execute elastic, active stretch. In this frame, all the muscles are both red and green because all muscles, strictly speaking, have the potential for both driving and damping. So, we haven't made any simplifying assumptions yet but rather just introduced the mechanical characters of our story.

In frame C, things are starting to get a bit simpler. Frame D is the same as C, with the silhouettes removed for ease of viewing. We have now removed all the green lines from the 'muscle chain' - that is, the sequence of muscles which connects the two partners' COMs. So, we have now removed the ability of these muscles to add energy through contractions. Rather, they can now only do two different things - elastic, active stretch, which transmits energy between partners, and damping, which removes energy from the partnership, as when a stop is led. In this frame, the leader's leg muscles can both add and subtract energy from the partnership; they are the ultimate drivers of the dynamics. At this point, the follower's legs retain the ability to add energy but this is intended only in the limited sense that she must continually add some energy to replace the energy naturally lost to friction as she moves. She's not trying to add energy to the partnership but rather only making sure that no energy is lost overall. It's a bit like driving a car down the highway - if the road were frictionless, the motor could be switched off and the car would cruise forever. But this is not the case and the motor must continually compensate for energy lost to friction. Indeed, I have heard several teachers encourage followers to 'keep motoring' once led; that is, conserve energy and don't slow down until a slow down is led.

In frame E, the legs remain the same but the detailed structure of the upper body is done away with and replaced by two springs, connecting their COMs in the middle (where the hands would be). As stated in the last post, it can be shown mathematically that a series of connected springs behaves like one big spring, so we can think of all the muscles of each dancer's upper body behaving simply as a single spring, connected to a single spring provided by the other partner.The capacity for damping remains in the connection here, though it should be emphasised that damping does not act constantly, but rather only when one or both partners wants to subtract some energy from their connection.

We extend this reasoning in frame F, representing the two partners' springs as a single spring (plus damper) connecting their COMs. The legs are again unchanged.

In frame G, we arrive at our final simplified model. Here, the legs have been replaced by wheels for further simplicity. The leader has one green and one red wheel, indicating that he can both add energy and subtract energy from the partnership's motion. The follower simply has two black/uncoloured wheels, indicating that her role is to simply conserve energy, neither adding nor subtracting it. The two partners are identified with their COMs and are connected to one another by a spring and a (voluntarily operable) damper. If we were to describe this model in technical language, we might do so like this: A pair of point masses connected by a damped spring with time-dependent spring constant and damping function. That is, both masses are acted upon by an elastic force and also by a damping force. In addition, one of the masses (the leader) is also subject to a time-dependent driving force and an additional time-dependent damping force.

Compared to the original dance partnership, in all its complex glory, this model might seem a bit yawn-inspiring. However, despite being a lot simpler than what we started with, describing it mathematically (which is what we must do in order to explore it completely and decide conclusively whether it's a good representation of a dance partnership) will take some doing. And I must confess that I haven't yet tried. When I do, I'll post some results. Moreover, this model only describes linear motion and says nothing about rotations. We will have to build up a complementary model for rotations down the line, and we will. But let's get started here first, without getting mathematical. We don't really need to go that far in order to use the model intuitively to talk through some familiar moves, which is what we will now do.

One move to describe them all: the sugar-push, in all its glory
Let's start with (arguably) the simplest move out there, the sugar-push. I love this move because it's so simple it's difficult. You can't fake it. If you're not on top of your movement and connection, there just ain't no sugar in your sugar-push, no matter how nice your lines or fancy your footwork. We will talk through the move in terms of our simple model - rather than talking about bodies, arms and steps, we will focus on masses, springs and forces. If at any point you find yourself unsure of how this translates back to bodies, take a look back at the above figure and discussion. There is something to keep in mind as we go along - the process of leading a sugar-push from start to finish contains absolutely everything that any linear move ever contains. This is it - understand this well and understand everything, do this well, and you can do anything. So, really, this is the only move we need to describe in detail. Everything else is just a variation on this.

Before we launch into it, it might be worth checking out the cool little interactive animation found at this site. It lets you watch the dynamics of two masses connected by a spring. I must stress that this model is not strictly the same as the one we're proposing for a dance partnership (because both springs here are also connected to two walls by two other springs and our dancers aren't doing that) but under certain conditions, it's close. The animation takes a while to get going but if you wait for about 5 oscillations, it starts to look remarkably like a dance couple doing some nice sugar-pushes. Well, their centres of mass, anyway. Think of the red mass on the right as the leader, and the blue one as the follower. You can play with the spring constants (the K values) if you like. Notice how things change if you do. Once you've got a visual feel for things, the below discussion might seem more intuitive. 

We assume that prior to the beginning of the move, both masses are stationary so there is no kinetic energy. The spring between them is at its rest length so it contains no potential energy. Overall, the partnership is a 'zero energy system'. To get things started, the leader adds some energy by applying a driving force with his legs, forcing his COM backwards, away from his follower. When this happens, the spring connecting the two COMs stretches immediately and immediately applies a force on the follower's COM. This means that her COM begins to move immediately. Let's address this right away. It is common for teachers to tell followers that they have to wait when they're being led, to force a delay between the leader's movement and their movement. Strictly, this is not true. Watch a couple of world-class dancers and you will see that it's not true. Good followers don't force anything; they follow. When the leader moves his centre and transmits energy through the connection to the follower's centre, it moves immediately (ok, well, strictly it takes a tiny fraction of a second for the energy to reach her since it travels through the connection, roughly at the speed of sound in water: about 1.4km/s). What is true for the follower is that she must not take the leader's giving of energy as a signal for her to add energy to her own movement. The energy of her centre will build gradually as it is transmitted to her from the leader. She will begin to move immediately but only because she is being moved, not moving herself. Remember, in our model, neither her legs nor her muscle chain can add energy. Ok, back to the sugar-push....

So, the leader's legs have moved his centre, causing the spring between the partners to stretch. The force which this stretch applies on the follower's centre is only small to begin with because the stretch is only small (remember F = - k.x from the last post). So, her centre begins to accelerate immediately (because a force causes acceleration by Newton's second law, F = m.a, where m is the mass) but only a little bit while the stretch is small. Bus as the leader's driving force moves his centre further and further away from hers, increasing the stretch length of the spring, the force on her centre increases and she accelerates ever more quickly. We see from this that the delay between partners is a natural consequence of elastic connection and has nothing to do with a deliberate action on behalf of the follower; she speeds up somewhat after he does because it takes time for the elastic force to build as a result of the stretch in the spring, which is caused by his movement. So, he moves, stretching the spring, which gradually moves her. Voila, a delay.

When the leader sees and feels that the follower is moving nearly as fast as he wants her to move, he shuts off the driving force from his legs, meaning that he stops accelerating is own centre. This means that he is no longer adding stretch to the spring, or even maintaining the stretch that's already there, so the spring then naturally returns to its rest length. ie. The follower catches up to the leader - the distance between them returns to what it was before the move started. The difference now is that the partnership is moving together; they have kinetic energy. If he were not in the midst of leading a sugar-push, the leader could simply choose to cruise across the floor with his partner. The spring would remain at its rest length, nice and relaxed, until he chose to reactivate the driving force to add energy in a new way. BUT, he is leading a sugar-push, so something different happens. He instead applies a new driving force equal and opposite to the one he applied originally, forcing his COM to move forward, towards his partner, which begins a compression in the spring (which is still mediated by active stretch in muscles, don't forget - just different muscles to those which were stretching during the spring stretch at the beginning of the move). Again, the follower's centre begins to slow immediately but the slow-down builds gradually as the spring compression builds. Eventually, the leader will have slowed himself to a stop and the follower will come to a stop shortly thereafter. However, the stop is only instantaneous and both partners move straight through it because the leader's driving force continues unchanged. It is only when both partners have achieved the desired speed in the opposite direction to before that the leader shuts off his driving force and begins a new driving force backwards (for him), to initiate a stretch and begin the whole process over again. In reality, the transitions between the forward and backward driving forces are not sharp and clunky like this but rather, gradual, ramping up and down like a wave.

In fact, everything about this whole process is like a wave, with different parts waving at different times, in a sequence. The diving force wave gets started first, the wave of the leader's speed follows shortly afterwards, which drives the wave of the elastic force applied to the follower. Coming slightly after that is the wave of the follower's acceleration and after that again, is the wave of her speed. So, we see the natural delay inherent in the process when everything is mediated by an elastic connection. To get a better feel for this sequence of waves, take a look back at this wiki page and take a close look at the third animation from the top, which has the arrows that grow and shrink over time. These arrows are like the waves for the quantities we've just discussed, but just for a single mass.

Right, there we have it! The sugar-push, cut to its bare bones! No magic, just some simple physics, which lets two bodies work together effectively. I promised it would be simple, didn't it? What's that?,"Then why is the above discussion so long and why does it require so much mind-bending visualisation?!" :-) Well, firstly, I guess I should say that simple doesn't necessarily mean easy! But that's not the whole story. Now that we've thought through everything in terms of forces, accelerations, speeds, etc, let's clear our heads for a second and then think of it in a new way, which will, hopefully, will be even simpler and tie this whole post together.

Tempo, frequency and spring constant: Resonance between music and dancers
 
Recall this, from the top of this post:

'if a mass of certain size, attached to a spring, is required to resonate at a particular frequency, the property of the system that must be carefully chosen is the spring constant (or elastic constant) k.'

Hold that thought as we discuss the following question: What is the most fundamental property of the music we dance to? This might seem like a controversial question to begin with but I don't think it is. I would say the answer (provided we restrict ourselves to a single genre of music, hot jazz, say) is tempo. Tempo is even more fundamental than rhythm. After genre, tempo is the only property of music which divides up events: contests have slow and fast divisions, exchanges have slow rooms and fast rooms. Tempo sets the energetic baseline of the dance.

Now, consider this: For all but a few exceptional dances, all dance movement is periodic - it is made up of waves. Everything that goes in one direction also must come back the other way in equal measure. You step out onto the floor with a partner and you move in all kinds of complicated ways but you (almost) always end up where you started, at (roughly) the same spot on the floor. You might move back a bit, but you always come forward again. You might move left but you return right. You bounce up and you drop down again. You end where you started. This means that, just like rhythms in the music you dance to, your movement is periodic; it goes in cycles and cycles within those cycles, and so on. Cycles always have a frequency - how fast they cycle/vibrate. When we dance, there are fast cycles (like bouncing with the beat), medium cycles (like the speed of repeated sugar-pushes or swingouts) and slow cycles (like drifting across the floor and back again). Most of those cycles - at least the fast and medium ones - are founded on the tempo. Let's tie this back in to our elastic model of connection.


Tying it all together
If a couple dancers repeat sugar-pushes or swingouts, say, at a particular tempo, their bodies are required to move back and forth - a kind of vibration - in sync with the music. In physics, when vibrations are in sync with each other, we say that they are on resonance. Now, remember that sentence from before again:
'if a mass of certain size, attached to a spring, is required to resonate at a particular frequency, the property of the system that must be carefully chosen is the spring constant (or elastic constant) k.'

Aha! If we assume that dancing is all about resonance with the music, and dance partnerships are held together by elastic connection, then it becomes clear that the tuning of the partnership's spring constant is of utmost importance. With this in mind, here's a genuinely simple recipe for good sugar-pushes, and indeed, any periodic, linear connected movement:

1. Listen to the tempo and use it to decide how quickly the dance partnership will have to resonate
2. Set your spring constant accordingly. Fast speeds need high constants (stiff springs), slow speeds, low constants (loose springs).
3. Apply the leader's leg driving force in time with the resonant frequency of the partnership, as determined by the spring constant. That is, like a father pushing his kid on the swing, the leader needs to add energy to the partnership's movement in time with its natural resonance to the music. All the rest of the nice, flowing dynamics is taken care of by the beautiful simplicity of the elastic connection; no fancy movements, no pushes or pulls, need to be added by either of the dance partners.

Of course, just repeating the same movement over and over again gets boring but there are lots of different movements that might be chosen and dancing is really just a matter of switching between them musically. Sometimes, not even one oscillation is completed before switching to a new movement but even a fraction of an elastic oscillation still follows the same rules:

1. Pick the speed of (part)oscillation required; how quickly does the leader want to follower to get to where he's going to send her?
2. Once he knows, he sets his spring constant and applies the driving force with his legs. A good follower will quickly detect her leader's spring constant and set her own to match it - this is the 'frame matching' that teachers sometimes talk about.
3. Simply hold an elastic relationship, obeying the rules of movement we've discussed earlier.
4. Change movement in interesting ways and repeat.
5. When a reduction in energy or a complete stop is desired, one or both partners apply damping, deliberately wasting their kinetic and potential energy so that they have less speed and/or less stretch, or stop completely, returning to a zero energy system.

Phew! Ok, that's it for this post. Now that we have thoroughly explored both movement and connection, at least for linear movements, in the next post we will discuss how even the most complex dance movements can be broken down in terms of just three directions of movement. If a pair of dance partners can move and connect in just those three directions, they can do literally anything, even the most complex 'moves'.