Mathematical solution needed for this biomechanics problem

[danlightbulb]

Hi all,

Wasn't sure whether this fits best in the maths or engineering forums, either way I'm hoping that a mathematical solution exists for my problem which I'll now explain.

My problem is one of biomechanics. One of the primary exercises I use is the barbell squat. Holding a loaded bar on your upper back and squatting down means that the skeleton of your body moves, the joints move and the bones and muscles transmit force through them down to the floor. I am looking for a mathematical model which can take the various measurements of the body and calculate the angles the joints are making with each other, the forces that are being transmitted through each member and the moments that exists around the joints.

Here is an image showing the position of the skeleton in the squat.

The load is positioned on the upper back, and L1 represents the length from the load to the hip joint. L2 is the length of the upper leg. The lower leg of length L3 pivots at the ankle and L4 represents the height of the heel on the shoe, which serves to lift the heel of the foot off the floor slightly. L5 is the length of the foot and the load must always remain on the centreline of the foot, as represented by the red centreline. The load always moves vertically on this centreline. The only inputs to the calculation are the lengths L1 to L5, and the size of the load in kilograms.

A1 is the angle formed between the spine and the upper leg at the hip. A2 is the angle formed between the upper leg and the lower leg at the knee. A3 is the angle formed between the lower leg and the foot at the ankle. What are these angles for any lengths of L1 to L5?

The load is stationary in this position, so the system must be in equilibrium. The load must be being transmitted to the ground through the bones. Given the size of the load, the various lengths L1 to L5, and the angles they are creating, what are the forces in each segment and what are the moments being created around each joint?Im looking for a set of equations that can describe this model which I can plug in various lengths of L1 to L5 and a size of load variable.

Many thanks and looking forward to any responses!

Dan

 

[paisiello2]

Do you know any trigonometry? Do you know any engineering mechanics?

 

[felmon38]

You can solve this problem by steps :

1. We isolete the spine and we apply the three dynamics equations on it to calculate the forces in A1 as the torque.
2. Do the same for L2, L3 and L5.

 

[danlightbulb]

Hi thanks for the replies.

The forum mods asked me to mention that this isn't a homework problem - I am not a student. Its just something I am trying to figure out for myself.

Regarding attempts at solving it. I know basic trigonomentry and how to use sine, cosine & tangent. But they aren't right angled triangles so I didn't think these functions worked.

For example I tried sin(A1)=O/H. But I can't work out O, because the triangle formed by L1 and L2 (and a hypothetical third side, O) isn't a right angled triangle.And felmon, thanks, its been so long since I've done any engineering study I don't know what the three dynamics equations are. And similar to the above, I struggled to break the load down into its vectors because they aren't right angled triangles.

 

[OldEngr63]

The load must be being transmitted to the ground through the bones.

Why do you say this? This seems to imply that the muscles are all slack when the system is under load, and I just don't believe that is true.

 

[felmon38]

Sorry danlight..., to solve this problem you must review the Newton Mechanic you studied
Regards

 

[Inventive]

A good start could be to review the law of sines and cosines. Your drawing shows a side view of a person. I am seeing this as a 3d figure compressed into the y-z plane with the thickness of the person and the barbell projecting into and out of the paper. Is the load perfectly balanced across the person? the components of L1 to L5 can be computed using the law of cosines since you are dealing with triangular shapes that are not right triangles. Once you know the magnitudes and the angles then you can form unit vectors who's length will be 1. If you take the vector representation of L1 as a unit vector for example then multiply it by a new length of the line represented by L1 will give you a new vector with a unique magnitude and same direction.
The torque can be computed by choosing a axis of rotation and taking the magnitude of the cross product of the two two vectors. depending on the direction of rotation the result will be pointing into or out of the paper
I hope this helps it has been a while worked mechanics problems

 

[danlightbulb]

Why do you say this? This seems to imply that the muscles are all slack when the system is under load, and I just don't believe that is true.

I was simplifying. The muscles are under tension and the bones under compression in a static position with the load stationary. Forces in the muscles equal and opposite to the forces in the bones, and around the joints, there are moments which are equal and opposite also, again because the load is stationary. I was correct in saying the load is being transmitted through the bones to the ground though, it can only be that way. In this static system, the muscles serve to cancel out the moments around the joints (otherwise the load would move), and the bones act in compression to transmit the force to the ground.

However what I am interested in is working out what these forces are. Felmon, I'm not capable of working this out for myself, its a complex problem, which is why I am posting here in the first place. I am afraid I need more than just a hint.

A good start could be to review the law of sines and cosines.

I just looked up the law of cosines. Unfortunately you need to know 2 sides and an angle to be able to compute other sides or angles. In this case we only know 2 sides (L1 and L2), not any of the angles between them. Unless I am misunderstanding this, I'm not sure if it can be used to solve the problem?

 

[Inventive]

Maybe this could help you too. would it help if someone setup one of the equations for you with some insight on how they accomplished this?

 

[danlightbulb]

Maybe this could help you too. would it help if someone setup one of the equations for you with some insight on how they accomplished this?

Yes it would because I really don't think I can solve it myself.

 

[paisiello2]

I'll do the easy one first:

M@A3= P×L5×cos(A5)
where
P = Load
A5 = Arcsin(L4/L5)

assuming we can ignore the weight of the lifter

 

[danlightbulb]

You've picked the only right angled triangle there haven't you lol.

So you're working out what A5 is (a new label representing the angle between the toe end of the foot and the ground), by taking the inverse sine of opposite (L4) / hypotenuse (L5), which only works for this bit because its a right angled triangle.

Then to calculate the moment you're taking the load, P, multiplying it by the length L5 and the cosine of A5 which is to somehow get the right vector is that correct?

Using some real life numbers.

P=100kg
L5=0.2m
L4=0.02m

M@A3=19.9 kg.m

So ok, not sure what that number really means in this context. Also I still don't know how to apply this to the other triangles which are not right angled triangles.

And I've spotted a misleading thing in the picture I posted as well. The foot is actually completely in contact with the ground through the shoe, its not raised off the ground like it looks there. The weightlifting shoe is shaped like a wedge, higher at the heel than the toe end, but the sole is completely in contact with the ground. The length L4 represents the height of the heel on the shoe, which during the lift movement is something which effects the knee position.

 

[paisiello2]

Maybe you could convert them into right angled triangles.

 

[danlightbulb]

You mean by subdividing them? I did think of that, but didn't know where to start with it.

 

[paisiello2]

Well to start with L3, the angle A3 - A5 makes a right angled triangle, dinnit?

 

[danlightbulb]

Well to start with L3, the angle A3 - A5 makes a right angled triangle, dinnit?

Im not sure what you mean sorry. Are you saying A3 minus A5 there? Or A3 to A5? A5 is the angle you labelled, I assumed it was the angle between the floor and the foot (L5) is that right? I don't see how anything there can be a right angled triangle.

 

[paisiello2]

Yes, I meant A3 minus A5.

It might help to communicate better if you draw a horizontal line through each joint on your sketch and label the angle each member makes with this horizontal line.

 

[danlightbulb]

Like this:

Ive labeled the angles between the horizontal lines as B1, B2 and B3.

Still don't understand what A3-A5 means.

 

[paisiello2]

Yes, that looks good. Hopefully you can see all the right angled triangles now.

Also, it should be clear that B3 = A5, yes?

 

[danlightbulb]

I can see that B3 = A5 yes, but how does that help figure out A3?

Im still not really sure on the right angled triangles. Are you saying that one of them might be if a line is dropped from the knee joint vertically down to the horizontal line above the foot? In that case L3 is the hypotenuse (side C in pythag), but the length of the hypothetical sides A and B is still unknown, and the angles A3 and A2 are still unknown so I don't see how that helps things.

 

[paisiello2]

But the angle A3 - A5 makes the right angled triangle for L3, n'est pas?

 

[danlightbulb]

Yes I can see that, but we don't know the value of A3 in the first place so how can we numerically calculate A3 just knowing L3, even if it is a right angled triangle. We still only know the numerical value of length L3 unless I am missing something?

Using my previous numbers, L5=0.2m, L4=0.02m and let's say L3=0.5m.

So we can work out A5 = B3 = Arcsin(0.02/0.2) = 5.7 degrees.

So we then know that L3 = 0.5m, and that A3-5.7 degrees = some angle which we still can't calculate.

 

[paisiello2]

Sorry, I see what you are getting at now. I was assuming that you were measuring all the angles and lengths.

The problem is statically indeterminate. You would need to know the rotational stiffness at each joint in order to solve the problem.

 

[danlightbulb]

Just the lengths are known because you would be measuring the limb lengths of people. The angles are controlled by the pull of the muscles around the joints, and are different for every person because limb proportions are different for everyone. However in this position the load is static, so the tensions being exerted by the muscles must create a single set of angles for that person based on their limb lengths. For example, when I squat I always squat to the same set of angles. I want to calculate what those angles are, the forces in the bones and tension in the muscles.

 

[paisiello2]

Yes, you are right, there would be a single set of angles created for a given load P. You would still need to know the rotational stiffness of each joint. The angles you make would depend on this stiffness.

 

[danlightbulb]

So in the human body, what stiffness does a joint have then? Its actually a range of none to lots, and that notional stiffnes is created by the muscles during the movement.

If I squat 50kg the angles are the same as if I squat 100kg, so I gather the muscles create a kind of dynamic stiffness to compensate exactly for the load?

Given then that the stiffness must be exactly compensating for the load, is there no way to calculate the solution? The stiffness must be whatever it needs to be to exactly balance the load.

 

[paisiello2]

I gave a little more thought on this. The lifter can assume any stance which is mostly arbitrary and only depends on 3 constraints:

1) the capability of each joint to form a given angle e.g. say the knee joint is capable of forming any angle A2 between 20 and 90 degrees.
2) the capability of each joint to sustain a given bending moment caused by the weight P
3) the angles must all take on values such that the weight P always lies directly over the "center" of the shoe (there is probably an equation to describe this but there would still be an infinite number of solutions)

So as long as the angles all fall within the above constraints then they can take on any value.

 

[danlightbulb]

Hmm i think in theory yes, many angles can fit the solution, but for a given individual for given limb lengths he only squats with one particular set of angles. So what is it that is making those angles what they are for a given individual?

There are possibly more contraints that can be given. For example, the upper leg must be below parallel, which typically means that B2 would be (I'd guess) between 5 and 10 degrees off the horizontal. Any less than that and its not a full squat, any more and the body won't allow you to hold your back muscles tight. So if we were to set the angle B2 at 5 degrees from the horizontal, does that then allow the rest of the system to be calculated?

 

[paisiello2]

I don't think so. I assume it would depend on each individual's specific physiology. So you would have to measure it.

 

[danlightbulb]

Guys I am struggling to believe there isn't a mathematical solution to this problem. There are surely enough known constraints to be able to work this out, for example:

1. The load can only travel vertically. Therefore the angle at A1 cannot get smaller on its own because L1 length is fixed. If A1 got smaller on its own then the load would move to the right of the vertical.

2. A1 could get bigger if A2 got smaller, but there are constraints on how small A2 can go because the leg flesh gets in the way. This is the same at A3, it can only go so far because of ankle mobility. A1 can only go to a certain smallness as well, constrained by the midsection contacting the thighs.

3. L2 must be below parallel with the ground.

There must be a way to use this information to constrain the system and calculate the solution.

Any more ideas?