Passive muscle forces

Hi guys,

my first thread here at the forum. How is summer in Denmark? As rainy and cold as it is in Germany…?

I have a general question: I modified the mandible model (clenching) from the AMMR. Applied a force at the two incisors of the mandible varying from 10 to 500 N (10 N per 0.2 seconds) and set the motion to zero at a postion of the previous motion. Then I looked at the forces of the tendon and muscles: I expected a linear behaviour of the forces, so double external forces would give you double internal forces. But actually it’s not like that. The graphs seem to be straight lines but especially in the beginning of the simulation you can see some non linearities. (see attached file clenchingFt.emf)
Then I looked at the passive forces of the muscle (see attached file clenchingFp.emf) and there you have some non linear behaviour. Since I didn’t change kinematics, there must be a relation to external forces.
I would like to know, how the mathematical formulation of the passive forces in the muscles look like and why?
Thank you very much for your help!

Best regards,

Hi Alex

June has been rather rainy until now, but now it seems to make a turn!

I will try to explain the behaviour of the muscle model.

The loading conditions in your case are that the muscle has a fixed length and force it nees to develop is increasing:

The muscle model is a three element hill type model where the passive element is in parallel with the contractile element.

In this loading case the muscle has initially high passive force in the parallel element, then when the muscle needs to develop more force the contratile elenment increase its forces. The contratile element is in parallel with the passive element so the passive element will gradually get unloaded when the activation increases, i think this is what you see.

The muscle model is itself a small mechanical system which finds it own equilibrium by changing the tendon length. So when the contratile element increase the activation the tendon will get longer and the passive element shorter which will mean a lower passive force. So it is not psosible to give the exact equations for the this since this equilibrim is solved numerically.

The stiffness model of the passive element and the tendon are both non-linear.

Please ask again if you have further question.

Best regards

Thank you for the answer!

Can you then give me an equation that describes the non linear behaviour of the tendon/ligament? Of course the muscle equilibrium is solved internally, but I would like to have have a look at the influence of the parameters you have to define in AnyBody (Jte, K1, K2 and so on). You could do this in AnyBody as well of course, but I guess I won’t find the equation with the parameters only seeing some graph…

Best regards,

Hi Alex,

Sorry abotu the slow reply.

I have attached the formula used for the tendon, the equation shows the force in the tendon as a function of the length:

Here is some explanation on the symbols:

K=J_T shapefactor
l=lt (current length of the tendon)

Hope it makes sense

Best regards

Hi Soeren,

thank you for the answer. I attached an Excel sheet where I tried to plot a graph what happens with the force if you stretch a tendon.
I was suprised by the extremly high forces when a strain of about 10% percent is appraoched. But of course after thinking about it, it makes sense in my opinion. Tendon is stiff tissue.

If you want to, just have a look ta the Excel file and give me some feedback if at least the graph looks correct to you;-)

If it is correct feel free to publish it on whenever you want.

Thanks and Best regards,

Hi Alex

Thanks for the excel file, i had a quick look at it and i thinks it looks reasonable.

It has a zero force at its slack length and about 1000N at 5% strain as the numbers also suggest it should have.

Best regards