# Stability analysis basics

I would like to see a rigorous treatment of stability analysis.

For example, a lot of high-school level texts give examples like this: https://courses.lumenlearning.com/physics/chapter/9-3-stability/

But they describe stability in very qualitative terms like "Hence, the chicken is in very stable equilibrium, since a relatively large displacement is needed to render it unstable. The body of the chicken is supported from above by the hips and acts as a pendulum between the hips. Therefore, the chicken is stable for front-to-back displacements as well as for side-to-side displacements."

What is the quantitative theory behind this?

If I give you an arbitrary object mesh, it's center of mass, and its position and orientation, can you give me a formula to determine if it is stable or not?

What if I give you $$n$$ objects?

What is the quantitative theory behind this?

The chicken is more stable the greater the restoring force or torque. See the right side figure below for an example of restoring torque where a force is applied from the left so chicken's left foot just leaves the ground.

An attempt to topple the chicken by tipping it on its right foot (facing the chicken) (point A) by the application of the force F through the center of gravity (COG) of the chicken giving it a clockwise torque (moment), is opposed by the counter clockwise torque about A due to the weight of the chicken acting through the COG. The sum of the moments about A is shown in the figure, where counter clockwise moments are considered positive. Equilibrium is when the sum is zero.

Note that for the chicken to topple over the torque due to the applied force has to exceed the restoring torque due to the chicken's weight, or

$$Fh>W\frac{L}{2}$$

When the force is removed, the counterclockwise torque acts to restore the chicken to standing on both feet. Note the following:

1. The lower the COG (lower the value of $$h$$) the greater the force $$F$$ needed to topple the chicken making the chicken more stable, all other things being equal.

2. The wider the chicken's feet are apart (the greater $$L/2$$), the greater the force $$F$$ needed to topple the chicken, making the chicken more stable, all other things being equal.

In both cases, the greater the stability of the chicken the greater the displacement of the COG to the right needed to topple the chicken.

If I give you an arbitrary object mesh, it's center of mass, and its position and orientation, can you give me a formula to determine if it is stable or not?

Not sure what you mean by "an arbitrary object mesh", but as your link explains a system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in a direction opposite to the direction of the displacement. So any formula will involve a net force or net torque.

Hope this helps.

• I don't understand what the W L / 2 is. How is that derived? – user3180 Jan 21 at 20:19
• What if the chicken has 3 legs? – user3180 Jan 21 at 20:19
• I think this paper: core.ac.uk/download/pdf/216155659.pdf describes what I want – user3180 Jan 21 at 20:20
• But my physics is not good enough to understand it – user3180 Jan 21 at 20:20
• How can I build my physics up to this level? – user3180 Jan 21 at 20:20

Today I'm in the mood of trying to be an artist 😂 :

Chicken is stable only partially, otherwise why it needs nails ? Humans uses somatosensory system in brains to control stability and balance in a stationary or moving positions. Chicken has small nervous system, thus nails for compensation.

The quantitative theory behind this is called "Dynamical systems". It is usefull to study large dimensionnal systems (n objects) and also non-linear systems.

If you have access to a university library, you can have a look to Strogatz, S. H. (2018). Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering. CRC press., which is a reference for this field.

• I've taken a bit of dynamical systems class, but as I recall it's a very different setting, where we consider simple systems and point masses – user3180 Jan 21 at 20:23
• I am more interested in things such as this statement "A sufficient condition for stability is if the object center of mass, when projected along the z-axis, ends up in the convex hull footprint of an object touching the table" – user3180 Jan 21 at 20:23