Looking for a book on non-holonomic constraints with worked examples

[PhDeezNutz]

A lot of the notes online about non-holonomic constraints are mathematically/theoretically heavy with no real worked examples. I feel like worked examples are a good place to start; it gives me an (incomplete) overview that helps me see the “forest from the trees”. If I can see the “forest from the trees” then that would help to motivate the theory. As it is right now all the notes online seem like “formula soup” to me. I’m looking for something that is digestible.To be clear I’m looking for the Lagrangian- treatment of general non-holonomic constraints.

$f_j \left(q_1,...,q_n, \dot{q}_1,..., \dot{q}_n\right) = c_j$

Depending on the problem at hand you can change the constraints to pure position constraints or pure velocity constraints but I’m trying to learn how to handle a most general situation.

thanks in advance for pointing me

 

[Frabjous]

Whittaker has a chapter. See attached.

 

[PhDeezNutz]

Whittaker has a chapter. See attached.

Thank you for posting. It’s funny that they should mention curvature of a surface. I always thought that

rolling without slipping on a surface with constant curvature = holonomic

and

rolling without slipping on a surface with non-constant curvature = non-holonomic

So at first glance it seems like Whitaker corroborates my notions.

I’ll check it out. Thank you again.

 

[Frabjous]

A control systems person recommended

https://www.amazon.com/gp/product/1493930168/?tag=pfamazon01-20

https://www.amazon.com/dp/082183617X/?tag=pfamazon01-20

 

[PhDeezNutz]

A control systems person recommended

https://www.amazon.com/gp/product/1493930168/?tag=pfamazon01-20

https://www.amazon.com/dp/082183617X/?tag=pfamazon01-20

Marsden is my kind of guy (R.I.P.). We used his vector calculus book in my multivariable calc class and I thoroughly enjoyed it. From perusing papers on non-holonomic dynamics it seems like Bloch is an important player so I’m glad to see they wrote a book together.

On a related note I’m finding that the notion of “virtual displacements” and “virtual work” as defined in Goldstein is sort of wrong or at least very narrow in scope.

https://arxiv.org/pdf/physics/0510204.pdf

Virtual displacements aren’t completely arbitrary as one would be led to believe from reading Goldstein. In fact they must be perpendicular to the forces of constraint even if the actual displacements are not.

that’s my take from reading the above paper.

 

[vanhees71]

The trick is then of course to introduce the Lagrange multipliers! Goldstein is nevertheless wrong in his use of the action principle with non-holonomous constraints (he's right about it using d'Alembert's principle).

 

[Frabjous]

You might find stuff of use in
Pars
https://www.amazon.com/dp/0435526901/?tag=pfamazon01-20

and

Papastavridis
https://www.amazon.com/dp/9814338710/?tag=pfamazon01-20
https://www.amazon.com/dp/0849385148/?tag=pfamazon01-20

 

[wrobel]

By the way nobody knows mechanical systems with constraints that are nonlinear in velocities besides a pair of pretty artificial constructions

 

[Haborix]

By the way nobody knows mechanical systems with constraints that are nonlinear in velocities besides a pair of pretty artificial constructions

What's the nature of problem with such systems (those with constraints non-linear in velocity)? Is the mathematical machinery not up to the task? Are the results nonsensical?

 

[wrobel]

What's the nature of problem with such systems (those with constraints non-linear in velocity)?

Problem is to bring an example of such a system

 

[Haborix]

Ah, I see, thanks.

 

[PhDeezNutz]

Problem is to bring an example of such a system

One of my grad classical mechanics test questions (in 2017) was the following

The constraints I come up with are

$f \left(x,y\right) = y - A\cos \left( k x \right) = 0$$g\left( \dot{x} , \dot{y} , \dot{\theta} \right) = \sqrt{\dot{x}^2 + \dot{y}^2} - R \dot{\theta} = 0$

The first being holonomic and the second being non-holonomic. I believe the second constraint must be put into the form of $g\left( \dot{x} , \dot{y} , \dot{\theta} \right)$ in order to answer part e)

For what it's worth I know that my professor did not use this question in later semesters.

 

[wrobel]

The first being holonomic and the second being non-holonomic.

If you mean that the problems sheet you brought contains nonholonomic problems that is wrong.

 

[PhDeezNutz]

If you mean that the problems sheet you brought contains nonholonomic problems that is wrong.

Isn’t rolling without slipping on a surface of non-constant curvature a non-holonomic constraint?

I believe in order to solve for the force of constraint in part e) you need to cast the second constraint as a non-holonomic constraint (i.e. a purely velocity constraint?)

$g\left( \dot{x} , \dot{y} , \dot{\theta} \right) = \sqrt{\dot{x}^2 + \dot{y}^2} - R \dot{\theta} = 0$The constraint may be linear in overall velocity

$v = \sqrt{\dot{x}^2 + \dot{y}^2}$

But it's not linear in $\dot{x}$ or $\dot{y}$ (Which again I think we have to keep $\dot{x}$ and $\dot{y}$ separate in order to solve for the force of constraint).

Unless there is a way to "linearize" this constraint that I don't know of.

 

[wrobel]

If a cylinder rolls on the floor of the form y=f(x) then the center of the cylinder describes a curve y=g(x). This is under assumption that the radius of the cylinder is smaller than 1/curvature of the floor. The function g is determined by f.

If the cylinder can slip then this is a system with two degrees of freedom. Generalized coordinates are the angle of cylinder's rotation and x-coordinate of the center of the cylinder. These two coordinates as well as corresponding generalized velocities are independent. The system is holonomic.