The meaning of the D'Alembert's Principle

[Loro]

The D'Alembert's Principle states that:

$\sum_s [\underline{ F_s^{applied}} - \frac{d}{dt} (\underline{p_s}) ] \cdot \underline{δr_s} = 0$
s - labels particles

That is when $F_s$ doesn't include the constraint forces, and the virtual displacement is reversible, and compatible with the constraints.

My question is - doesn't it just say that:

- if there are no constraints, Newton's laws are obeyed, with force being $F_s$ - (the parenthesis is zero)
- if there are holonomic constraints, we can only displace the object perpendicular to the constraint forces - (the dot product is zero)
?

Does this principle also say something about non-holonomic constraints? And if so, can anyone give an example?

And what exactly is the difference between reversible and irreversible virtual displacement? If a displacement is virtual, and if displacing by dx is possible, then also displacing back by -dx should be possible. So how can we have an irreversible displacement at all?

 

[Studiot]

Whenever you use virtual displacements you introduce another condition into the system.

You introduce (geometric) compatibility.
This condition limits what virtual displacements you can employ.

D'Alembert's principle allows you to add imaginary forces(not virtual forces they are different) to a non equilibrium system to employ the equations of equilibrium.

 

[Loro]

I understand the bit about virtual displacements.

Let's think of a ball rolling off a solid sphere.

https://dl.dropbox.com/u/94695102/fizyka/kula.jpg

One way of looking at it is that there is a reaction force $F_R$, etc.

But we can also think that the ball is constrained to be outside of the sphere, and that puts a limitation on the possible virtual displacements.
In this picture $F_R$ is a constraint force, and $F_G$ an applied force. Also: $\frac{d}{dt}p = F_{net}$ And in my book there's this formula that I've given in the previous post.

When the ball doesn't touch the sphere, $F_{net} = F_G$ so clearly the parenthesis is zero so it works.

When the ball still rolls on the sphere, the formula apparently works only is we displace it along its surface - this would be the only possibility if the constraints were holonomic - then the displacement is perpendicular to the parenthesis which equals $F_R$.

But if we raise it outwards from the surface, which is compatible with the constraints, the dot product isn't zero, and neither is the parenthesis. Where's my mistake?

And before I try to digest your second point - is "imaginary force" the same thing as "fictitious force"?

 

[Studiot]

And before I try to digest your second point - is "imaginary force" the same thing as "fictitious force"?

That's a quick yes.

For instance one way to tackle a body, mass m, traveling in a circle is to introduce an imaginary or fictitious centrifugal force F = mass x central acceleration = mrω²

 

[PJC01]

I would respond by saying that the D'Alembert's Principle is a fundamental principle in classical mechanics that helps us understand the motion of objects under various constraints. It states that the sum of the applied forces and the time derivative of the momentum of each particle in a system, multiplied by the virtual displacement of each particle, is equal to zero.

Your understanding of the principle is correct. If there are no constraints, then the principle reduces to Newton's laws of motion. If there are holonomic constraints (constraints that can be described by equations involving only the positions of the particles), then the virtual displacement must be perpendicular to the constraint forces. This means that the particles can only move in directions that are allowed by the constraints.

The D'Alembert's Principle does not directly address non-holonomic constraints (constraints that cannot be described by equations involving only the positions of the particles). These types of constraints can make the virtual displacement irreversible, meaning that the particles cannot be displaced back to their original positions. An example of this would be a ball rolling on a rough surface. The friction between the ball and the surface would cause an irreversible virtual displacement.

The difference between reversible and irreversible virtual displacement lies in the constraints present in the system. In a reversible virtual displacement, the particles can be displaced back to their original positions without violating any constraints. In an irreversible virtual displacement, the constraints prevent the particles from returning to their original positions.

In summary, the D'Alembert's Principle is a powerful tool for analyzing the motion of particles under various constraints. It helps us understand how forces and constraints affect the motion of objects and can be applied to both holonomic and non-holonomic systems.