# Virtual work and D'alembert's principle

CrazyNeutrino
I can't for the life of me figure out what virtual work or D'alemberts principle mean and what the intuition behind them is. As far as I'm concerned D'alemberts principle is just a restatement of Newton's second law but considering the work instead of just the forces. What am I missing? I'm trying to understand how Lagrange developed his variational principles and how Hamilton's principle of least action arose from Virtual work and D'alemberts principle.

Dr.D
First, I would suggest that you mentally separate Virtual Work (VW) and D'Alembert's Principle. They are different things.

In many respects, you are correct about VW being equivalent to Newton's Law; both give the equations of static equilibrium. The advantage of VW is often found that only active forces (forces that do work in a virtual displacement) need to be considered. This eliminates all the reactions at fixed supports and similar. The Principle of VW is often very useful for systems where the geometry is the thing to be determined, such as a linkage hanging under the influence of gravity. In a case like that, the linkage angles are all unknown (they are to be determined), and the force summations become awkward.

D'Alembert's Principle is an entirely different ball of wax. At the risk of possibly stepping on some toes, I see this as an outright contradiction to the Law of Action and Reaction. D'Alembert postulates a force (-M*a) that does not exist. As a force that does not exist, it therefore has no reaction to be found anywhere. This becomes extremely confusing, and I will never use D'Alembert. The "advantage" of D'Alembert is that it reduces the problem to a statics problem, or so they say. What advantage is that, when you must still find the acceleration? It usually has the effect of causing people to not pay adequate attention to the kinematics problem, and thus to incorrectly express the acceleration. Overall, it is a grand source of confusion.

Virtual Work is very powerful and useful; D'Alembert not so much!

CrazyNeutrino
CrazyNeutrino
Thank you! I'm starting to get a sense of what these actually mean. Thus far I have just seen derivations and statements of both principles.
First, I would suggest that you mentally separate Virtual Work (VW) and D'Alembert's Principle. They are different things.

In many respects, you are correct about VW being equivalent to Newton's Law; both give the equations of static equilibrium. The advantage of VW is often found that only active forces (forces that do work in a virtual displacement) need to be considered. This eliminates all the reactions at fixed supports and similar. The Principle of VW is often very useful for systems where the geometry is the thing to be determined, such as a linkage hanging under the influence of gravity. In a case like that, the linkage angles are all unknown (they are to be determined), and the force summations become awkward.

D'Alembert's Principle is an entirely different ball of wax. At the risk of possibly stepping on some toes, I see this as an outright contradiction to the Law of Action and Reaction. D'Alembert postulates a force (-M*a) that does not exist. As a force that does not exist, it therefore has no reaction to be found anywhere. This becomes extremely confusing, and I will never use D'Alembert. The "advantage" of D'Alembert is that it reduces the problem to a statics problem, or so they say. What advantage is that, when you must still find the acceleration? It usually has the effect of causing people to not pay adequate attention to the kinematics problem, and thus to incorrectly express the acceleration. Overall, it is a grand source of confusion.

Virtual Work is very powerful and useful; D'Alembert not so much!

Thank you! Thus far I've only seen insipid derivations and statements of both principles but I think I'm starting to understand what they actually mean. Also, are you Dr, D as in Dr. Dynamics on youtube?

Dr.D
Also, are you Dr, D as in Dr. Dynamics on youtube?

No, I have nothing on YouTube. I'm just a retired ME Prof, who continues to work in kinematics, dynamics, and vibrations on a daily basis. I'm currently revising a textbook I wrote almost 30 years ago, and there is much to update.

CrazyNeutrino
Oh alright, thank you nonetheless!