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D'Alembert's Principle
By Philip Ellison on Thursday, February 21, 2002 - 09:57 pm:

In his answer to one of the STEP questions I posted on this board, Arun refers to "D'Alembert's Principle" and told me that it involves using "inertial forces" on bodies in static equilibrium. Could somebody please explain this principle or direct me to a website with information on it (as I cannot find it on the "World of Mathematics" site). If possible, please reply tonight.
Thanks for any help

By Kerwin Hui on Friday, February 22, 2002 - 01:17 am:

D'Alembert's principle is usually known as the principle of virtual work. It states that if a system is in equilibrium, then the work done (by external forces) in a virtual (infinitesimal) displacement is zero.

Kerwin

By Philip Ellison on Friday, February 22, 2002 - 06:27 pm:

Could you please elaborate on this definition (or give examples of how to use it, etc.)? The principle seems self-evident, but how to apply it less so.
Thanks

By Arun Iyer on Saturday, February 23, 2002 - 12:14 pm:

Phillip,
I will try and explain D'Alembert's principle to the best of my abilities...

Consider a particle P of mass m having acceleration a when acted upon by several forces(say F1 and F2).Let SF be the resultant of these forces....

Now,by Newton's second law,SF=ma
where the acceleration of the particle is in the direction of the resultant force SF.
This equation is referred to as the "equation of motion of particle P"

This equation can be written in the form...
SF-ma=0
which means that the resultant of the external forces(SF) and the force(-ma) is zero.The force(-ma) is called the inertia force.(The inertia of the body can be defined as resistance to the change in the condition of rest or of uniform motion of the body.)

Now the equation is written as...
SF+(inertia force)=0
where inertia force =-ma
this equation is called as the equation of dynamic equilibrium of the particle P.

Now the concept of D'Alemberts principle is that..."to write the equation of dynamic equilibrium of a particle, add a fictitious force equal to the inertia force to the external force acting on the particle and equate the sum to zero."

Hope this helps...

love arun

By Arun Iyer on Saturday, February 23, 2002 - 12:16 pm:

kerwin,
i don't know how D'Alembert's principle is related to the principle of virtual work...i think you may have go it wrong!!(Please do correct me if i am wrong!!!)

love arun

By Kerwin Hui on Saturday, February 23, 2002 - 02:08 pm:

Arun,

There are two versions of D'Alembert's principle (and different books have different nomenclature, as always). One version, which is also called the principle of virtual work, states that the condition for equilibrium of a system is that the virtual work of the applied forces is zero. The other version, which is also known as the dynamic principle of virtual work, is roughly what you had stated. More precisely, this version states that

dW=S i(Fi-dPi/dt)• dri=0

where Fi, Pi and dri are the external force on the ith particle, the momentum of the ith particle, and the virtual displacement of the ith particle respectively. This can be deduced from Newton's second law Fi+fi=dPi/dt and assuming the internal forces fi do no work (which is often the case). If the particle is in equilibrium, then Pi is constant for every i, so we recover the principle of virtual work.

Kerwin

By Arun Iyer on Saturday, February 23, 2002 - 05:32 pm:

Hmmm!!!i see!!
i know the principle of virtual work but never knew that its also called the D'Alembert's principle!!(thanks for the info Kerwin!!)

love arun
P.S->i really like principle of virtual work ,it makes so many problems of mechanics simple...!!