The document covers the principles of virtual displacement, virtual work, and D'Alembert's principle in classical mechanics. It distinguishes between virtual and actual displacement, discusses the conditions under which virtual work is zero, and introduces generalized coordinates and their relationships. The principles are mathematically formalized, showing how forces and motion relate within a system of particles in equilibrium.

1. D’ Alembert’s Principle and
Generalized Force.
“A lecture on basics of classical mechanics.”
M. S. Ramaiah University of Applied
Sciences.
Faculty of Science and Humanities
Manmohan Dash

2. Virtual Displacement
• We define a Virtual Displacement of a system
of particles described by the coordinates ri at
a given instant of time t.
• Lets say the system has an infinitesimal
amount of change in its configuration due to
an arbitrary change in its coordinates; ri , in
accordance with the applied forces Fi
(a) and
constraint forces fi.

3. Virtual Displacement
• A virtual displacement is different from an
actual displacement in that in actual
displacement there is a time lapse dt during
which applied forces and constraint forces
may change.
• If the system is in equilibrium the force on the
ith particle is zero, so Fi = 0. As a result virtual
work done on the ith particle is zero or Fi.ri=0.
Considering all particles; iFi.ri = 0

4. Principle of virtual work
• The total force on ith particle can be written
into the applied force and the constraint force.
So; Fi = Fi
(a) + fi
• From i Fi.ri = 0; i Fi
(a).ri + i fi.ri = 0.
• We assume the “net virtual work done by the
forces of constraints” to be zero so that
i Fi
(a).ri = 0. This is known as “Principle of
virtual work”.

5. Virtual displacement
• In the above equation the virtual coordinates
ri are not independent of each other and as a
consequence all of the applied forces are not
zero.
• The virtual coordinates ri need to be
transformed into generalized coordinates qi
which are then independent of each other.

6. Conditions on Virtual Work
Under what conditions i fi.ri = 0 is valid;
o Rigid Body constraints are one such example.
o A particle moving on a surface. Here the
forces of constraints and displacement of
particle are perpendicular to each other.
o Rolling friction, as a constraint force; during
virtual displacement the point in contact with
surface is momentarily at rest.

7. Conditions on Virtual Displacement
Under what conditions i fi.ri = 0 is valid;
o If the surface on which a particle is in motion
is moving, the condition that “zero virtual
work due to forces of constraints” is still valid
in the instant of motion.

8. Equation of Motion
• The condition of virtual work being zero is not
sufficient for a general description of motion
• We need to introduce the equation of motion;
Fi = tPi where the t stands for differentiation
wrt time. We read tPi as Pi dot.
• We thus have Fi - tPi = 0. It states that the
system of particles is in a state of equilibrium
under the application of a force Fi and a
reverse effective force - tPi .

9. D’ Alembert’s Principle.
• The principle of virtual work now becomes;
i (Fi - tPi).ri = 0.
Since Fi = Fi
(a) + fi this becomes;
i (Fi
(a) - tPi).ri + i fi.ri = 0.
• By applying “zero net virtual work by the
forces of constraints” as discussed already we
have; i (Fi
(a) - tPi).ri = 0.
• i (Fi
(a) - tPi).ri = 0 is known as “ D’ Alembert’s
Principle “.

10. Generalized Coordinates.
• We transform the ordinary “dependent
coordinates ri ” into holonomic “generalized
coordinates qj” which are independent of each
other;
ri = ri (q1, q2, q3, qj, …, qn, t)
• There are n such generalized coordinates. (and
N dependent coordinates ri ).

11. Generalized velocity and
displacement.
• We apply the chain rule of partial
differentiation to get the generalized velocity.
vi  dri/dt = k ∂ri/∂qk .tqk + ∂ri/∂t.
This shows ordinary velocity vi is related this
way to the generalized velocity tqk.
• Also arbitrary virtual displacement ri is
connected to the generalized virtual
displacement qj; ri = j (∂ri/∂qj).qj

12. Generalized work and force.
• Virtual Work of Fi; we dropped index of Fi
(a)
i Fi.ri = i, j Fi.∂ri/∂qj qj = i Qjqj
with Qj = i Fi. ∂ri/∂qj
• The q is generalized coordinates, Q is
“generalized force” and q are “generalized
virtual displacement”. Also Qq are
“generalized work”.
• Q’s and q’s do not have the dimension of force
and length respectively but Qq have the
dimension of work necessarily.