3.2 Lagrange’s Equations of Motion 121
cussed in Chapter 1. These are a set of second order, ordinary differential
equations involving the generalized displacement variables for the system.
Given an appropriate set of initial conditions these differential equations can
be solved to find the trajectory of the system.
The application of equation (3.9) requires:
1. The selection of a set of the n generalized displacements q = [q
1
,q
2
,···,q
n
]
T
,
and the corresponding generalized flows f = [f
1
, f
2
, ···, f
n
]
T
.
2. The formulation of the kinetic coenergy, T
∗
(f, q, t), the potential energy,
V (q), and the dissipation function, D(f), in terms of the generalized dis-
placement and flow variables.
3. The determination of the virtual work done by the applied efforts δW =
P
n
i=1
e
s
i
δq
i
, which yields the generalized efforts due to effort sources, e
s
i
,
i = 1, 2, ···, n.
Note that the generalized efforts do not include the applied efforts due to
capacitors and resistors, since these applied efforts are already accounted for
by V (q) and D(f ), respectively.
The Lagrangian approach to constructing the equation of motion does
not require the determination of the constraint efforts. (This is because the
constraint efforts do no virtual work.) The exclusion of the constraint efforts
greatly simplifies the modeling process since, the resolution of the constraint
efforts is often a cumbersome task.
The significance of the terms in (3.9) can be illustrated by rearranging the
equation as follows. Let e
C
i
= −∂V/∂q
i
be the efforts applied to the system
by the ideal capacitors, e
R
i
= −∂D/∂ ˙q
i
be the efforts applied to the system by
the ideal resistors, and e
I
i
= d/dt(∂T
∗
/∂ ˙q
i
) − ∂T
∗
/∂q
i
be the inertia efforts.
Then, (3.9) can be rewritten as
d
dt
∂T
∗
∂ ˙q
i
−
∂T
∗
∂q
i
= −
∂D
∂ ˙q
i
−
∂V
∂q
i
+ e
s
i
,
e
I
i
= e
R
i
+ e
C
i
+ e
s
i
= e
i
, i = 1, 2, ···, n.
where e
i
are the applied efforts due to the capacitors, resistors and sources.
Reinserting the last equation into (g) gives
n
X
i=1
(e
I
i
− e
i
) δq
i
= 0. (3.10)
In analytical dynamics equation (3.10) is known as D’Alembert’s Principle.
Which is an application of Bernoulli’s Principle of Virtual Work to dynamic
systems.
The principle of virtual work states that; A system is in equilibrium if
and only if the virtual work done by the applied efforts is zero, i.e., δW =
P
n
i=1
e
i
δq
i
= 0.