Page 175 - The Mechatronics Handbook

0066-frame-C09 Page 49 Friday, January 18, 2002 11:01 AM

If the transforming relations are restricted to be holonomic, the constraint conditions are implicit in
the transforming relations, and independent coordinates are assured. Consequently, all the terms in Eq.
(9.30) must vanish for independent virtual displacements, δq j , resulting in the n equations:


d ∂T  ∂T
----- ------- – ------- = Q j (9.31)


dt ∂q ˙ j ∂q j
These equations become Lagrange’s equations through the following development. Restrict all the applied
forces, Q j , to be derivable from a scalar function, U, where in general, U = U(q j , q ˙ j ), and

d ∂U
------- +
Q j = – ∂U ----- -------  

∂q j dt ∂q ˙ j
The Lagrangian is defined as L = T − U, and substituted into Eq. (9.31) to yield the n Lagrange equations:

d ∂L  ∂L
----- ------- – ------- = Q j (9.32)


dt ∂q ˙ j ∂q j
This formulation yields n second-order ODEs in the q j .
Dealing with Nonconservative Effects
The derivation of Lagrange’s equations assumes, to some extent, that the system is conservative, meaning
that the total of kinetic and potential energy remains constant. This is not a limiting assumption because the
process of reticulation provides a way to extract nonconservative effects (inputs, dissipation), and then
to assemble the system later. It is necessary to recognize that the nonconservative effects can be integrated
into a model based on Lagrange’s equations using the Q i ’s. Associating these forces with the generalized
coordinates implies work is done, and this is in accord with energy conservation principles (we account
for total work done on system). The generalized force associated with a coordinate, q i , and due to external
forces is then derived from Q i = δW i /δ q i , where W i is the work done on the system by all external forces
during the displacement, δq i .

Extensions for Nonholonomic Systems
In the case of nonholonomic constraints, the coordinates q j are not independent. Assume you have m
nonholonomic constraints (m ≤ n). If the equations of constaint can be put in the form

∑ --------dq k + ------- dt = ∑ a lk dq k + a lt dt = 0 (9.33)
∂a l
∂a l
∂q k ∂t
k k
where l indexes up to m such constraints, then the Lagrange equations are formulated with Lagrange
undetermined multipliers, λ l . We maintain n coordinates, q k , but the n Lagrange equations are now
expressed [11] as

d  ∂L  ∂L
----- -------- – -------- = ∑ λ l a lk , k = 1, 2,…,n (9.34)

dt ∂q ˙ 
k ∂q k l
However, since there are now m unknown Lagrange multipliers, λ l , it is necessary to solve an additional
m equations:

∑ a lk q ˙ + a lt = 0 (9.35)
k
k

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