ME 4590 Dynamics of Machinery
Lagrange’s Equations for Multi-Degree-of-Freedom, Planar Systems
The configuration of systems with N degrees of freedom (DOF) can be defined in
terms of N generalized coordinates, say qk (k  1, , N ) . The differential equations
of motion (EOM) of the system can be derived using Lagrange's equations
d  K

dt  qk
 K
 Fqk

 qk
(k  1,
,N )
where
K
 m v
2
1
i
Gi


 vGi  I ii2  kinetic energy of system
bodies
Fqk  generalized force associated with generalized coordinate qk
Note that it is important that K and Fqk be written only in terms of qk and qk and no
other variables.
If some of the forces and torques are conservative, then we can define their
contribution to the equation of motion in terms of a potential energy function. In
this case, the differential EOM can be derived from the equations
d  L 
L
  Fqk 

 
nc
dt  qk 
qk
(k  1,
,N )
where L  K  V is the Lagrangian of the system, V is the potential energy
function for the conservative forces and torques, and  Fqk nc is the generalized force
associated with qk for the nonconservative forces and torques, only.
Kamman – ME 4590: page 1/1