ME 4590 Dynamics of Machinery
Lagrange's Equations for One Degree-of-Freedom, Planar Systems
The configuration of systems with one degree of freedom (DOF) can be
defined in terms of a single generalized coordinate, say q . The differential
equation of motion (EOM) of the system can be derived using the single
Lagrange's equation
d  K  K
 Fq


dt  q  q
where Fq is the generalized force associated with q and K is the kinetic energy of
the system
K  12
 m v
i
Gi

 vGi  I Gi i2
bodies
(i )

Note that it is important that K and Fq be written only in terms of q and q and no
other variables.
If some of the forces and torques are conservative, then we can define their
contribution to the EOM in terms of a potential energy function. In this case, the
EOM can be derived from the single Lagrange's equation
d  L  L
  Fq 
 
nc
dt  q  q
where L  K  V is the Lagrangian of the system, V is the potential energy
function for the conservative forces and torques, and  Fq nc is the generalized force
associated with q for the non-conservative forces and torques, only.
Kamman – ME 4590: page 1/1