ME 4590 Dynamics of Machinery
Natural Frequencies and Mode Shapes
To calculate the natural frequencies and mode shapes for multiple degree-offreedom (DOF), rigid-body systems, we first linearize the equations of motion
(EOM) and express them in the matrix form:
[ M ] { q }  [C ] { q }  [ K ] {q}  {F (t )}
Here, [ M ], [C ], and [ K ] represent the system's mass, damping, and stiffness
matrices, the vector {q} represents the changes in all the generalized coordinates
from their equilibrium values, and the vector {F} represents all the forces acting
on the system. Then, we set the damping matrix and force vector to zero to get
[ M ] { q }  [ K ] {q}  {0}
This equation describes the free, undamped response of the system.
Following the pattern for single DOF systems, we look for a solution to this
equation of the form
{q}  e st {u}
with s  j . This equation describes a steady-state, undamped solution, with {u}
representing the mode shape of the oscillations. Substituting this into the
differential EOM gives
[ K ]   [M ] {u}
2
 {0}
The problem now is to find  2 and {u} that satisfy this algebraic equation. This
is called an eigenvalue problem.
For this equation to have a non-zero solution for {u} , we require
det [ K ]   2[ M ]  0
If the matrices [ M ] and [ K ] are N  N matrices, the solution to this equation
yields N values for  2 , and hence N values for  . These are the N natural
frequencies of the system. Associated with each of these frequencies is a mode
shape {u} . The  2 are called the N eigenvalues of the system, and the {u} are
called the N eigenvectors of the system.
Kamman – ME 4590: page 1/1