Page 280 - Finite Element Modeling and Simulations with ANSYS Workbench

Structural Vibration and Dynamics 265

1 , A, L 2
AL AL
m = m =
2
1
2 2
u 1 u 2
FIGURE 8.5
The lumped mass for a 1-D bar element.

 ρAL 
 2 0 
m =  
 0 ρAL 
  2  

which is a diagonal matrix and thus is easier to compute.
In general, we apply the following element consistent mass matrix:

m = ∫ ρ N N dV (8.9)
T
V
where N is the same shape function matrix as used for the displacement field and V is the
volume of the element.
Equation 8.9 is obtained by considering the kinetic energy within an element:

1 1
Κ= umu (cf. mv )
T
2

2 2
1 1
2
T

()
= ∫ ρ ρ udV = ∫ ρ u udV
2 V 2 V
1
)
= ∫ ρ(Nu T dV
) (Nu
2 V
1
T

= u T ∫ ρN N ddV u (8.10)
2

V
m
For the bar element (linear shape function), the consistent mass matrix is
T
m = ∫ V ρ N N dV = ∫ V ρ 1 −ξ   1 −ξ ξ AL d ξ


ξ


/
13/ 16 u 1
=ρAL    (8.11)
/
/

 16 13 u 2
which is a nondiagonal matrix.
Similar to the formation of the global stiffness matrix K, element mass matrices are
established in local coordinates first, then transformed to global coordinates, and finally
assembled together to form the global structure mass matrix M.