Page 131 - Mechatronic Systems Modelling and Simulation with HDLs

120 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

z x r r z1
z0 u y0
y u y1
Figure 6.8 Degrees of freedom of the shear-resistant beam element at the nodes k and l: u y
(deflection in y-direction), r z (rotation about the z-axis)
the beam section i. The beam load is concentrated by p i0 and p i1 on the nodes 0
and 1 of the beam element. In the shear-resistant case and for small deflections
th
the stiffness matrix K i , the mass matrix M i and the load vector p i of the i beam
element are independent of the deflection. If we select the interpolation functions
h 1 ... h 4 in the variables ξ for the approximation of the continuous displacements
as follows:

2
h 1 (ξ) = 1 − 3ξ + 2ξ 3
2
h 2 (ξ) =−ξ(1 − ξ) l i
(6.35)
2
h 3 (ξ) = 3ξ − 2ξ 3
2
h 4 (ξ) = ξ (1 − ξ)l i
then we find the following element matrices and vectors, see Gasch and Knothe [113]:
 
12 −6l i −12 −6l i
B i  −6l 4l 2 6l i 2 

i
K i = i 2l 
3
l  −12 6l i 12 

i 6l i 
−6l 2l 2 6l i 4l 2
i i
 
156 −22l i 54 13l i
 2 2 
µ i l i  −22l i 4l i −13l i −3l 
i
M i =   (6.36)
54 −13l i 156
420  22l i 
13l i −3l 2 22l i 4l 2
i i
   
7/20 3/20
   
 −l i /20   −l i /30 
p i = p i0 l i   + p i1 l i  
 3/20   7/20 
l i /3 l i /20
The equation system for such an element thus takes the form:
(6.37)
M i ¨u i + K i u i = p i
where
u i = [u y0 , r z0 , u y1 , r z1 ] T

where u i represents the element displacement vector, and thus the degrees of freedom.