Download ASEN5022.HW06.2005.pdf

Homework #6 (ASEN5022, Spring 2005)
Due at Start of Class on Tuesday, 15 March
6.1 FEM Modeling Vibration of Bar
A free-free bar is to be modeled by four linear bar elements asshown below.
Partition a free-free bar or beam into finite elements
(1)
element
global node 3
(2)
4
1
5
elemental nodes 1
global nodes
(4)
(3)
2
2
4
1
element 2
Figure for Problem 6.1
The global node numbering was generated by a random-walk algorithm as shown below. The map
between the elemental local nodes and the global node numbers for the four elements are given as

element number
1


2


3
4
element first node
3
1
4
5

element second node
1


4


5
2
(1)
The elemental mass matrix, m, and elemental stiffness matrix, k, are given by
1/3
m=
1/6
6.1.1
6.1.2
1/6
,
1/3
1
k=
−1
−1
1
(2)
Assemble by hand (or write an assembly routine for this case) the global mass and stiffness
matrix. Show step by step how you are accomplishing the assembly task.
Indicate the process of applying the following boundary conditions in your assembled
matrix:
•
free-free boundary conditions.
•
fixed-fixed boundary conditions.
•
fixed-free boundary conditions.
Over
6.1.3
Implement the necessary vibration analysis of the above four-element FEM bar model and
obtain the first two modes and mode shapes for the three boundary conditions. You may
want to express the frequencies in terms of the β L as given by
βL =
ρ ∗ b ∗ h/E A ∗ ωL
where ρ = b = h = E A = 1 for the present case.
Hint: one possible way is to overwrite Kg and Mg on the FEM bar vibration analysis code I handed
out by your appropriate matrix. In addition, if you want to be elegant in programming, you can
build a boundary condition cases for your problem into the code.
However, the use of your hand-derived K g and Mg for the above three cases is also acceptable.