# ME 522 Problem 1: Homework 5: Modal Analysis and Buckling Homework

```ME 522
Homework 5: Modal Analysis and Buckling Homework
Distributed: April 8, 2014
Due: April 15, 2014
Problem 1: Solve the eigenproblem
1 −1
−1
1
u1
u2
=λ
2 0
0 1
u1
u2
Find all the eigenpairs. Make the eigenvectors M-orthonormal.
Problem 2: Solve the eigenproblem
⎡
⎤⎡
⎤
⎡
⎤⎡
⎤
5
0
0
u1
1 0 0
u1
⎢
⎥⎢
⎥
⎢
⎥⎢
2 −2 ⎦ ⎣ u2 ⎦ = λ ⎣ 0 2 1 ⎦ ⎣ u2 ⎥
⎣ 0
⎦
0 −2
2
0 1 2
u3
u3
Find all the eigenpairs. You do not need to make the eigenvectors M-orthonormal.
Problem 3: Consider the dynamic equations of motion
1 −1
−1
1
u1
u2
+
2 1
1 2
ü1
ü2
=
0
0
Find the resonant frequencies and mode shapes for the system. You do not need to make
the mode shapes M-orthonormal.
Problem 4: Using a “Modal” analysis system in Workbench, find the first three resonant
frequencies and mode shapes for the beam we studied in Homework 1. Use 3D solid elements
to model the beam. (We want the actual dynamic resonances, not the buckling loads.) Print
out a picture of the each mode, making sure that the frequency is visible on the plot. Using
the Blevins reference posted on the course website, check the values that you obtained for
each of the modes.
Problem 5: Get the nx_paint_can2 .igs file from the course web site, and perform a
linearized buckling analysis in ANSYS Workbench to determine how many cans we can
stack one on top of the other.
(a) The geometry from the web site is a solid object. Use “Concept → Surfaces from Faces”
to create a part made of the five faces that form the cap. Give the cap a thickness of 0.125”.
Now select the rest of the faces and extract the surfaces to form another part. Make that
one 0.063” thick. Now select both of the surface body parts and form a new part. Suppress
the solid body. You can use Workbench’s polyethylene properties for the material.
(b) Create a shell model of the surfaces using the default mesh (even the default is going to
take a while to run.) On the bottom face, use a fixed support. (This isn’t a terribly good
representation of the true boundary condition, but it is nice and simple.)
(c) Now apply a hydrostatic pressure to the “Top” surface of the shells. Make sure that
the pressure values are reasonable and print a page that shows the hydrostatic pressure
distribution. This is actually a pressure on the outside of the can, which isn’t what we
want. Switch the setting to apply the pressure to the “Bottom” surface of the shells. Run a
static analysis and confirm that the can walls move outward. Print a page that shows the
deformations due to the pressure loading. Make sure that the deformations are reasonable.
(d) Next, suppress the hydrostatic pressure and apply a compressive 10 lb force to the curved
edges on the top circular cap. Print the page that shows the deformations due to this loading.
Make sure that the deformations are reasonable.
(e) Unsuppress the hydrostatic pressure, apply both loads to the can, and solve. Transfer
this information to the buckling analysis, and solve for the buckling load. How much is the
buckling factor? What does this mean for the buckling load? Does the buckling mode seem
reasonable?
(f) The buckling factor applies to both loads– the compressive force and the hydrostatic
pressure. We really only wanted it to apply to the compressive force. The only way around
this is to increase just the compressive force in the static solution until the buckling factor
becomes “exactly” one. Do this. Print a picture of the buckling mode and make sure that
the eigenvalue is legible. What is the static compressive force now? If a can of paint weighs
8 pounds, how many cans can we stack on top of this one before this one buckles? Would
we expect the real number of cans that can be stacked on top to be larger or smaller than this
prediction?
(g) Now suppress the hydrostatic pressure and rerun the static and buckling analyses. Print
a picture of the buckling mode and make sure that the eigenvalue is legible. Was the
hydrostatic pressure important?
```