@UEBER = 5.8 BEISPIEL 7: ROHR UNTER INNENDRUCK, TORI NR.7

5.8 MOTORCYCLE CRANKSHAFT, TETRAHEDRON NO. 16

Copy the sample file B11_G.COS to the Z88 input file Z88G.COS.

We want to compute a crankshaft for a monocylinder motorcycle engine and put a force of -5,000 N onto the piston. The meshing will do Pro/ENGINEER.

The boundary conditions are a bit tricky for this example: Put a reference (or datum) point to the center of the face of the crankshaft. We'll need this point to fix the crankshaft in Z direction, i.e. lengthwise.

The ball bearings, which allow always some angular movement, and, thus, should be regarded as moment- free supports, are fastened to the larger shaft axises. The flange facings of the shaft axises are to be fixed in X and Y direction. Because whole surfaces are fixed, don't allow one ore more of these surfaces to be fixed in Z direction, too. This would result in blocking the angular movement - try it, if you won't believe it.

A total force of -5,000 N will be put onto the peripheral surface of the crankshaft journal.

The mesh is automatically generated by Pro/MECHANICA featuring parabolic tetrahedrons. After storing the COSMOS file, a Z88 session may start:

Copy B11_G.COS to Z88G.COS, the COSMOS file for the converter Z88G

Start converting Z88G.COS with Z88G

(Windows: COSMOS converter Z88G. Looks quite similar on UNIX machines)

and proceed with the Cuthill- McKee algorithm Z88H, because we'll expect a very bad node- numbering for the parabolic tetrahedrons.

(Windows: Cuthill- McKee program Z88H. Looks quite similar on UNIX machines)

The first line of Z88I1.TXT tells you the following values:

- 6,826 nodes
- 3,941 elemente
- 20,478 dof

MAXKOI must have as a minimum 3,941 elements * 10 nodes per element = 39,410.

Thus, Z88.DYN should look as follows:

MAXGS when starting, any value
MAXKOI minimum 39410
MAXK minimum 6826
MAXE minimum 3941
MAXNFG minimum 20478
MAXNEG minimum 1

Proceed with a look at the structure with Z88O (or with Z88P ).

The computing time with Z88F is about 1,5 minutes on a PC (900 MHz AMD- processor, 512 MByte memory). Enter a value of about 11,400,000 for MAXGS..

See the deflected structure with Z88O. The angular deflection of the axises is quite amazing. Now you would read off the deflections of distinguished nodes, multiply with the appropriate lever arms and check with the bearing catalogue if your ball bearings will allow this angular movement without problems.

(Windows: Computing deflections with Z88F. Looks quite similar on UNIX machines)

(Windows: Plot programm Z88O, undeflected structure)

(Windows: Plot programm Z88O, deflected structure)

Now we'll launch the iteration solver Z88I1 and Z88I2. To begin with, we'll try some values for MAXSOR and MAXPUF in Z88.DYN (you may also enter, for example, 50,000,000 for MAXSOR and 5,000,000 for MAXPUF, if you want):

`COMMON START
MAXGS 11500000 has for Z88I1 no meaning !
MAXKOI 40000 must always be large enough !
MAXK 7000 read off from Z88I1.TXT
MAXE 4000 read off from Z88I1.TXT
MAXNFG 21000 read off from Z88I1.TXT
MAXNEG 1 read off from Z88I1.TXT
MAXSOR 2000000 important for Z88I1
MAXPUF 500000 important for Z88I1
COMMON END
`

Our entries did work properly (otherwise, you would have to increase MAXSOR and MAXPUF) and the sorting times was about 15 seconds on a PC (900 MHz AMD- processor, 512 MByte memory).

Read off for MAXGS: 768,687, rounded up 770,000. This looks fairly better than the direct Cholesky solver Z88F with its need of 11,381,064 8- Byte elements = 87 Mbyte. The second part of the iteration solver, i.e. Z88I2, will only need 768,687 8- Bytes elements = 6 MByte.

Thus, we would adjust the memory in Z88.DYN as follows (feel free to enter even bigger values):

`COMMON START
MAXGS 770000 important !
MAXKOI 40000 must always be large enough !
MAXK 7000 read off from Z88I1.TXT
MAXE 4000 read off from Z88I1.TXT
MAXNFG 21000 read off from Z88I1.TXT
MAXNEG 1 read off from Z88I1.TXT
MAXSOR 2000000 not used by Z88I2
MAXPUF 500000 not used by Z88I2
COMMON END
`

If you adjust the iteration parameters in Z88I4.TXT (chapter 3.6) as follows:

`10000 1e-7 1.
`

i.e. a maximum of 10,000 iterations, EPS with 1E-7 and RP (here Omega) with 1, then this results in a computing time of about 1 minute on a PC (900 MHz AMD- processor, 512 MByte memory).

In this case, both the iteration solver and the direct Cholesky solver need about the same time, but the iteration solver needs fewer than one tenth of memory. For large structures, things get even worse for the Cholesky solver! But pay attention to the fact, that you can't really compare the computing times. Try other entries for EPS, for example 1E-5 (resulting in 303 iterations and 45 seconds) or 1E-10 (resulting in 474 iterations and 1:08 minutes), and see the different computing times.

(Windows: The iteration solver Part 2, i.e. Z88I2)

However, a very nice experiment is this:

Start from the very beginning, run Z88G, but not the Cuthill- McKee algorithm Z88H. Launch directly after Z88G a test run with Z88F (UNIX: z88f -t):

(Windows: The direct Cholesky solver in test mode)

Gee, see the faces falling: now we would need 184,122,663 8- Byte elements = 1,4 GByte. Absolutely no need for this!

However, run again the iteration solver part 1, i.e. Z88I1. This will again result in only 768,687 elements for the total stiffness matrix. Calculate, please:

184,122,663 : 768,687 = 240 : 1

The second part of the iteration solver, i.e. Z88I2, needs now some more iterations (451 in contrary to 415 with an equal EPS of 1E-7), because the matrix features the same number of non- zero elements, though, but the condition is worse because of the very bad node- numbering of Pro/MECHANICA. That means: When using the iteration solver you don't need to run the Cuthill- McKee algorithm Z88H for reducing the storage needs of the iteration solver (in contrary to the direct Cholesky solver Z88F, which may depend heavily on Z88H for larger structures!). However, Z88H may improve the matrix condition anyway.