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Example Problems
AutoCAD Analyzer:
CHAPTER 6: Stacked Blocks
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6.0 Stacked blocks: An introduction to the multi-loop problem.

This example demonstrates how multiple vector loops are used to solve for assembly variations. Three blocks of various shapes and sizes are stacked together to form the following assembly (Figure 6.1).

Figure 6.1. Assembly of stacked blocks.

Because of the number of parts and contact points (joints) in the assembly, network graph theory dictates that we use three vector loops to solve for all of the unknown assembly variations of this problem. As explained in the Modeler Manual, we will use the three loops shown in Figure 6.2. The initial tolerances were determined by the type of manufacturing processes to be used to produce the parts. This chapter will present procedures for only basic analysis of this assembly. To review other and more advanced analysis procedures, see Chapter 5. Various results are shown in the verification manual.


Figure 6.2. Vector loops used to solve for assembly variations of stacked blocks.

Independent Variables

A - CYLINDER/ 1-4 and 1-7 = 6.620 ± 0.2mm
B - GROUND/ 8-9 = 4.060 ± 0.15 mm
C - BLOCK/ 2-5 = 6.805 ± 0.075mm
D - GROUND/ 3-8 = 3.905 ± 0.125mm
E - GROUND/ 10-11 = 10.675 ± 0.125mm
F - GROUND/ 3-10 = 28.125 ± 0.35mm

Dependent Variables (lengths only)

D1 - GROUND/3-4 = 18.718 ± ? mm
D2 - GROUND/3-5 = 10.048 ± ? mm
D3 - BLOCK/ 6-7 = 8.671 ± ? mm
D4 - BLOCK/ 2-9 = 2.189 ± ? mm
D5 - BLOCK/ 2-11 = 27.297 ± ? mm

6.1 Start Up

After creating a vector-loop model of the stacked blocks assembly and a CATS neutral file with the AutoCATS 2-D Modeler, execute the 2-D Analyzer and open the neutral file by typing F O STKBLKS Y <CR> from the Main Menu. Select the 2D ANALY option to start your analysis. For more specific instructions on getting started, review section 5.1.

6.2 Equating Variables

The two vectors representing the radii of the cylinder (CYLINDER/1-4 and CYLINDER/1-7) must be equivalenced, since they represent the same dimension. The same process produces them and they are of the same batch. They may be equivalenced from the MODIFY submenu.

PARAM MODIFY KINEMAT DESNSPEC ALLOCATE %CONTRIB SENSDIAG XMATSAVE
.... GLOBAL COMMANDS ....
/HELP /UP /TOP /BLOCK /REF < > /SYSTEM
2D> M <CR>

LOOP FEATURE DESNSPEC ALOCDATA COSTDATA EQUATE
.... GLOBAL COMMANDS ....
/HELP /UP /TOP /BLOCK /REF < > /SYSTEM
MODIFY> E <CR>

Notice that BLOCK/2-5 and BLOCK/2-6 have already been identified as equivalent. Although these are two different names, they represent the same dimension because JOINT5 and DATUM6 are at the same location.

The following sets of dimensions have been identified as equivalent:

NO. 1: BLOCK/2-5 BLOCK/2-6

(A)dd a set (D)elete a set (S)ave changes (Q)uit A

IDENTIFY EQUIVALENT DIMENSIONS
NO. DIM. NAME DIMENSION
1 BLOCK/2-5 6.8050
2 BLOCK/2-6 6.8050
3 CYLINDER/1-7 6.6200
4 CYLINDER/1-4 6.6200
5 GROUND/3-8 3.9050
6 GROUND/8-9 4.0600
7 GROUND/10-11 10.6750
8 GROUND/3-10 28.1250
Identify a set of equivalent dimensions by entering the numbers
above that identify them (separated by spaces).
<CR> TO END SEQUENCE: 3 4 <CR>

CYLINDER/1-7
CYLINDER/1-4
Make the following dimensions equivalent? (Y/n) <CR>

The following sets of dimensions have been identified as equivalent:
NO. 1: BLOCK/2-5 BLOCK/2-6
NO. 2: CYLINDER/1-7 CYLINDER/1-4

(A)dd a set (D)elete a set (S)ave changes (Q)uit S

6.3 Kinematic Assembly Variations

Choose the 2D ANALY option from the Main Menu and then select KINEMAT to see the kinematic assembly variations determined by the three closed loops. The results can be saved to the file STKBLKS.OUT for later viewing.

PARAM MODIFY KINEMAT DESNSPEC ALLOCATE %CONTRIB SENSDIAG XMATSAVE
.... GLOBAL COMMANDS ....
/HELP /UP /TOP /BLOCK /REF < > /SYSTEM
2D> K <CR>

< CLOSED LOOP ANALYSIS RESULTS >
Assembly Name: STKBLKS No. Closed Loops: 3
Feature Tolerances: Applied
Target Acceptance Fraction (Zasm): 3.00
Loop Names: LOOP_3 LOOP_2 LOOP_1

Controlled Dimensions:
Nominal Symmetric Standard
Part Name/Dimen Dimension Tolerance Deviation (RSS)
---------------:-------------:-----------:-----------
BLOCK/2-5 : 6.80500 : .07500 : .02500
CYLINDER/1-4 : 6.62000 : .20000 : .06667
GROUND/3-8 : 3.90500 : .12500 : .04167
GROUND/8-9 : 4.06000 : .15000 : .05000
GROUND/10-11 : 10.67500 : .12500 : .04167
GROUND/3-10 : 28.12500 : .35000 : .11667
---------------:-------------:-----------:-----------

Kinematic Assembly Variables:
Variations
Variable Name DEG of FDM Worst Case RSS Case 6-SIG Case
--------------:-------------:-----------:-----------:-----------
GROUND/3-5 : TRANSLATION : .33854 : .18741 : .24845
JOINT5 : ROTATION : 1.00509 : .49113 : .65096
BLOCK/6-7 : TRANSLATION : .52248 : .27956 : .36880
DATUM1 : ROTATION : 1.00509 : .49113 : .65096
GROUND/3-4 : TRANSLATION : .60910 : .30109 : .40018
BLOCK/2-9 : TRANSLATION : .27859 : .14004 : .18303
JOINT9 : ROTATION : 1.00509 : .49113 : .65096
BLOCK/2-11 : TRANSLATION : .43693 : .36347 : .48322
JOINT11 : ROTATION : 1.00509 : .49113 : .65096
--------------:-------------:-----------:-----------:-----------

Results to file STKBLKS.OUT (y/N)? Y<CR>

Three tolerance accumulation models are compared side-by-side: Worst Case, Statistical (RSS), and Six Sigma. The Six Sigma model includes a default "mean drift" factor of 0.25 for each component tolerance. Notice that the variation of the critical dependent LENgth that we're interested in (GROUND/3-4) will be ±.60911mm at the most, but that it will be ±.30110mm 99.73% of the time (Zasm=3.00). Also notice that the angular variation at all four dependent angles is equal (~1.005deg. for Worst Case).

6.4 Analysis of Design Specifications

Since our design spec (dependent LENgth) is a closed loop spec, we can see from selecting the KINEMAT option that its RSS 3[[sigma]] variation will be ±.30110mm. Our design limits (as defined in the Modeler) were ±.300mm. The two numbers are very close. How many more rejects than 2700 ppm will this small difference result in? The DESNSPEC option allows you to see how many parts per million are outside of the specified design limits. The results can be saved to the file STKBLKS.OUT for later viewing.

PARAM MODIFY KINEMAT DESNSPEC ALLOCATE %CONTRIB SENSDIAG XMATSAVE
.... GLOBAL COMMANDS ....
/HELP /UP /TOP /BLOCK /REF < > /SYSTEM
2D> D <CR>

< DESIGN SPEC ANALYSIS RESULTS >
Assembly Name: STKBLKS
Analysis Model: RSS
Target Acceptance Fraction (Zasm): 3.00

-----< Spec # 1 >------------------------------------<< OUTPUTS >>---------
Spec Type : DEP. LENGTH : With Feature :Without Feature
References: GROUND/3-4 nil : Tolerances : Tolerances
--------------------------------------------:--------------:---------------p> :
Mean: 18.71820 Computed Variation: ± .30109: ± .29889
Nominal: 18.71820 :
Upper Tol: +.30000 Upper Tail Z (sigma): 2.99: 3.01
Lower Tol: -.30000 Lower Tail Z (sigma): 2.99: 3.01
Max Limit: 19.01820 USL Rejects (ppm): 1398.9: 1301.4
Min Limit: 18.41820 LSL Rejects (ppm): 1398.8: 1301.3
--------------------------------------------:--------------:---------------p>
TOTAL REJECTS (PPM): 2797.7: 2602.7
--------------------------------------------:--------------:---------------p>
Do you want this written to STKBLKS.OUT (y/N)? Y<CR>

The analysis predicts that there will be nearly 2800 ppm rejects--100 ppm more than that specified by our target acceptance fraction. This is probably acceptable; however, section 6.7 will illustrate how much the tolerances will need to be tightened to bring the acceptance fraction to exactly 3.00 sigma.

6.5 Percent Contribution Chart

What contributes to the ±.30110mm variation in GROUND/3-4? A graphical representation of how component dimensions and their tolerances affect the default dependent LENgth design spec can be obtained by choosing %CONTRIB.

PARAM MODIFY KINEMAT DESNSPEC ALLOCATE %CONTRIB SENSDIAG XMATSAVE
.... GLOBAL COMMANDS ....
/HELP /UP /TOP /BLOCK /REF < > /SYSTEM
2D> % <CR>

Figure 6.3 Percent Contribution to dependent LENgth design spec (GROUND/3-4)

Notice that the major contributor to the variation of the critical dependent LENgth GROUND/3-4 is CYLINDER/1-4. It is the source of over 75% of GROUND/3-4's variance.6.6 Sensitivity Diagram

In section 6.5, we saw that CYLINDER/1-4 was the largest contributor to the ±.30110mm variation in GROUND/3-4. Was this due to its relatively large tolerance (.20 mm), GROUND/3-4's sensitivity to it, or both? A graphical representation of how sensitive design specifications are to component dimensions can be obtained by choosing SENSDIAG.

PARAM MODIFY KINEMAT DESNSPEC ALLOCATE %CONTRIB SENSDIAG XMATSAVE
.... GLOBAL COMMANDS ....
/HELP /UP /TOP /BLOCK /REF < > /SYSTEM
2D> S <CR>


Figure 6.4 Sensitivities of dependent LENgth design spec (GROUND/3-4)

Notice that GROUND/3-4 is most sensitive to CYLINDER/1-4. It is almost equally sensitive to BLOCK/2-5 and the two form tolerances (flatness and roundness) at JOINT7. If tolerances need to be tightened, those on CYLINDER/1-4 and BLOCK/2-5 will have the most effect. The designer might wish to base his choice of allocation weight factors for these dimensions on this sensitivity information.

6.7 Tolerance Allocation

Let's perform a simple proportional scaling tolerance allocation with the default PARAMeter settings. Select ALLOCATE from the 2D ANALY submenu.

PARAM MODIFY KINEMAT DESNSPEC ALLOCATE %CONTRIB SENSDIAG XMATSAVE
.... GLOBAL COMMANDS ....
/HELP /UP /TOP /BLOCK /REF < > /SYSTEM
2D> A <CR>

< ALLOCATION RESULTS >
Assembly: STKBLKS
Analysis Model: RSS
Target Acceptance Fraction (Zasm): 3.00
Tolerance Allocation Option: Proportional Scaling
Nominal Allocation Option : None
-----< Spec # 1 >---------------------------------------------------------
Spec Type : DEP. LENGTH Nominal: 18.71820
References: GROUND/3-4 nil Max Limit: 19.01820
Min Limit: 18.41820
--------------------------------------------------------------------------<> :
PECIFIED VALUES : ALLOCATED VALUES : Optimized Perc
Part Name/Dimen : Nominal ±Tol : Nominal ±Tol : Std Dev Cont
--------------- : --------- ------- : --------- ------- : --------- ------
BLOCK/2-5 : 6.8050 .07500 : 6.8050 .07472 : .02491 6.67
CYLINDER/1-4 : 6.6200 .20000 : 6.6200 .19926 : .06642 75.68
GROUND/3-8 : 3.9050 .12500 : 3.9050 .12454 : .04151 .71
GROUND/8-9 : 4.0600 .15000 : 4.0600 .14945 : .04982 13.66
GROUND/10-11 : 10.6750 .12500 : 10.6750 .12454 : .04151 1.15
GROUND/3-10 : 28.1250 .35000 : 28.1250 .34871 : .11624 .67
--------------- : -----------Form Tolerances----------- : --------- ------
JOINT5 : 0.0 .04000 : 0.0 .04000 : .01333 .00 *
JOINT7 : 0.0 .02500 : 0.0 .02500 : .00833 .75 *
JOINT7 : 0.0 .01000 : 0.0 .01000 : .00333 .12 *
JOINT4 : 0.0 .04000 : 0.0 .04000 : .01333 .13 *
JOINT4 : 0.0 .01000 : 0.0 .01000 : .00333 .01 *
JOINT9 : 0.0 .02500 : 0.0 .02500 : .00833 .41 *
JOINT11 : 0.0 .02500 : 0.0 .02500 : .00833 .05 *
--------------- : ----------------- : ----------------- : --------- ------
COMPUTED ASSEMBLY VARIATION -- DEP. LENGTH .10000 100.00
Mean ± Var : 18.7182 .30109 : 18.7182 .30000 :
: ----------------- : ----------------- :
Min. Max. : 18.4171 19.0193 : 18.4182 19.0182 :
----------------:-------------------:-------------------:-----------------
REJECTS Z PPM : Z PPM :
Upper: 2.99 1398.9 : 3.00 1349.9 :
Lower: 2.99 1398.8 : 3.00 1349.8 :
Total: 2797.7 2699.7
* = Fixed Tolerance

Do you want this written to STKBLKS.OUT (y/N)?

Notice how little the tolerances need to be tightened to meet the default 3-sigma acceptance fraction specified in the PARAMeter table. For example, CYLINDER/1-4's tolerances changed by only 73 ten-thousandths of a millimeter and BLOCK/

 


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