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Sign Convention: Myth Exploded by Law of Signs Addendum 1 PDF Print E-mail
  
Friday, 21 January 2005 15:19
Reference:

Sign Convention: Myth Exploded by Law of Signs ID - 0
Line references in the text are to line numbers in ID - 0 (excluding blank lines) in the printer version of 6dof.com of ID-0

INTRODUCTION

In [ID - 0] the basic (global) coordinate system adopted in the MSC.Nastran Primer [see ID - 0 - (line 6)] was examined. Postponing, for the time being, the promised correlation of I/O signs of Force distribution in bar elements [see ID - 0 - (lines 21-24)], the effect of altering the basic coordinate system on displacements, and forces of single point constraint will be considered. After computing the required changes in INPUT signs the expected output signs will be forecast. Interested readers are invited to execute the MSC.Nastran runs and challenge the author if discrepancies are detected. With permission from 6dof web administrator author may be contacted. The author is willing to scrutinize input and output free of charge to point out and explain discrepancies, should these occur.
For easy comparison of "output signs" (for that is the only entity which will change) basic X - axis will be made to coincide with the center line of the cantilever ( E -W), Y - axis will lie in the horizontal plane (N-S) and Z - axis will remain vertical (U-D). This judicious selection will preserve the designations T1, T2, T3, R1, R2, R3 in [ID - 0[ and input criteria for all three cases.

ALTERNATE BASIC SYSTEMS

1. Origin at grid point 2 X axis positive E Y axis positive S.
2. Origin at grid point 1 X axis positive W Y axis positive N
3. Origin at grid point 1 X axis positive W Y axis positive S
As MSC.Nastran users know, the third axis Z is chosen by the software. How does the software accomplish this task? What the brilliant NASTRAN programmers did, the author is not privy to - but a simple explanation by the law of signs is provided. The author believes MSC.Nastran programmers could not have done otherwise.
Apparently, no one would seriously suggest that the widely used three finger rule, thumb, index and middle finger, of the right hand for the right-hand rule, or of the left hand for the left-hand rule, or even the palm contortion rule was some how coded.

COMPUTING Z-AXIS BY THE LAW OF SIGNS

The rule to determine the Z - axis can be stated as follows:
If the rotation from positive X - axis to positive Y - axis through the smaller angle (i.e. 90 degrees) is CLOCKWISE, erecting positive Z - axis pointed toward the observer [that is on the positive face, this will be discussed further for the element (local) coordinate system] normal to the plane, creates a LEFT-HANDED SYSTEM:, while erecting Z - axis away from the observer [obviously the negative face] generates a RIGHT-HANDED SYSTEM.
The exact opposite is true for COUNTERCLOCKWISE rotation from positive X - axis to positive Y - axis, through 90 degrees.
While it is not possible for the author to prove it mathematically, any reader can verify all possible 48 (and no more) systems by either the three finger or the palm contortion rule.

By the above rule, for Right-handed system:
X (E) to Y (S) rotates CLOCKWISE through 90 degrees, hence Z - axis points DOWN i.e. ESD
X (W) to Y (N) rotates CLOCKWISE through 90 degrees, hence Z - axis points DOWN i.e. WND
X (W) to Y (S) rotates COUNTERCLOCKWISE through 90 degrees hence Z - axis points UP i.e. WSU

CODING CONVENIENCE USING ABOVE RULES

Assigning the numbers 1, 2, 3, 4, 5, and 6 to N, S, E, W, U, D it is easy to create two array of 48 constants, which define 24 Right-handed and 24 Left-handed coordinate systems. Matching the product XY [ES, i.e. 32, (3X10 + 2)] to the integer quotient of 48 stored constants 325 and 326 [since 325/10 = 32] will be obtained. Clearly 321, 322, 323, and 324 cannot be members of the table for the obvious reason that 2 cannot be repeated for a triad, and 1 and 2, and 3 and 4 are collinear hence cannot constitute an orthogonal triad. ESD (326) is the obvious choice for the proposed Right-handed system. Obviously by the three finger rule, and by the proposed rule, 325 is left-handed.
Another array of constants can be created pre-defining the clockwise and counterclockwise sense of rotation from X to Y. For example ESD (326) 32 is CLOCKWISE, 26 is COUNTERCLOCKWISE, and 63 is also COUNTERCLOCKWISE. All three would appear in the array of constants and can be easily matched.
It ought to be carefully noted that handedness of the coordinate system has no effect on the two-element rotation: i.e. 32 in either the 326 Right-handed system or 325 Left-handed system has identical CLOCKWISE sense of rotation,
Again, the author is unable to provide any mathematical proof - but, as pointed out in [ID - 0 (line 80)], comparison of MSC.Nastran outputs verifies this assertion.

THREE ALTERNATE RIGHT HANDED SYSTEMS

1. ESD (326) 32 (+)ve CLOCKWISE 26 (+)ve COUNTERCLOCKWISE 63 (+)ve COUNTERCLOCKWISE
2. WND (416) 41 (+)ve CLOCKWISE 16 (+)ve CLOCKWISE 64 (+)ve CLOCKWISE
3. WSU (425) 42 (+)ve COUNTERCLOCKWISE 25 (+)ve CLOCKWISE 54 (+)ve COUNTERCLOCKWISE

REQUIRED CHANGES TO INPUT TABLE

These changes will be referenced to the Input Table in Harry G. Schaeffer's book noted in [ID - 0 (lines 1 - 6)] section 12.12.3 pages 382 and 383. The location of changes would refer to the rows numbered from 0001 thru 0036 and columns 1 through 9

In all cases there are no changes to rows 0001 through 0021. There is no change in row 0022 either since in GRDSET entry "14" (T1 and R1) do not change

No changes will occur in rows 0034 thru 0036 as these do not pertain to physics of the system

CASE ESD

Rows 0023 thru 0026 do not change for origin is still at grid 2, CBAR is oriented from grid 2 to grid 1, and PBAR Moment of Inertia properties do not change. Change will occur in row 0027. Note that in the original run [ID - 0] the axes lying in the plane of the cross-section were Y (N), and Z (U). In this run, the axes are Y (S) and Z (D) hence all signs in locations [0027 - 2] thru [0027 - 9] will be reversed. The entries are then:

[0027 - 2] 0.125 [0027 - 3] 0.25 [0027 - 4] -0.125 [0027 - 5] 0.25 [0027 - 6] -0.125
[0027 - 7] -0.25 [0027 - 8] 0.125 [0027 - 9] -0.25

Rows 0028 - 0030 will remain unchanged as material properties remain constant.

No changes will occur in row 0031 since grid point 2 is restrained in T2 (2) , T3 (3), R2 (5) and R3 (6) directions

Rows 0032 and 0033 will both change as the loads directed N and U are now to be refereed to axes directed S and D. Thus the changes are

[0032 - 7] -5.0 [0032 - 9] -5.0 [0033 - 7] -8.6603 [0033 - 9] -8.6603

CASE WND

Origin is now at grid 1, CBAR is to be oriented from grid 1 to grid 2 Thus the following changes are needed in rows 0023 through 0025

[0023 - 4] 0.0
[0024 - 4] 10.0
[0025 - 4] 1 [0025 - 5] 2 No other changes since element (local) Y - axis coincides with basic Y - axis

No change in row 0026 since PBAR Moment of Inertia properties do not change. Change will occur in row 0027. Note that in the original run [ID - 0] the axes lying in the plane of the cross-section were Y (N), and Z (U). In this run, the axes are Y (N) and Z (D) hence signs of Z coordinates in locations [0027 - 3], [0027 - 5], [0027 - 7] and [0027 - 9] will be reversed. The entries are then:

[0027 - 3] 0.25 [0027 - 5] 0.25 [0027 - 7] -0.25 [0027 - 9] -0.25

Rows 0028 - 0030 will remain unchanged as material properties remain constant.

No changes will occur in row 0031 since grid point 2 is restrained in T2 (2) , T3 (3), R2 (5) and R3 (6) directions. Note that the fixed end 2 does not change

Row 0032 will not change as the load directed N is now defined by positive Y (N) axis
Row oo33 will change as load directed U is now defined by positive Z (D) axis.

[0033 - 7] -8.6603 [0033 - 9] -8.6603

CASE WSU

Origin is now at grid 1, CBAR is to be oriented from grid 1 to grid 2 Thus the following changes are needed in rows 0023 through 0025

[0023 - 4] 0.0
[0024 - 4] 10.0
[0025 - 4] 1 [0025 - 5] 2 No other changes since element (local) Y - axis coincides with basic Y - axis

No change in row 0026 since PBAR Moment of Inertia properties do not change. Change will occur in row 0027. Note that in the original run [ID - 0] the axes lying in the plane of the cross-section were Y (N), and Z (U). In this run, the axes are Y (S) and Z (U) hence signs of Y coordinates in locations [0027 - 2], [0027 - 4], [0027 - 6] and [0027 - 8] will be reversed. The entries are then:

[0027 - 2] 0.125 [0027 - 4] -0.125 [0027 - 6] -0.125 [0027 - 8] 0.125

Rows 0028 - 0030 will remain unchanged as material properties remain constant.

No changes will occur in row 0031 since grid point 2 is restrained in T2 (2) , T3 (3), R2 (5) and R3 (6) directions. Note that the fixed end 2 does not change

Row 0032 will change as the load directed N is now defined by positive Y (S) axis

[0032 - 7] -5.0 [0032 - 9] -5.0

Row 0033 will not change as load directed U is now defined by positive Z (U) axis.




FORECAST OF OUTPUT CHANGES

I. DISPLACEMENTS

These data are tabulated on page 387 of the book noted in [ID - 0, (lines 1 thru 6)].

(a) Subcase 1

T2 and R3 are the only non-zero results. T2 is displacement in Y (N - S) direction and R3 is the rotation in the XY (EN) plane. Positive displacement of grid point 2 is in the N direction. T2 is tabulated as (+) 0.3201971 ins, i.e. it displaces North.
Positive rotation from X to Y (E to N) is counterclockwise, hence the tabulated result
(+) 0.0426677 rads indicates counterclockwise rotation of grid point 2

In all three cases we should have identical results tabulated as T2, and R3 except ofcourse for the signs. The signs would be dictated by the positive properties of the basic coordinate systems.

CASE ESD

Positive Y is South hence forecast T2 is (-) 0.3201971 ins to point North

Positive rotation XY (ES) is clockwise hence forecast R3 is (-) 0.0426677 rads to indicate physical counterclockwise rotation of grid point 2

CASE WND

Positive Y is North hence forecast T2 is (+) 0.3201971 ins to point North

Positive rotation XY (WN) is clockwise hence forecast R3 is (-) 0.0426677 rads to indicate physical counterclockwise rotation of grid point 2

CASE WSU

Positive Y is South hence forecast T2 is (-) 0.3201971 ins to point North

Positive rotation XY (WS) is counterclockwise hence forecast R3 is (+) 0.0426677 rads to indicate physical counterclockwise rotation of grid point 2


(b) Subcase 2

T3 and R2 are the only non-zero results. T3 is displacement in Z (U - D) direction and R2 is the rotation in the ZX (UE) plane. Positive displacement of grid point 2 is in the U direction. T3 is tabulated as (+) 0.1389341 ins, i.e. it displaces Up.
Positive rotation from Z to X (U to E) is clockwise, hence the tabulated result
(-) 0.01847649 rads indicates counterclockwise rotation of grid point 2

In all three cases we should have identical results tabulated as T3, and R2 except ofcourse for the signs. The signs would be dictated by the positive properties of the basic coordinate systems.

CASE ESD

Positive Z is Down hence forecast T3 is (-) 0.1389341 ins to point Up

Positive rotation ZX (DE) is counterclockwise hence forecast R2 is (+)0.01847649 rads to indicate physical counterclockwise rotation of grid point 2

CASE WND

Positive Z is Down hence forecast T3 is (-) 0.1389341 ins to point Up

Positive rotation ZX (DW) is clockwise hence forecast R2 is (-)0.01847649 rads to indicate physical counterclockwise rotation of grid point 2

CASE WSU

Positive Z is Up hence forecast T3 is (+)0.1389341 ins to point Up

Positive rotation ZX (UW) is counterclockwise hence forecast R2 is (+)0.01847649 rads to indicate physical counterclockwise rotation of grid point 2

II. FORCES OF SINGLE POINT CONSTRAINT

These data are tabulated on page 387 of the book noted in [ID - 0, (lines 1 thru 6)].

(a) Subcase 1

T2 and R3 are the only non-zero results. T2 is reaction in Y (N - S) direction and R3 is the fixing moment in the XY (EN) plane. Positive force at grid point 2 is in the N direction. T2 is tabulated as (-) 50.00000 lbs, i.e. it points South.

Positive rotation from X to Y (E to N) is counterclockwise, hence the tabulated result
(-) 250.0000 in-lbs indicates clockwise fixing couple at grid point 2

In all three chosen cases we should have identical results tabulated as T2, and R3 except ofcourse for the signs. The signs would be dictated by the positive properties of the basic coordinate systems

CASE ESD

Positive Y is South hence forecast T2 is (+)50.00000 lbs, i.e. it points South.

Positive rotation XY (ES) is clockwise hence forecast R3 is (+)250.0000 in-lbs to indicate clockwise fixing couple at grid point 2

CASE WND

Positive Y is North hence forecast T2 is (-) 50.00000 lbs, i.e. it points South.

Positive rotation XY (WN) is clockwise hence forecast R3 is (+)250.0000 in-lbs to indicate clockwise fixing couple at grid point 2

CASE WSU

Positive Y is South hence forecast T2 is (+)50.00000 lbs, i.e. it points South.

Positive rotation XY (WS) is counterclockwise hence forecast R3 is (-)250.0000 in-lbs to indicate clockwise fixing couple at grid point 2


(b) Subcase 2

T3 and R2 are the only non-zero results. T3 is reaction in Z (U - D) direction and R2 is the fixing moment in the ZX (UE) plane. Positive force at grid point 2 is in the U direction. T3 is tabulated as (-) 86.60300 lbs, i.e. it points Down..

Positive rotation from Z to X (U to E) is clockwise, hence the tabulated result
(+) 433.0150 in-lbs indicates clockwise fixing couple at grid point 2

In all three chosen cases we should have identical results tabulated as T3, and R2 except ofcourse for the signs. The signs would be dictated by the positive properties of the basic coordinate systems

CASE ESD

Positive Z is Down hence forecast T3 is (+)86.60300 lbs, i.e. it points Down.

Positive rotation ZX (DE) is counterclockwise hence forecast R2 is (-)433.0150 in-lbs to indicate clockwise fixing couple at grid point 2

CASE WND

Positive Z is Down hence forecast T3 is (+)86.60300 lbs, i.e. it points Down..

Positive rotation ZX (DW) is clockwise hence forecast R2 is (+)433.0150 in-lbs to indicate clockwise fixing couple at grid point 2

CASE WSU

Positive Z is Up hence forecast T3 is (-)86.60300 lbs, i.e. it points Down...

Positive rotation ZX (UW) is counterclockwise hence forecast R2 is (-)433.0150 in-lbs to indicate clockwise fixing couple at grid point 2

ANOTHER FALL OUT FROM THE LAW OF SIGNS

If, as asserted, positive rotations in the component planes of any three-dimensional coordinate system are uniquely defined, then, there must exist an algorithm which, given, arbitrarily, any one of the 48 systems, should be capable of drawing a sketch comprising two orthogonal axes, draw the directed arc to identify positive rotation, and to instill confidence, compute the signed cross product and interpret physical sense of rotation of the moment about the origin, of a force chosen by the user, parallel to either one of the two chosen axes, and pointed either in the positive or negative direction of that axis.
Such an algorithm has been derived, using the interactive capability of Visual Basic 3, and producing sketches, and physically interpreting the signed cross product. This may be examined free of charge with permission from 6dof administrator. This examination is possible, since a windows environment can run an executable without supporting VB3 software.
As will be seen in subsequent articles, sign of positive couple is an essential ingredient in the determination of the sign of bending moment.
 
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