Modal assurance criterion for comparing two mode shapes of a multi-component structure

1. A method for comparing two mode shapes of a multi-component structure, comprising:

forming, by a validation tool residing in memory, a first mode shape vector and a second mode shape vector for a first mode shape and a second mode shape of the multi-component structure, respectively, by grouping modal displacement at each node associated with each component in the multi-component structure:

squaring sum of magnitudes of correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure by the validation tool;

computing a product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure by the validation tool;

computing by the validation tool, modal assurance criterion between the first mode shape vector and the second mode shape vector of the multi-component structure by dividing the squared sum of magnitudes of the correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure with the computed product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure; and

comparing, by the validation tool, the first mode shape and second mode shape of the multi-component structure using the computed modal assurance criterion to identify similarity between the two mode shapes of the multi-component structure.

2. The method of claim 1 , wherein the first mode shape vector and the second mode shape vector for the first mode shape and the second mode shape of the multi-component structure, respectively, are formed using equations:

Ψ

r

=

{

Ψ

r

,

comp

-

1

Ψ

r

,

comp

-

2

⋮

Ψ

r

,

comp

-

n

}

⁢

⁢

and

⁢

⁢

Ψ

s

=

{

Ψ

s

,

comp

-

1

Ψ

s

,

comp

-

2

⋮

Ψ

s

,

comp

-

n

}

wherein, ψ r is the first mode shape vector of the multi-component structure, ψ s is the second mode shape vector of the multi-component structure, ψ r,comp-n is the first mode shape vector of a component n, ψ s,comp-n is the second mode shape vector of the component n, and n is the number of components in the multi-component structure.

3. The method of claim 1 , wherein the square of the sum of magnitudes of the correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure is computed using an equation:

∥ψ r,comp-1 H ψ s,comp-1 |+|ψ r,comp-2 H ψ s,comp-2 |+ . . . +|ψ r,comp-n H ψ s,comp-n ∥ 2

wherein, ψ r H ,comp-n is hermitian transpose of ψ r,comp-n .

4. The method of claim 1 , wherein the product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure is computed using an equation:

(ψ r H ψ r )(ψ s H ψ s )

wherein, ψ r H , is hermitian transpose of ψ r and ψ s H is hermitian transpose of ψ s .

5. The method of claim 1 , wherein the modal assurance criterion between the first mode shape vector and the second mode shape vector of the multi-component structure is computed using an equation:

MAC

⁡

(

Ψ

r

,

Ψ

s

)

=





Ψ

r

,

comp

-

1

H

⁢

Ψ

s

,

comp

-

1



+



Ψ

r

,

comp

-

2

H

⁢

Ψ

s

,

comp

-

2



+

…

+



Ψ

r

,

comp

-

n

H

⁢

Ψ

s

,

comp

-

n





2

(

Ψ

r

H

⁢

Ψ

r

)

⁢

(

Ψ

s

H

⁢

Ψ

s

)

wherein, ∥ψ r,comp-1 H ψ s,comp-1 |+|ψ r,comp-2 H ψ s,comp-2 |+ . . . +|ψ r,comp-n H ψ s,comp-n ∥ 2 is the square of sum of the magnitudes of the correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component: structure and (ψ r H ψ r )(ψ s H ψ s ) is the product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure.

6. The method of claim I further comprising:

scaling the first mode shape vector and the second mode shape vector using strain energy weighting factors at each component in the multi-component structure.

7. The method of claim 6 , wherein the strain energy weighting factor is computed by dividing strain energy at each component with total strain energy of the multi-component structure.

8. The method of claim 6 , wherein the first mode shape vector and the second mode shape vector at each component in the multi-component structure are scaled using equations:

ψ r′,comp-n =SE comp-n ψ r,comp-n and ψ s′,comp-n =SE comp-n ψ s,comp-n

wherein, SE comp-n is the strain energy weighting factor of component n, ψ r′,comp-n is the scaled first mode shape vector of the component ψ s′,comp-n is the scaled second mode shape vector of the component n, and n is the number of components in the multi-component structure.

9. The method of claim 8 , wherein the scaled first mode shape vector and scaled second mode shape vector for the first mode shape and the second mode shape of the multi-component structure, respectively, are formed using equations:

Ψ

r

′

=

{

Ψ

r

′

,

comp

-

1

Ψ

r

′

,

comp

-

2

⋮

Ψ

r

′

,

comp

-

n

}

⁢

⁢

and

⁢

⁢

Ψ

s

′

=

{

Ψ

s

′

,

comp

-

1

Ψ

s

′

,

comp

-

2

⋮

Ψ

s

′

,

comp

-

n

}

wherein, ψ r′ is the scaled first mode shape vector of the multi-component structure, ψ s′ is the scaled second mode shape vector of the multi-component structure, ψ r′,comp-n is the scaled first mode shape vector of the component a, ψ r′,comp-n is the scaled second mode shape vector of the component n, and a is the number of components in the multi-component structure.

10. The method of claim 9 , further comprising:

squaring sum of magnitudes of correlation of the modal displacement at each component for the scaled first mode shape vector and the scaled second mode shape vector in the multi-component structure.

11. The method of claim 10 , wherein the square of the sum of magnitudes of the correlation of the modal displacement at each component for the scaled first mode shape vector and the scaled second mode shape vector in the multi-component structure is computed using an equation:

∥ψ r′,comp-1 H ψ s′,comp-1 |+|ψ r′,comp-2 H ψ s′,comp-2 |+ . . . +|ψ r′,comp-n H ψ s′,comp-n ∥ 2

wherein, ψ r′,comp-1 H is hermitian transpose of ψ r′,comp-n .

12. The method of claim 10 , further comprising:

computing a product: of squared magnitudes of the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure.

13. The method of claim 12 , wherein the product of squared magnitudes of the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure is computed using an equation:

(ψ r′ H ψ r′ )(ψ s′ H ψ s′ )

wherein, ψ r′ H is hermitian transpose of ψ r′ and ψ s′ H is hermitian transpose of ψ s′ .

14. The method of claim 12 , further comprising:

computing scaled modal assurance criterion between the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure by dividing the squared sum of magnitudes of the correlation of the modal displacement at each component for the scaled first mode shape vector and the scaled second mode shape vector in the multi-component structure with the computed product of squared magnitudes of the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure.

15. The method of claim 14 , wherein the scaled modal assurance criterion between the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure is computed using an equation:

SCMAC

⁡

(

Ψ

r

′

,

Ψ

s

′

)

=





Ψ

r

′

,

comp

-

1

H

⁢

Ψ

s

′

,

comp

-

1



+



Ψ

r

′

,

comp

-

2

H

⁢

Ψ

s

′

,

comp

-

2



+

…

+



Ψ

r

′

,

comp

-

n

H

⁢

Ψ

s

′

,

comp

-

n





2

(

(

Ψ

r

′

H

⁢

Ψ

r

′

)

⁢

(

Ψ

s

′

H

⁢

Ψ

s

′

)

)

wherein, ∥ψ r′,comp-1 H ψ s′,comp-1 |+|ψ r′,comp-2 H ψ s′,comp-2 |+ . . . +|ψ r′,comp-n H ψ s′,comp-n ∥ 2 is the square of sum of the magnitudes of the correlation of the modal displacement at each component for the scaled first mode shape vector and the scaled second mode shape vector in the multi-component structure and (ψ r′ H ψ r′ )(ψ s′ H ψ s′ ) is the product of squared magnitudes of the sealed first mode shape vector and the scaled second mode shape vector of the multi-component structure.

16. A modal analysis system to compare two mode shapes of a multi-component structure, comprising:

a processor; and

memory coupled to the processor, wherein the memory includes a validation tool to:

form a first mode shape vector and a second mode shape vector for a first mode shape and a second mode shape of the multi-component structure, respectively, by grouping modal displacement at each node associated with each component m the multi-component structure;

compute a square of sum of magnitudes of correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure;

compute a product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure:

compute modal assurance criterion between the first mode shape vector and the second mode shape vector of the multi-component structure by dividing squared sum of magnitudes of the correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure with the computed product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure; and

compare the first mode shape and the second mode shape of the multi-component structure using the computed modal assurance criterion to identify similarity between the two mode shapes of the multi-component structure.

17. The modal analysis system of claim 16 , wherein the validation tool is residing in a modal analysis tool in the modal analysis system.

18. The modal analysis system of claim 16 , wherein the validation tool forms the first mode shape vector and the second mode shape vector for the first mode shape and the second mode shape of the multi-component structure, respectively, using equations:

Ψ

r

=

{

Ψ

r

,

comp

-

1

Ψ

r

,

comp

-

2

⋮

Ψ

r

,

comp

-

n

}

⁢

⁢

and

⁢

⁢

Ψ

s

=

{

Ψ

s

,

comp

-

1

Ψ

s

,

comp

-

2

⋮

Ψ

s

,

comp

-

n

}

wherein, ψ r is the first mode shape vector of the multi-component structure, ψ s is the second mode shape vector of the multi-component structure, ψ r,comp-n is the first mode shape vector of a component n, ψ s,comp-n is the second mode shape vector of the component n, and n is the number of components in the multi-component structure.

19. The modal analysis system of claim 16 , wherein the validation tool computes the square of the sum of magnitudes of the correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure using an equation:

∥ψ r,comp-1 H ψ s,comp-1 |+|ψ r,comp-2 H ψ s,comp-2 |+ . . . +|ψ r,comp-n H ψ s,comp-n ∥ 2

wherein, ψ r,comp-n H is hermitian transpose of ψ r,comp-n .

20. The modal analysis system of claim 16 , wherein the validation tool computes the product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure using an equation:

(ψ r H ψ r )(ψ s H ψ s )

wherein, ψ r H , is hermitian transpose of ψ r and ψ s H is hermitian transpose of ψ s .

21. The modal analysis system of claim 16 , wherein the validation tool computes the modal assurance criterion between the first mode shape vector and the second mode shape vector of the multi-component structure using an equation:

MAC

⁡

(

Ψ

r

,

Ψ

s

)

=





Ψ

r

,

comp

-

1

H

⁢

Ψ

s

,

comp

-

1



+



Ψ

r

,

comp

-

2

H

⁢

Ψ

s

,

comp

-

2



+

…

+



Ψ

r

,

comp

-

n

H

⁢

Ψ

s

,

comp

-

n





2

(

Ψ

r

H

⁢

Ψ

r

)

⁢

(

Ψ

s

H

⁢

Ψ

s

)

wherein, ∥ψ r,comp-1 H ψ s,comp-1 |+|ψ r,comp-2 H ψ s,comp-2 |+ . . . +|ψ r,comp-n H ψ s,comp-n ∥ 2 is the square of sum of the magnitudes of the correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure and (ψ r H ψ r )(ψ s H ψ s ) is the product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure.

22. The modal analysis system of claim 16 , wherein the validation tool further scales the first mode shape vector and the second mode shape vector using strain energy weighting factors at each component in the multi-component structure.

23. The modal analysis system of claim 22 , wherein the strain energy weighting, factor is computed by dividing strain energy at each component with total strain energy of the multi-component structure.

24. The method of claim 22 , wherein the validation tool further scales the first mode shape vector and the second mode shape vector at each component in the multi-component structure using equations:

ψ r′,comp-n =SE comp-n ψ r,comp-n and ψ s′,comp-n =SE comp-n ψ s,comp-n

wherein, SE comp-n is the strain energy weighting factor of component n, ψ r′,comp-n is the scaled first mode shape vector of a component n, ψ s′,comp-n is the scaled second mode shape vector of the component n, and n is the number of components in the multi-component structure.

25. The modal analysis system of claim 24 , wherein the validation tool further forms the scaled first mode shape vector and scaled the second mode shape vector for the first mode shape and the second mode Shape of the multi-component structure, respectively, using equations:

Ψ

r

′

=

{

Ψ

r

′

,

comp

-

1

Ψ

r

′

,

comp

-

2

⋮

Ψ

r

′

,

comp

-

n

}

⁢

⁢

and

⁢

⁢

Ψ

s

′

=

{

Ψ

s

′

,

comp

-

1

Ψ

s

′

,

comp

-

2

⋮

Ψ

s

′

,

comp

-

n

}

wherein, ψ r′ is the scaled first mode shape vector of the multi-component structure, ψ s′ is the scaled second mode shape vector of the multi-component structure, ψ r′,comp-n is the scaled first mode shape vector of the component n, ψ r′,comp-n is the scaled second mode shape vector of the component n, and n is the number of components in the multi-component structure.

26. The modal analysis system of claim 25 , wherein the validation tool further computes a square of sum of magnitudes of correlation of the modal displacement at: each component for the scaled first mode shape vector and the scaled second mode shape vector in the multi-component structure

27. The modal. analysis system of claim 26 , wherein the validation tool further computes the square of sum of magnitudes of the correlation of the modal displacement at each component for the scaled first mode shape vector and the scaled second mode shape vector in the multi-component structure using an equation:

∥ψ r′,comp-1 H ψ s′,comp-1 |+|ψ r′,comp-2 H ψ s′,comp-2 |+ . . . +|ψ r′,comp-n H ψ s′,comp-n ∥ 2

wherein, ψ r′,comp-1 H is hermitian transpose of ψ r′,comp-n .

28. The modal analysis system of claim 26 wherein the validation tool further computes a product of squared magnitudes of the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure.

29. The modal analysis system of claim 28 , wherein the validation tool further computes the product of squared magnitudes of the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure using an equation:

(ψ r′ H ψ r′ )(ψ s′ H ψ s′ )

wherein, ψ r′ H is hermitian transpose of ψ r′ and ψ s′ H is hermitian transpose of ψ s′ .

30. The modal analysis system of claim 28 , wherein the validation tool further computes scaled modal assurance criterion between the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure by dividing the squared sum of magnitudes of the correlation of the modal displacement at each component for the scaled first mode shape vector and the scaled second mode shape vector in the multi-component structure with the computed product of squared magnitudes of the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure.

31. The modal analysis system of claim 30 , wherein the validation tool further computes the scaled modal assurance criterion between the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure using an equation:

SCMAC

⁡

(

Ψ

r

′

,

Ψ

s

′

)

=





Ψ

r

′

,

comp

-

1

H

⁢

Ψ

s

′

,

comp

-

1



+



Ψ

r

′

,

comp

-

2

H

⁢

Ψ

s

′

,

comp

-

2



+

…

+



Ψ

r

′

,

comp

-

n

H

⁢

Ψ

s

′

,

comp

-

n





2

(

(

Ψ

r

′

H

⁢

Ψ

r

′

)

⁢

(

Ψ

s

′

H

⁢

Ψ

s

′

)

)

wherein, ∥ψ r′,comp-1 H ψ s′,comp-1 |+|ψ r′,comp-2 H ψ s′,comp-2 |+ . . . +|ψ r′,comp-n H ψ s′,comp-n ∥ 2 is the square of sum of the magnitudes of the correlation of the modal displacement at each component for the scaled first mode shape vector and the scaled second mode shape vector in the multi-component structure and (ψ r′ H ψ r′ )(ψ s′ H ψ s′ ) is the product of squared magnitudes of the scaled first mode shape vector and the scaled second mode shape vector of the multi-component structure.

32. A non-transitory computer-readable storage medium for comparing two mode shapes of a multi-component structure having instructions that, when executed by a computing device, cause the computing device to:

form a first mode shape vector and a second mode shape vector for a first mode shape and a second mode shape of the multi-component structure, respectively, by grouping modal displacement at each node associated with each component in the multi-component structure;

square sum of magnitudes of correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure;

compute a product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure;

compute modal assurance criterion between the first mode shape vector and the second mode shape vector of the multi-component structure by dividing the squared sum of magnitudes of the correlation of the modal displacement at each component for the first mode shape vector and the second mode shape vector in the multi-component structure with the computed product of squared magnitudes of the first mode shape vector and the second mode shape vector of the multi-component structure; and

compare the first mode shape and second mode shape of the multi-component structure using the computed modal assurance criterion to identify similarity between the two mode shapes of the multi-component structure.

33. The non-transitory computer-readable storage medium of claim 32 , wherein the validation tool further having instructions to:

scale the first mode shape vector and the second mode shape vector using strain energy weighting factors at each component in the multi-component structure.

34. The non-transitory computer-readable storage medium of claim 33 , wherein the strain energy weighting factor is computed by dividing strain energy at each component with total strain energy of the multi-component structure.