PSI - Issue 5

M. Braz-César et al. / Procedia Structural Integrity 5 (2017) 347–354 Braz-César M. et al./ Structural Integrity Procedia 00 (2017) 000 – 000

352

6

    

    

9 1,1      12 1,2 10 2,2 c c 12 2,1

c c c

0

,

c c

(7)

C

, ( ,

,      ,

)



13 2,3

9

10

11

12

13

0

13 3,2  

11 3,3

where α i represent tuning coefficients for each element of these matrices. To reduce the number of optimization parameters, the damping can be assumed as a linear combination of the mass and stiffness matrices. Thus, the optimization problem will be formulated with the Caughey damping matrix defined as

1 10 11 C M K KM K        9

(8)

The objective function also includes a cost function related with the damping coefficients. Thus, it follows that

(9)

( ) ( ) ( ) f x f x f x f x       ( )

where f ω ( x ) accounts for the difference between numerical and reference frequencies, f φ ( x ) is related with the correlated mode shapes and f ξ ( x ) is related with the damping ratio. These functions are given by

  

  

* 2

  

N



f x 

( )

, 0

1

i 

i   



(10)

i

i

* 

i

1



i

2

      

   

1 MAC 

N



i

f x 

(11)

( )

, 0

1

i 

i   



MAC

i

1



i

  

* 2

  

N



*  (12) where and ∗ are the th analytical and measured frequencies, respectively, and δ i are the weight factors and ξ i and ξ i * are the i th experimentally measured and analytically predicted damping ratios, respectively. MAC represents the th Modal Assurance Criterion (MAC) defined as               2 * * * MAC T i i T T i i i i        (13) th experimentally measured and theoretically predicted mode shapes, respectively. Values close to unity indicate good correlation between the between measured and predicted mode shapes while zero means no correlation at all. The optimization parameters were constrained to ensure that the resultant matrices have physical meaning and also to be compatible with the actual range of values of the experimental model (Table 2). 1 ( ) , 0 1 i i i  i    i i f x    where φ i and φ i * are the i

Table 2. Model parameters to be updated. Num. Parameter

Initial value

Upper limit

Lower limit

m i (kg)

1 to 3 4 to 8

4.50 6000

6.65 2500

3.65 7200

k 1 (N/m)