PSI - Issue 19

Giovanni M. Teixeira et al. / Procedia Structural Integrity 19 (2019) 175–193 Author name / Structural Integrity Procedia 00 (2019) 000–000

182

8

(SDOF) system as shown in Fig. 8. The SDOF in the picture refers to a base motion case, where x is the harmonic displacement of the base and y is the harmonic relative displacement of the mass. The output/input ratio (y/x) is described by the several curves in the diagram on the right. They differ by the amount of damping (ξ) only. The horizontal axis represents the ratio between the frequency of the base load and the system resonant frequency (Ω= ω ⁄ ω n ).

Fig. 8. Structural Stresses from nodal forces and moments.

In a multiaxial scenario there are multiple outputs and the transfer function is replaced by a matrix of transfer functions, often called “Dynamic Response Matrix” Q :

    ,1 ,2 f f

S S

     

      

S

(12)

S

Q

 

  f

M

  , f N

S

Nx

1

S

Where N is the number of load channels and S S is the equivalent structural stress. This matrix Q is then used to transform the PSD of loads (G LOAD ) into the PSD of stresses G σ :      t LOAD G f Q G Q  (13) where G LOAD is defined as:

G

G

      

       

11

1

N

(14)

O

 

LOAD G f

 

G

G

NN x N N

N

1

and G ii represents the auto-PSD components and G ij (i ≠ j) the cross- PSD components (describing the relationship between loading channels). As an example, in the case of a chassis on 4 wheels (Fig. 9) the number of load channels would be at least 12 as accelerations need to be defined in 3 different directions ( X,Y,Z ) at each wheel.