PSI - Issue 8

Davide Zanellati et al. / Procedia Structural Integrity 8 (2018) 92–101 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

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A prototype of the new system layout, described in Section 2, is mounted to the shaker head with screws and nuts, as shown in Fig. 7b. The measurement instruments used consist of a tri-axis accelerometer (#1) placed on shaker head to measure the input accelerations in closed loop control, and other two accelerometers (#2 and #3) for monitoring the system responses at the extremity of both the specimen and the cantilever beam. As exhibited below, a harmonic analysis was performed in laboratory to measure the system response, with the aim to confirm if, at low frequencies, bending and torsion are fully uncoupled, as obtained in numerical simulations. Therefore, the tests were focused on a narrow frequency range (from 25 to 50 Hz) below the first resonance. Moreover, the influence of the phase shift ( φ ) between input accelerations was also investigated. A first harmonic analysis applied, to the shaker head, two in-quadrature ( φ = π/2 ) harmonic accelerations of 1g amplitude, with frequency varying from 25 to 50 Hz, in 1 Hz steps. The first acceleration x 1 ( t ) is along the vertical direction, the second x 2 ( t ) is along the horizontal direction perpendicular to the specimen axis (Fig. 7a). The other horizontal acceleration longitudinal to the specimen axis, not required to have bending and torsional loading, was significantly smaller and equal to 0.1g. A second similar harmonic analysis was also conducted by using in-phase ( φ =0°) input accelerations. The output accelerations were measured at the specimen extremity along the vertical direction (accelerometer #2 in Fig. 7b) and at the cantilever beam extremity along the horizontal direction (accelerometer #3).

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

specimen extremity in phase specimen extremity in quadrature cantilever beam extremity in phase cantilever beam extremity in quadrature

specimen extremity FEM specimen extremity experimental cantilever beam extremity FEM cantilever beam extremity experimental

|Y i ( f )| [g]

|Y i ( f )| [g]

20

25

30

35

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45

50

55

20

25

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f [Hz]

f [Hz]

(a)

(b)

Fig. 8. (a) Output acceleration amplitudes | Y i ( f )| with in-quadrature and in-phase input accelerations; (b) comparison between numerical and experimental output acceleration amplitudes | Y i ( f )| for in-quadrature input accelerations.

Fig. 8a compares the acceleration amplitudes | Y i ( f )| measured by accelerometers #2 and #3, for both in-quadrature and in-phase input accelerations. In both cases, a higher response amplitude results for in-phase input accelerations. This trend can be explained by using the theory of Multi-Input/Multi-Output (MIMO) model, in which x 1 ( t ) and x 2 ( t ) are the input accelerations at the shaker head along vertical and horizontal directions, respectively, while y 1 ( t ) is the acceleration response at specimen extremity along vertical direction and y 2 ( t ) is the acceleration response at

cantilever beam extremity along the horizontal direction (symbol t indicates time). In matrix notation, the relationship in frequency domain between inputs and outputs is:

2 Y f Y f 1

21 H f H f 11

22 H f H f 12

2 X f X f 1

( ) ( )

( ) ( )

( ) ( )

( ) ( )

  

   

  

  

  

  

  ( ) f f Y H X 

  ( ) ( ) f



→

(2)

where the column vectors { X ( f )}, { Y ( f )} represent the Fourier Transform of x ( t )={ x 1 ( t ), x 2 ( t )} y 2 ( t )} T , while [ H ( f )] is the transfer function matrix, in which the term H ij ( f )= Y i / X j represents the FRF when the system only has one input “ j ” and the response is measured at coordinate “ i ”. By matrix product, each output response yields: T , y ( t )={ y 1 ( t ),