PSI - Issue 19

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537

531

Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Table 1. Heavy vehicle schedule breakpoints [19]. Wideband Random Spectrum

Harmonic Swept Narrowbands

Amplitude ( 2 / ) 0.001

Narrowband

f1

f2

f3

Frequency (Hz)

Bandwidth (Hz)

5

10

15

20 170

40-340

60-510

5

Swept BW (Hz)

energy per frequency ( 2 / ) Moving rate (Hz/sec) # Narrowbands Sweep Cycles

0.15 0.15

0.15

20

0.01

510

0.01

1

2

3

2000

0.001

2

2

2

The PSD data based on combination of the wideband and three swept narrowbands in Table 1 is used an input PSD base excitation in the modeling approach. Fig. 2 shows a schematic representation of the input PSD function as a function of the time.

Fig. 2. Schematic representation of time-varying input PSD.

2.2. Response PSD

Dynamic response of a simple structure subjected to random loads can be mathematically represented as a single degree-of-freedom (SDOF) system provided that the first mode shape is dominant mode. The assumption of a SDOF can be employed to simplify the modeling approach [22,23]. The system equation of a damped SDOF is the equation of the base-excited function shown in Eq. (1): mÿ + ( ̇ − ̇) + ( − ) = 0 (1) Where m is the mass of the system, and c and k are the damping and elastic constants of the SDOF system, respectively. The transfer function expressed in Eq. (2) can be defined as a gain function for the base excitation. T f = | | = √ 1 + (2 ) 2 (1 − 2 ) 2 + (2 ) 2 (2) Where damping ratio ζ = c c cr , critical damping c cr = 2 , natural frequency f n2 = / and frequency ratio β = f/f n . The response PSD for a given specific input PSD can be computed by the transfer function of the defined SDOF