Fatigue life estimation algorithm of structures with high frequency random vibration based on hybrid energy finite element method

2.1. The direct field and reverberant field

Fig. 1 shows the wave field components in a plate: where $S$ is the exciting position, $\overrightarrow{q_d}$ and $\overrightarrow{q_r}$ represent direct and reverberant power flow intensity. $\mathrm{\Omega }$ represent domain, $\mathrm{\Gamma }$ represent boundary.

Fig. 1Wave field components in a plate

The total energy density and power flow intensity in the domain can be written as [7]:

where, the subscript $d$ represents the direct field, r represents the reverberant field.

The energy density of the direct field is:

where, $r$ represents the distance from the excitation point, and $r_0$ is expressed as $r_0=1/2{\pi }^2/\lambda +m$.

The reverberant field can be written as [8]:

2.2. Coupling boundary

P. Hardy deduces the bending wave energy flow at the junction of two plates using the energy transfer coefficient and power flow balance [8]. According to P. Hardy’s thought, the Hybrid-EFEM, which only consider the role of bending wave can be extended to general coupling conditions. Considering the space limitations, here we do not present the detailed derivation.

2.3. An example of hybrid-EFEM

Consider the two square plates that are coupled as Fig. 2, and the material parameters and dimensions of the two plates are shown in the Table 1.

Fig. 2L-type coupling plate

Considering a unit force and $f=$ 1000 Hz, $\eta =$ 0.1. The result on the dotted line calculated by FEM, Hybrid-EFEM, EFEM and SEA is shown in Fig. 3.

It can be seen that compared with EFEM and SEA, Hybrid-EFEM can better simulate the responses especially places near the excitation point. This is of great significance to predict the location and stress level of the structure’s dangerous point.

Fig. 3Comparison of bending energy results in four methods

3.1. Zero order moment stress spectrum theory

Y. Y. Wang [4] analyzes the influence of the stress PSD shape on the fatigue damage and obtains that the fatigue damage is insensitive to the shape of the stress PSD, and only the stress RMS is enough for the fatigue life estimation, as for broad-band processes, the whole band can be divided into several partitions, and the stress RMS of each partition is used to estimate the fatigue damage.

3.2. Application of hybrid-EFEM in zero order moment stress spectrum method

From the Parseval theorem we can see: The zero order moment of a physical quantity’s PSD is equal to the mean square of the physical quantity. Thus, we can use the Hybrid-EFEM to determine the sturacture’s total energy in the 1/3 octave, then, A formulation for estimating the equivalent stress from the energy density is deduced like SEA [5]. Then, the zero order moment of the stress PSD in the octave band is obtained through Parseval theorem. Ultimately, the structural fatigue life prediction is carried out by combining the zero order moment stress spectrum method.

Considering the type of three coupling plate as shown in the Fig. 4. Which usually work as a support structure, it is significant to study its fatigue life. each plate is aluminum plate structure of 1 m×1 m length and 3 mm thickness, Poisson’s ratio is 0.32, The internal loss factor of each band is 0.0063. Assuming that the plate 2 is randomly excited in the center point. The force is shown in Fig. 5. Time domain force is translated into frequency domain by Fourier transform. The broad-band is divided into 10 1/3 octaves from 1 kHz to 8 kHz. The zero-order moment of the stress PSD in each octave is calculated respectively, and then the fatigue life is estimated.

Fig. 4Three coupling plate structure

Fig. 5Force – time domain curve

Fig. 6Bending wave energy nephogram in center frequency f= 1000 Hz

a) EFEM

b) Hybrid-EFEM

The bending wave energy nephogram in center frequency $f=$ 1000 Hz calculated separately by EFEM and Hybrid-EFEM was shown in Fig. 6.

From Fig. 6 we can see the energy is symmetrical due to the symmetrical structure and force. And the dangerous point is the center of plate 2. Most of all, Fig. 6 shows the stress level of the dangerous point simulated by Hybrid-EFEM is much larger than that of EFEM.

Assumed the two coefficients of the S-N curve is $k=$ 3.015, $c=$ 1.268×10¹³, The fatigue life calculated by the zero-order moment stress spectrum is shown in Table 2.