Experimental HALTs with sine-on-random synthesized profiles

2.1. Set up

An extensive experimental campaign was performed to further investigate the effectiveness of the proposed SoR synthesis procedure [2] with respect to the traditional PSD synthesis [1]. To this aim many laboratory tests were carried out by means of an electromechanical shaker (Dongling ES-2-150), which imposed the reference signal and different synthesized profiles on purpose-built specimens (Fig. 2). The idea was to compare the fatigue damage actually induced by the synthesized excitations with the damage due to the reference one. The specimen was a flat beam, made of the Aluminum alloy EN AW-6060, fixed to a rigid support in a cantilevered configuration. The thickness was 2 mm and the critical section 10 mm wide; a lumped mass was fixed to its tip (the center of mass being about 66.5 mm from the rigid support) in order to present the natural frequency of the first bending mode at about 32.65 Hz, i.e. corresponding to the main peak of the base excitation signal. A preliminary experimental characterization [3] permitted to determine the material actual mechanical properties (Fig. 2), in particular the value of the Basquin‘s law exponent $b$, which is particularly important in the Mission Synthesis algorithms [1, 4, 5]. Besides the control accelerometer placed at the specimen support base, a second sensor was fixed under the appended mass in order to measure the specimen tip acceleration.

Fig. 2Purpose-built specimen used for the experimental validation

2.2. Test procedure

The helicopter measured vibrations were kept as the reference for this experimental investigation. They were replicated as base excitation to induce on the specimen a certain fatigue damage. Due to both the signal and specimen characteristics, failure breakdown was not achievable (in a reasonable time) so that a “conventional Time-to-Failure”, cTTF, was introduced considering the effect of accumulated fatigue damage to decrease the specimen natural frequency [6, 7]. In particular, cTTF is here defined as the time necessary to decrease the specimen natural frequency from the value $f_{in}=$ 32.67 Hz to the final value $f_{fin}=$ 31.04 Hz (5 % decrement). The value $f_{in}=$ 32.67 Hz was chosen since (i) is bigger than and very close to the excitation peak at 32.65 Hz and (ii) most specimens exhibit an initial value $f_{n0}\ge$ 32.67 Hz (Table 1).

Five kinds of tests were carried out, each one repeated for three different specimens for the sake of data reliability, corresponding to the application of the following base excitations:

s0. reference signal (specimens and tests denoted as $s_{01}$, $s_{02}$, and $s_{03}$);

s1. synthesized PSD ($s_{11}$, $s_{12}$, $s_{13}$) computed to induce the same damage in the same test duration of the reference signal, that is to present the same FDS and cTTF;

s2. synthesized SoR ($s_{21}$, $s_{22}$, $s_{23}$) computed to have the same FDS and cTTF of the reference signal;

s3. synthesized PSD ($s_{31}$, $s_{32}$, $s_{33}$) computed to have the same FDS and half the duration cTTF of the reference signal (accelerated tests);

s4. synthesized SoR ($s_{41}$, $s_{42}$, $s_{43}$) computed to have the same FDS and half the duration cTTF of the reference signal (accelerated tests).

The FDS functions, i.e. the target in the PSD and SoR profile syntheses, were computed considering the following parameters: bandwidth 5-200 Hz, frequency resolution $df=$ 0.05 Hz; $Q$ Factor $Q=$ 1/2, $\zeta =$ 69.4; Wöhler curve slope $b=$ 6.3; material constants appearing in Eq. (11) of ref. [2] $K=C=$1.

Among the three data sets for each kind of test, the results corresponding to the specimen exhibiting the median value of cTTF were considered as representative for a direct comparison. In particular, the cTTF associated with the reference signal, necessary to synthesize the other four signals, was 484 minutes ($s_{02}$, Section 2.3).

The shaker controller was run by means of the LMS Test.Lab modules Single Axis Waveform Replication (for tests $s_{0i}$, $s_{2i}$, and $s_{4i}$, $i=$ 1, 2, 3, where the input profile was sequentially replicated until the achievement of cTTF) and Random Control (for tests $s_{1i}$ and $s_{3i}$, $i=$1, 2, 3) with a sampling frequency of 800 Hz (automatically fixed by the software): the results were then low-pass filtered at 200 Hz and down sampled at 400 Hz, in order to exactly match the signal characteristics of the original measurements.

The natural frequency $f_n$ of the specimens was monitored by computing the Frequency Response Function (FRF) between the mass and base accelerations. The FRFs were computed for signal windows of 30 seconds (with NFFT = 4096 points per signal-block, an overlap of 66.67 %). The initial value of the natural frequency, $f_{n0}$, was computed for each specimen with a running average approach performed for 3 values of $f_n$. The same approach is used to compute the final value of the natural frequency that determines the cTTF. Table 1 reports the values of $f_{n0}$ (as well as the mean values of the damping factor ${\zeta }_{mean}$ computed for the entire test duration) for each specimen. It can be noted that specimens $s_{21}$ and $s_{22}$ exhibit an initial frequency $f_{n0}$ smaller than $f_{in}=$ 32.67 Hz. For these specimens cTTF is computed as the time necessary to decrease the corresponding $f_{n0}$ of an amount of 5 %.

2.3. Results

Table 2 reports the most important results of the experimental campaign. It can be noted that for both the equal-time tests and the HALTs (time reduction factor equal to 0.5) the proposed mission synthesis procedure performs better than the traditional algorithms. Indeed, even though both the PSD and SoR profiles were synthesized in order to match the FDS of the reference signal (specifically the target one corresponding to $s_{02}$), the SoR profiles prove a superior suitability to represent the reference signal in terms of induced fatigue damage, as it can be appreciated by comparing the actual median cTTFs with respect to the corresponding desired value cTTF_target. The “errors” in terms of actual cTTF vs. target cTTF are in fact +3.5 % and +17.4 % for $s_{23}$ and $s_{43}$ (SoR profiles), in spite of +101.7 % and +79.3 %, for $s_{12}$ and $s_{31}$ (PSD profiles), respectively.

As an example, Fig. 3 reports the variation of the specimen $s_{43}$ natural frequency over time: for the other cases, similar trends were observed.

Fig. 3Specimen s43 natural frequency

Fig. 4FDSs computed for the measured base excitations (close-up)

Fig. 4 reports a close-up view of the actual FDS functions computed from the measured base acceleration signals referring to the time interval which defines the cTTFs. Fig. 5 reports the PSD of the measured inputs (base acceleration) and responses (relative mass-base acceleration). It can be noted that the PSD synthesized profiles (red and black curves) were not able to perfectly match the very narrow peak at 32.65 Hz. The damage potential was thus spread over the adjacent frequencies to compensate for the absence of the deterministic sinusoid components, but the final effect proved not meaningful in this application where the specimen natural frequency was set up on purpose just to be resonant, Fig. 5(b).

Fig. 5PSD computed for the a) measured base (close-up) and b) relative mass-base accelerations

a)

b)

Finally, Table 3 reports the peak and RMS values of input and response accelerations measured for the representative specimens of the five tests. These data further prove the better consistency of the SoR synthesized profiles with the reference signal than the PSD profiles.