Dynamic Experiments
6.1 Experiments on the shaker
6.1.1 Experimental Setup
Figure 6.1: The experimental setup on the shaker
Figure 6.2: A wire rope spring on the shaker
44 6.1 Experiments on the shaker
Three springs are placed together on the shaker table. A dummy mass, M, of 148.4 kg, equally distributed over the three springs, is placed on top of the wire rope springs. Hence, the center of mass of the dummy mass lies exactly above the geometrical center of the equilateral frame formed by the three springs, which is schematically depicted in figure 6.3.
n Accelerometer
t
Accelerometershakertable Actuator
Figure 6.3: Schematic representation of the shaker experiments
Subsequently, the system can be regarded as a single degree of freedom system and it is sufficient to measure the response directly above the center of mass. The excitation ac- celeration is measured on the shaker table. For both measurements SCADAS-systems have been used to measure the acceleration. The sample frequency, f,, equals 333.33 Hz. At the maximum frequency, f,,,, which is 30 Hz, still at least ten points are available to describe one excitation period. Additional experiments with a lower dummy mass are carried out as well.
The frequency sweep is divided into two parts: a sweep-up and a sweep-down. It has been decided to carry out both sweep-ups as sweep-downs to find out whether the system behaves the same during a sweep-up and a sweep-down. During a sweep-up the excitation frequency increases, while the frequency decreases during a sweep-down. A slow frequency sweep of 1 octave/minute is used, implying that during a sweep-up the excitation frequency doubles per minute, see equation (6.1), where fo = 1.6 Hz for a sweep-up and fo = 30 Hz for a sweep-down.
f = fo
*
2h for a sweep-up- t
f = fo
*
260 for a sweep-downWhile the main application of the wire rope springs lies in the naval field, see section 2.2, the shaker table excites the system with constant velocity amplitudes according to standard regulations [12]. Every experiment is carried out three times to get an indication about the repeatability. Furthermore accidental failures during one sweep can be corrected with the results from another sweep. So if during one sweep something unusual happens or a failure is made, the results from two other sweeps can be used to compare the data and if necessary correct the measurement. It can be concluded that the repeatability seems pretty good since the various responses resemble each other well.
6.1.2 Results of the shaker experiments
A fifth-order low-pass filter with a cut-off frequency, f,, of 35 Hz has been used to filter the collected data, because a signal with a frequency of 50 Hz is superimposed on the measure-
ments. Figure 6.4 shows the absolute magnitude of the FFT, X ( f ) , for a measured response before and after filtering. It confirms that the 50 Hz signal is present and has vanished after filtering.
Figure 6.4: Magnitude of the FFT a measured response before and after filtering Various velocity amplitudes have been used as excitation leading to frequency amplitude curves for the wire rope spring system, shown in figure 6.5.
Figure 6.5: Maximum experimental acceleration, x M ( t ) for different excitation levels
Figure 6.5 shows the nonlinear dynamic behavior of the wire rope spring. Clearly an amplitude dependent harmonic resonance can be seen. Furthermore, a second peak occurring at a slightly higher frequency, can be distinguished. Possibly, this peak is a second harmonic resonance since all the springs have a slight different behavior. However, this does not seem very reasonable as the difference between both frequencies is quite large. A second explanation can be that the springs have not been placed exactly straight under the mass and it is a
46 6.1 Experiments on the shaker
resonance peak of the shear or roll mode, but this does not seem logical given that the stiffness in those directions is smaller than in the tension-compression mode and hence the resonance frequency is expected to occur at a lower frequency, which is also shown by Ni et al. [2]. Another option can be the presence of a l / k t h subharmonic response implying that the period time is k times the period time of the excitation. By inspecting the period time of the excitation and the response it has been determined that both period times are the same, which is also confirmed by figure 6.6.
Figure 6.6: Comparison between the shaker excitation and response around 13.6 Hz
3
Figure 6.7 shows the excitation velocity for vamp = 20 mm/s. Although this signal ought to have a constant amplitude it is clear that between the frequencies 8 and 17 Hz this is not true, probably due to the interaction between the mass and the shaker excitation. This might be the reason for the second peak.
- shaker exc~tat~on
.
-
ww7w:
Figure 6.7: The excitation signal for vamp = 20 mm/s.
Figure 6.6 compares the response and the excitation around 13.6 Hz. The excitation even after filtering still has a strange form. Figure 6.8 depicts the power spectral density, S,,, of the excitation signal to see which frequencies are present in the signal. It can be seen that a
frequency of 27.5 Hz is superimposed on the signal. This presence of this frequency cannot be explained.
Figure 6.8: Power Spectral Density for the excitation around 13.6 Hz
For weakly damped nonlinear systems, as e.g. the Duffing system, it is known that there can exist more than one solution at the same frequency. In this case three solutions are found, two stable at the points a and c(-), and one unstable solution (- -), see figure 6.9. Two cyciic-foid bifurcations lead to jumps which impiies that the responses for an increasing or decreasing frequency are different.
Figure 6.9: A cyclic-fold bifurcation
Figure 6.9 depicts the cyclic fold bifurcation, where the solid line represents the stable solutions and the dashed line the unstable solution. When the frequency is increased the response will jump from a to b. Similarly, the response will jump from c to d for a decreasing frequency. Hence, the responses for increasing and decreasing excitation frequency differ.
So the responses of a sweepup and a sweep-down have to be compared. However, it is not likely that the responses will differ due to the large amount of hysteretic damping. This is confirmed by figure 6.10 which shows the responses for the sweep-up and sweep-down for a velocity amplitude of 15 mm/s. Both responses are very similar and do not show jumps at different frequencies.
Next, a closer look is taken at the frequency range from 1.6 till 5 Hz to find out whether superharmonic resonances can be found. The remark must be made that it is very hard to
48 6.1 Experiments on the shaker
Figure 6.10: Difference between the responses of a sweep-up and a sweep-down
distinguish a superharmonic resonance. At approximately 5 Hz a peak can be distinguished in figure 6.5, which might be a superharmonic resonance. Figure 6.11, which shows the excitation and response signal around 4.8 Hz, indicates that the small peak can be caused by the excitation. However, it can also be the case that the peak in the excitation is caused by the superharmonic resonance.
[ I - measured response I
Figure 6.11: The excitation and response signai around 4.8 Hz
Figure 6.12 shows two small resonances at respectively 2.15 and 1.70 Hz, which are both measured during a sweep-down which assures that no transient behavior is present. These might be two superharmonic resonances.
Finally, a comparison is made between the excitation and response signal for vamp = 20 mm/s to learn whether the wire rope spring damps the signal. Figure 6.13 shows the maximum values of both signals against the excitation frequency. It can be seen that for frequencies above 15 Hz the wire rope springs show a very good damping performance when loaded with a mass of 50 kg per spring. Due to the resonance frequency the amplitude of the excitation is larger for frequencies below 15 Hz.
Additional measurements are carried out with less mass attached to the system. Figure 6.14 shows the response for three different excitation levels, which qualitatively do not differ
Figure 6.12: Experimental responses around two different frequencies
rnassl, v = 20 rnrnls
amp
Figure 6.13: Comparison between the excitation and response signal
50 6.1 Experiments on the shaker
very much from the frequency amplitude curves in 6.5. Again an amplitude dependent har- monic resonance can be distinguished, but the resonance frequency as well as its amplitude has changed in comparison with the original setup. Furthermore the second peak is present as well.
Figure 6.14: Experimental shaker response with less mass attached to the wire rope springs
A strange peak can be seen just before the resonance peak. Figure 6.15 shows the excita- tion and response signal around 11 Hz for vamp = 4 mm/s. Nothing unusual can be seen for the response and the excitation
Figure 6.15: Measured excitation and response for mass 2 around 11 Hz
A second small peak can be distinguished around 6.1 Hz which may be caused by the excitation as well. However comparing the excitation and response signals does not lead to a result as has been shown in figure 6.11. Both signals are shown in figure 6.16 but nothing unusual can be distinguished for the excitation. So perhaps this might be a superharmonic resonance but it cannot be assured, because figure 6.16 shows that nothing unusual can be distinguished for the response as well.
v = 15 mmls. mass 2 amp
Figure 6.16: Excitation and response signal around 6.1 Hz