de Mayo de 1915.

x t X_e e^{i t}d

f

³

1

2S : ^: : [4.18]

With X^_e : being a real positive function and x t a real even function, an   input, thus defined, will resemble a SHOC waveform (Chapter 8). This input is independent of the characteristics of the test item and thus eliminates the need to define the transfer function H : . The only necessary parameter is the module of the Fourier transform (or the undamped residual shock spectrum). A series of tests showed that this approach is reasonable [SMA 72].

4.4.6. Restitution of an SRS by a series of modulated sine pulses

This method, suggested by D.L. Kern and C.D. Beam [KER 84], consists of applying a series of modulated sine wave shocks sequentially. The retained waveform resembles the response-versus-time of the mass of a one-degree-of-freedom system base-excited when it is subjected to an exponentially decayed sine wave excitation; it has as an approximate equation:

x t A t e^{ K :t} sin : for tt t 0  

x t 0 elsewhere [4.19]

where : 2Sf:

K = damping of the signal;

A : K  ; e x_m

x_m amplitude of x t ;  

e Neper number.

Figure 4.13. Shock waveform (D.L. Kern and C.D. Hayes)

The choice of K must meet two criteria:

– to be close to 0.05, a value characteristic of many complex structures;

– to allow the maximum of x t (the largest peak) to take place at the same time   as the peak of the envelope of x t .  

Figure 4.14. Coincidence of the peak of the signal and its envelope

The interesting thing about this approach (which again takes a proposal of J.T. Howlett and D.J. Martin [HOW 68] containing purely sinusoidal impulses) is the facility of determination of the characteristics of each sinusoid, since each one of them is considered separately, contrary to the case of a control-per-spectrum (Chapter 8). The shocks are easy to create and realize.

The adjustable parameters are the amplitude and possibly the number of cycles.

The number of frequencies is selected so that the intersection point of the spectra of two adjacent signals is not more than 3 dB lower than the amplitude of the peak of the spectrum (plotted for a damping equal to 0.05). Like the slowly swept sine, this method does not make it possible to excite all resonances simultaneously.

We will see in Chapter 8 how this waveform can be used to constitute a complex drive signal restoring the whole of the spectrum.

4.5. Interest behind simulation of shocks on shaker using a shock spectrum The data of a shock specification for a response spectrum has several advantages:

– the response spectrum should be more easily exploitable for dimensioning of the structure than the signal x t itself;  

– this spectrum can result directly from measurements of the real environment and does not require us, at the design stage, to proceed to an often-delicate equivalence with a signal of simple shape;

– the spectrum can be treated in a statistical way if we have several measurements of the same phenomenon; it can be the envelope of several different transitory events and can be increased by an uncertainty coefficient;

– the reference most commonly allowed to judge quality of the shock simulation is a comparison of the response spectra of the specification with the shock carried out.

In a complementary way, when the shock tests can be carried out using a shaker, we can have direct control from a response spectrum:

– The search for a simple form shock of a given spectrum, compatible with the usual test facilities, is not always a simple operation, according to the shape of the reference spectrum resulting from measurements of the real environment.

– The shapes of the specified spectra can be very varied, contrary to those of the spectra of the usual shocks (half-sine, triangle, square pulse, etc.) carried out on the shock machines. We can therefore improve the quality of simulation and reproduce shocks which are difficult to simulate with the usual means (case of the pyroshocks for example) [GAL 73] [ROT 72].

– Taking into account the oscillatory nature of the elementary signals used, the positive and negative spectra are very close, which makes a reversal of the test item [PAI 64] useless.

– In theory, simple-shaped shocks created on a shock machine are reproducible, which makes it possible to expect uniform tests from one laboratory to another. In practice, we were obliged to define tolerances on the shapes of the time history signals to take account of the distortions of the signal that are really measured and difficult to avoid. The limits are rather broad (+15  20%) and can result in accepting two shocks, included within these limits, that are likely to have very different effects (which we can evaluate with the shock spectra) [FAG 67].

Figure 4.15. Nominal half-sine and its tolerances Figure 4.16. Shock located between the tolerances

Figure 4.17. SRS of the nominal half-sine and the tolerance limits

Figures 4.15 and 4.17 show an example of a nominal half-sine (100 m/s², 10 ms) and its tolerance limits, as well as the shock spectra of the nominal shock and of each lower and upper limit. Figure 4.16 represents a shock made up of the sum of

the nominal half-sine and of a sinusoid of amplitude 15 m/s² and frequency 250 Hz.

The spectrum of this signal is superimposed on the spectra of the tolerance limits in Figure 4.18. Although this composite signal remains within the tolerances, it is noted that it has a very different spectrum from those of the tolerance limits for small [ in a frequency band around 250 Hz and that the negative spectra of the tolerance limits intersect and thus do not delimit a well-defined domain [LAL 72].

Figure 4.18. SRS of the shock of Figure 4.16 and of the tolerance limits

NOTE: Several current standards specify the tolerance limits on the SRS with values in the order of ±6 dB [NAS 99].

With this, some practical advantages are added:

– sequence of the shock and vibration tests without disassembly and with the same test fixture (saving of time and money);

– maintenance of the test item with its normal orientation during the test.

These last two points are not, however, specific to spectrum control, but more generally relate to the use of a shaker. Control by the spectrum, however, increases the capacities of simulation because of the possibility of the choice of the shape of the elementary waveforms and of their variety.

The main control methods are described in Chapter 8.