3. Properties of Shock Response Spectra

Chapter 3

Properties of Shock Response Spectra

3.1. Shock response spectra domains

Three domains can be schematically distinguished in shock spectra:

-An impulse domain at low frequencies, in which the amplitude of the spectrum (and thus of the response) is lower than the amplitude of the shock. The shock here is of very short duration with respect to the natural period of the system. The system reduces the effects of the shock. The properties of the spectra in this domain will be detailed in section 3.2.

- A static domain in the range of the high frequencies, where the positive spectrum tends towards the amplitude of the shock whatever the damping. Everything occurs here as if the excitation were a static acceleration (or a very slowly varying acceleration), the natural period of the system being small compared to the duration of the shock. This does not apply to rectangular shocks or to the shocks with zero rise time. The real shocks necessarily have a rise time different from zero, this restriction remains theoretical.

-An intermediate domain in which there is dynamic amplification of the effects of the shock, the natural period of the system being close to the duration of the shock. This amplification, more or less significant depending on the shape of the shock and the damping of the system, does not exceed 1.77 for shocks of traditional, simple shape (half-sine, versed sine and terminal peak sawtooth (TPS)). Much larger values are reached in the case of oscillatory shocks, made up, for example, by a few periods of a sinusoid.

96 Mechanical Shock

3.2. Properties of SRS at low frequencies

3.2.1. General properties

In this impulse region ( 0 I fo T I 0,2 ):

-the form of the shock has little influence on the amplitude of the spectrum. We will see below that only (for a given damping) the velocity change AV associated with the shock, equal to the algebraic surface under the curve x(t), is important;

-the positive and negative spectra are, in general, the residual spectra (it is necessary sometimes that the frequency of spectrum is very small and there can be exceptions for certain long shocks in particular). They are nearly symmetric so long as damping is small;

2 - the response (pseudo-acceleration wo zsup or absolute acceleration ys,) is

lower than the amplitude of the excitation; there is an “attenuation”. It is consequently in this impulse region that it would be advisable to choose the natural frequency of an isolation system to the shock, from which we can deduce the stiffness envisaged of the insulating material:

2 2 2 k = m w o = 4 n f o m

(with m being the mass of the material to be protected);

-the curvature of the spectrum always cancels at the origin (fo = O Hz) [FUN 571.

The properties of the SRS are often better demonstrated by a logarithmic chart or a four coordinate representation.

3.2.2. Shocks with zero velocity change

For the shocks that are simple in shape (AV f 0), the residual spectrum is larger than the primary spectrum at low frequencies.

For an arbitrary damping 5, it can be shown that the impulse response is given by:

Properties of Shock Response Spectra 97

dz(t) where z( t ) is maximum for t such that - = 0, i.e. for t such that: dt

lil - t2 wo J x t = arctan- 5

yielding:

The SRS is thus equal at low frequencies to:

1.e.:

If 6 -+ 0, (~(5) -+ 1 and the slope tends towards AV. The slope p of the spectrum at the origin is then equal to:

The tangent at the origin of the spectrum plotted for zero damping in linear scales has a slope proportional to the velocity change AV corresponding to the shock pulse.

If damping is small, this relation is approximate.

98 Mechanical Shock

Example 3.1.

Half-sine shock pulse 100 m/s2, 10 ms, positive SRS (relative displacements).

The slope of the spectrum at the origin is equal to (Figure 3.1):

120

30 p = - = 4 d s

yielding:

a value to be compared with the surface under the half-sine shock pulse:

2 2 - 100 0.01 = - d s n n

I Figure 3.1. Slope of the SRS at the origin

With the pseudovelocity plotted against coo, the spectrum is defined by

coo zsup = AV dt)


When w0 tends towards zero, w0 zsup tends towards the constant value

AV cp( 5). Figure 3.2 shows the variations of cp( 5) versus 5.

1 .o 0.9

0 8

rl 0.7 s 4 0.6

0 5

0 4

0.3 10” 10-2 1 00

5

Figure 3.2. Variations ofthe function p(4

Zxample 3.2.

TPS shock pulse 100 m/s2, 10 ms.

Pseudovelocity calculated starting from the positive SRS (Figure 3.3).

Figure 3.3. Pseudovelocity SRS o fa TPS shock pulse

It can be seen that the pseudovelocity spectrum plotted for 5 = 0 tends towards 1.5 at low frequencies (area under TPS shock pulse).

100 Mechanical Shock

The pseudovelocity oo zsup tends towards AV when the damping tends towards

zero. If damping is different from zero, the pseudovelocity tends towards a constant value lower than AV .

The residual positive SRS of the relative displacements (mi zsup) decreases at

low frequencies with a slope equal to 1, i.e., on a logarithmic scale, with a slope of 6 dB/octave (5 = 0).

The impulse absolute response of a linear one-degree-of-freedom system is given by relation [4.74] (Volume 1). It can also be written:

where:

0, = OoJl - t2

‘p = arc tan 1-252

If damping is zero:

h(t) = oo sin oo t

AV = r Ix ( t )d t = 6 x(t)dt

The “input” impulse can be represented in the form:

x(t) = AV 8(t)

as long as oo 2: << 1. The response which results is:

[3 9 1

[3.10]

[3.11]

[3.12] 2 w0 z(t) = AV h(t)

The maximum of the displacement takes place during the residual response, for:

h(tmax 1 a 0 [3.13]

Properties of Shock Response Spectra 10 1

[3.14]

yielding the SRS:

S = A V o o

and:

log(S) = log(wo) + log(AV) [3.15]

A curve defined by a relation of the form y = a f" is represented by a line of slope n on a logarithmic grid:

log y = n log f + log a [3.16]

The slope can be expressed by a number N of dB/octave according to:

N dB/octave = 20 loglo 2n = 20 n (loglo 2)

N dB/octave = 6 n

[3.17]

[3.18]

The undamped SRSplotted on a log-log grid thus has a slope at the origin equal to 1, i.e. 6 dB/octave.

Terminal peak sawtooth pulse 10 ms, 100 m / s 2

AV = 0.5 m / s

Figure 3.4. TPSshockpulse


I TPS. - 1 0 0 m l s ~ - lOrns

Frequency (Hz)

Figure 3.5. Residualpositive SRS (relative displacements) of a TPS shock pulse

The primary positive SRS zsup always has a slope equal to 2 (12 dB/octave)

[SMA 851.

Example 3.4.

HaEsiiie - 100 mls2 - 10 ins lo2

2 10‘

25 100

ffi 10-1

g 10”

z 10.3

10.“

1 0 6

10”

m

i?

a,

109 10-l i o n 10’ lo2 103 Frequency (Hz)

Figure 3.6. Primary positive SRS of a ha2f-sine shock pulse

The relative displacement zsup tends towards a constant value zo = xm equal to

the absolute displacement of the support during the application of the shock pulse (Figure 3.7). At low resonance frequencies, the equipment is not directly sensitive to accelerations, but to displacement:

zsup -+1 xm


Support - - - - E l

- - Mass - U

Figure 3.7. Behaviov of a Yesonator at very low vesonancefvequency

The system works as soft suspension which attenuates accelerations with large displacements [SNO 681.

This property can be demonstrated by considering the relative displacement response of a linear one-degree-of-freedom system given by Duhamel’s equation (Volume 1, Chapter 3):

z(t) = - 1 Sfx(a) e-Soo(t-a) sin[wod? (t -a)] d a mod? O

If 0 0 + 0 , e-6 (t-a) + 1 and

z(t) = -c x(a) (t -a ) d a

After integration by parts we obtain:

z ( t )= t v(0)-x(t)+x(O) [3.19]


If x(0) = 0 and v(0) = 0 ,

z(t) -+ - x(t) [3.20]

The mass m of an infinitely flexible oscillator, and therefore of infinite natural period (f, = 0), does not move in the absolute reference axes. The spectrum of the relative displacement thus has as an asymptotic value the maximum value of the absolute displacement of the base.

Example 3.5.

Figure 3.8 shows the primary positive SRS %p(fo) of a shock of half-sine ihape 100 m/s2 , 10 ms plotted for 5 = 0 between 0.01 Hz and 100 Hz.

Figure 3.8. Primary positive SRS of a half-ine (relative displacements)

The maximum displacement x, under shock calculated from the expression 2( t) for the acceleration pulse is equal to:

.. 2

7T x, =*=3.18 m

The SRS tends towards this value when fn -+ 0 .


For shocks of simple shape, the instant of time tp at which the first peak of the 7t

response takes place tends towards - as wo tends towards zero [FUN 571. 2 0 0

The primary positive spectrum of pseudovelocities has a slope of 6 dB/octave at the low frequencies.

Example 3.6.

Frequency (Kz)

Figure 3.9. Primary positive SRS of a TPSpulse four coordinate grid)

3.2.3. Shocks with AV = 0 and AD it 0 at the end of apulse

In this case, for 5 = 0 :

- the Fourier transform of the velocity for f = 0, V( 0) , is equal to

V(0) = dt= AD [3.21]

Since acceleration is the first derivative of velocity, the residual spectrum is equal to wo AD for low values of coo. The undamped residual SRS thus has a slope equal to 2 (i.e. 12 dB/octave) in this range.

2


Example 3.7.

10 ms. Shock made up of one sinusoid period of amplitude 100 d s 2 and duration

Frequency (Hz)

Figure 3.10. Residual positive SRS of a “sine I period” shockpulse

- the primary relative displacement (positive or negative, according to the form of the shock) zsup tends towards a constant value equal to x,, absolute

displacement corresponding to the acceleration pulse x(t ) defining the shock:


Cxample 3.8.

Let us consider a terminal peak sawtooth pulse of amplitude 100 m / s 2 and luration 10 ms with a symmetric square pre- and post-shock of amplitude 10 d s 2 . The shock has a maximum displacement given by (Chapter 7):

At the end of the shock, there is no change in velocity, but the residual lisplacement is equal to:

Using the numerical data of this example, we obtain:

x, =-4.428 mm

We find this value of x, on the primary negative spectrum o I.

:Figure 3.1 1). In addition:

Xresidual = -0.9576 mm

Frequency (Hz)

Figure 3.11. Primary negative SRS (displacements) of a TPSpulse with square pre- andpost-shocks

..is shoc,.


3.2.4. Shocks with AV = 0 and AD = 0 at the end of apulse

For oscillatory shocks, we note the existence of the following regions [SMA 851 (Figure 3.12):

-just below the principal frequency of the shock, the spectrum has, on a logarithmic scale, a slope characterized by the primary response (about 3);

-when the frequency of spectrum decreases, its slope tends towards a smaller value of 2;

-when the natural frequency decreases further, we observe a slope equal to 1 (6 dB/octave) (residual spectrum). In a general way, all the shocks, whatever their form, have a spectrum of slope of 1 on a logarithmic scale if the frequency is quite small.

ZERDpulse- 1 0 0 m l s ~ - l00Hz - q=005 3

1 w3 102

10‘

100

10-1

1 o-2 10”

1 o - ~ 10”

10.; 1v3 1v2 1Q.l 1 OD 10’

Frequency (Hz)

Figure 3.12. Shock response spectrum (relative displacements) of a ZERD pulse (AD = 0, AV = 0) [FIS 771 [LAL 901 [SMA 8.51

2 The primary negative SRS oo zsup has a slope of 12 dB/octave; the relative displacement zsup tends towards the absolute displacement xm associated with the shock movement x( t) .


Example 3.10.

Half-sine - Half-sine pre and post-shocks 103

A 102 %. . g 101

2 100

Q) 10-1

2 10-2

p E tr: 10"s

M

i? 10.3

- 10.6

1 OD 10' l o 2 1c Frequency (Hz)

Figure 3.13. Primary negative SRS of a half-sine pulse with half-sine pre- andpost-shocks

Half-sine - Half-sine pre and post-shocks Half-sine - Half-sine pre and post-shocks 2

1 w2 A

3, 10-3

p: tc1

3 P

h

2 1 w2

I 0-3

.- E Is;

Frequency (Hz)

Figure 3.14. Primary negative SRS (displacements) of a half-sine pulse with half-sine pre- andpost-shocks

If the velocity change and the variation in displacement are zero at the end of the shock, but if the integral of the displacement has a non-zero value AD, the undamped residual spectrum is given by [SMA 851:


3 S,(o0) = o0 AD

for small values of oo (slope of 18 dB/octave).

[3.22]

Example 3.11.

Half-sine - Half-sine pre and post-shocks 103

Q lo2

.s 10’

2 1c-’

2 10”

3 10-3

P

M Qi

100 d 3

0. - J

1 o - ~ 10” 35 1 co 10‘ 102

Frequency (Hz)

Figure 3.15. Residualpositive SRS of a half-sine pulse with half-sine pre- andpost-shocks

3.2.5. Notes on residual spectrum

Spectrum of absolute displacements

When oo is sufficiently small, the residual spectrum of an excitation x(t) is identical to the corresponding displacement spectrum in one of the following ways [FUN 611:

a) X ( T ) = 0 X(T) # 0

b) X(T) = x(Z) = 0 but 5,; x(t) dt # 0

X ( T ) = X(Z) = I;x(t) dt = 0 but x(t) dt f Ofor 0 < t < X(T)

Properties of Shock Response Spectra 11 1

w0 11: x(h) dl.1 and w’, 1 1;‘ x(h) h dh

However, contrary to the case (c) above, if:

where t = tp is the time when the integral

x(z) = x(z) = 1: x(t) dt = 0

but if there exists more than one value tp of time in the interval 0 < tp I z for

which 5,” x(t) dt = 0 , then the residual spectrum is equal to a’, I J;x(t) dtl while

the spectrum of the displacements is equal to the largest values of a’, 11;‘ x(t) dtl

[FUN 611.

If AV and AD are zero at the end of the shock, the response spectrum of the absolute displacement is equal to 2 x( z) where x( z) is the residual displacement of the base. If X(T) = 0, the spectrum is equal to the largest of the two quantities

5,;’ x(h) dh is cancelled. The absolute displacement of response is not limited if the

input shock is such that AV # 0.

Relative displacement

When w0 is sufficiently small, the residual spectrum and the spectrum of the displacements are identical in the following cases:

a) if X ( T ) # 0 at the end of the shock;

b) if x(z) = 0, but x( t) is maximum with t = T.

If not, the residual spectrum is equal to x(z), while the spectrum of the displacements is equal to the largest absolute value of x( t) .

3.3. Properties of SRS at high frequencies

The response can be written, according to relation [2.16]:

2 - 0 0 I t X ( @ ,-5w, (t-a) sin . wo Jz( t - a) d a JS wo z(t) = ~

i.e., while setting u = t - a


2

We want to show that lim o$Z(t) = -x(t) . Let us set: wo +-

w(t)=- X(t) ,-5 0 0 u ~ , m u du

Integrating by parts:

w( t) tends towards -x ( t ) when oo tends towards infinity. Let us show that:

lim [w;z( t ) -w( t ) ]=O, i.e., b f & > O , 3 Q > O such that V ’ O ~ Q , wo +- la; z(t> - w(tj I & . constant .

-x( t - u) sin( oo J’_52u)] e-‘ dul

-x( t - u) sin( wo d z u ) l e-‘ Oat du


If the function x( t) is continuous, the quantity

tends towards zero as u tends towards zero. There consequently exists ?'l E [0, t] such that V u E [ 0, q] , f( u) I E and we have:

The function x( t) is continuous and therefore limited to [ 0, t] : 3 M > 0 when, for all u E [ 0, t] , lx( u)I 5 M , and we have:

Thus, for o 2 Q

a0 z( t ) - w( t)l I ~ p f ( u) e-' ' du JZ O


0 0 0 0 t z( t) - w( t)l I ~ p f ( u) e-' wo du + ~ I f ( u) e-' O0 du J? O J1-52 rl

2 At high frequencies, coo z( t) thus tends towards x( t) and, consequently, the SRS tends towards x,, a maximum x( t) .

3.4. Damping influence

Damping has little influence in the static region. Whatever its value, the spectrum tends towards the amplitude of the signal depending on time. This property is checked for all the shapes of shocks, except for the rectangular theoretical shock which, according to damping, tends towards a value ranging between once and twice the amplitude of the shock.

In the impulse domain and especially in the intermediate domain, the spectrum has a lower amplitude when the damping is greater. This phenomenon is not great for shocks with velocity change and for normal damping (0.01 to 0.1 approximately). It is marked more for oscillatory type shocks (decaying sine for example) at frequencies close to the frequency of the signal. The peak of the spectrum here has an amplitude which is a function of the number of alternations of the signal and of the selected damping.

3.5. Choice of damping

The choice of damping should be carried out according to the structure subjected to the shock under consideration. When this is not known, or studies are being carried out with a view to comparison with other already calculated spectra, the outcome is that one plots the shock response spectra with a relative damping equal to 0.05 (i.e. Q = 10). No justification of this choice is given in the literature. A study of E.F. Small [SMA 661 gives the distribution function (Figure 3.16) and the probability density (Figure 3.17) of Q-factors observed on electronic equipment (500 measurements).


D

Figure 3.16. Distribution function of Qlfactors measured on electronic equipment

Figure 3.17. Probability density of Q-factors measured on electronic equipment

The value Q = 10 appears completely acceptable here, since the values generally recorded in practice are lower than 10. Unless otherwise specified, as noted on the curve, it is the value chosen conventionally. With the spectra varying relatively little with damping (with the reservations of the preceding section), this choice is often not very important. To limit possible errors, the selected value should, however, be systematically noted on the diagram.

NOTE: In practice, the most fiequent range of variation of the Q factor of the structures lies between approximately 5 and 20. Larger values can be measured if the sensor is fixed on a plate or a cap [HAY 721, but measurement is not very significant since we are instead interested here in structural responses. There is no exact relation which makes it possible to obtain a SRS of a given Q factor starting from a spectrum of the same signal calculated with another Q factor.

1 16 Mechanical Shock

Q 5 10 20 30 40 50

M.B. Grath and W.F. Bangs [GRA 721 proposed an empirical method deduced @om an analysis of spectra ofpyrotechnic shocks to carry out this transformation. It is based on curves giving, depending on Q, a correction factor, equal to the ratio of the spectrum for the quality factor Q to the value of this spectrum for Q = I 0 (Figure 3.18). The first curve relates to the peak of the spectrum, the second the standard point (non-peak data). The comparison of these two curves confirms the greatest sensitivity of the peak to the choice of Q factor.

Standard points Peaks 0.085 0.10

0 0.00 0.10 0.15 0.15 0.24 0.19 0.30 0.21 0.34

These results are compatible with those of a similar study carried out by W.P. Ruder and KF. Bangs [RAD 701, which did not however distinguish between the peaks and the other values.

Q factor

Figure 3.18. SRS correction factor of the SRS versus Q factor

To take account of the dispersion of the results observed during the establishment of these curves and to ensure reliability, the authors calculated the standard deviation associated with the correction factor (in a particular case, a point on the spectrum plotted for Q = 20; the distribution of the correction factor is not normal, but near to a Beta or type I Pearson law).

Table 3.1. Standard deviation of the correction factor


The results show that the average is conservative 65% of the time, and the average plus one standard deviation 93%. They also indicate that modifiing the amplitude of the spectrum to take account of the value of Q factor is not suficient for fatigue analysis. The correction factor being determined, they proposed to calculate the number of equivalent cycles in this transformation using the relation developed by J.D. Crum and R.L. Grant [CRU 701 (see section 4.4.2) giving the expression for the response z ( t ) depending on the time during its establishment under a sine wave excitation as:

m; z ( t ) = Q x, ( 1 - e nN/Q)cos ( 2 7z f o t ) [3.23]

(where N = number of cycles carried out at time t).

c

w 9 d2

Q Figure 3.19. SRS correction factor versus Q factor

This relation, standardized by dividing it by the amount obtained for the particular case where Q = 10 , is used to plot the curves of Figure 3.19 which make it possible to read N, for a given correction factor and a given Q factor. They are not reliable for Q < 10 , relation [3.23] being correct only for low damping.

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