525.2Method 525.2 contents - Continued

FIGURES

525.2 CONTENTS - Continued Paragraph

525.2 CONTENTS - Continued

525.2 TIME WAVEFORM REPLICATION

NOTE: Tailoring is required. Select methods, procedures, and parameter levels based on the tailoring process described in Part One, paragraph 4.2.2, and Annex C. Apply the general guidelines for laboratory test methods described in Part One, paragraph 5 of this Standard.

525.2 ANNEX A

525.2 ANNEX A

confidence (dependent upon the accuracy of the covariance function estimate). Figure 525.2A-4 provides a biased cross-covariance function estimate between the band-limited reference and control time traces.

Figure description:

Bandlimited Reference and Control Time Traces

(H Biased Cross-Covariance Estimate

(S Biased Cross-Covariance Estimate

Figure 525.2A-4. Cross-covariance function estimates between reference and control time traces.

By examining the cross-correlation estimate region near a lag of zero seconds, it is apparent that the reference and control time traces are in phase, and no shifting of one time trace relative to the other is necessary.

525.2 ANNEX A

525.2 ANNEX A

Figure 525.2A-13 provides an overlay of envelopes of r and c in terms of root-mean-square g's for a short-time averaging increment of 0.1 seconds (STA assessment). If the product model can be assumed, the differences in root- mean-square envelope levels are a maximum of 2 percent.

Figure description:

Transient Vibration RMS Envelope

Transient Vibration RMS Envelope

Legend

• ref — color: black; symbol: Dashed line
• ctl — color: black; symbol: Solid line

Figure 525.2A-13. Composite root-mean-square envelope estimates for r and c.

Figure 525.2A-14 provides a composite of normalized ASD estimates for r and c. The estimates were determined by one-sixth octave band frequency averaging. The normalized ASD estimates differ by less than 2 dB.

Figure description:

Transient Vibration NASD Analysis

Transient Vibration NASD

Legend

• ref — color: gray; symbol: Solid line
• ctl — color: black; symbol: Solid line

Transient Vibration NASD Difference

Figure 525.2A-14. Composite normalized ASD estimates for r and c.

From the above statistics, it can be concluded that no valid distinction can be made between r and c under the product model assumption, even though the non-Gaussian distribution of errors is difficult to interpret. It would appear that tolerance for this particular segment could be established as less than 0.2 grms amplitude for 90 percent of the time

525.2 ANNEX A

trace envelope, and 2 dB for the normalized ASD estimates, based on the information in Figures 525.2A-13 and 525.2A-14. This concludes replication error processing and tolerance specification for the transient vibration sub- event.

525.2 ANNEX A

Annex B defines the FOT distribution for difference time trace assessment. This assessment empirically defines the

fraction of time the error lies outside (or inside) given error amplitude bounds. This assessment is mathematically equivalent to a probability density (or distribution) assessment but more transparent and easier to interpret for an allowable error tolerance specification. Since TWR is time based, an allowable error of x-percent of the time the error amplitude may exceed y-percent of the root-energy-amplitude level (REA) of the deterministic reference time trace is easily visualized. Figures 525.2A-18a,b,c display the time-varying error in g’s for the stationary segment along with the REA percentage error plotted against the FOT quantiles. For the example under consideration the REA for the reference is 1.85 g-rms. Both two-sided and one-sided analyses are considered. The FOT ranges from 0.0 to 1.0 over approximately plus and minus 10% of the REA. Figure 525.2A-18a displays FOT quantiles for 10% to 10% REA error percentage. Figure 525.2A-18b displays the REA random error -5% to 5% for FOT quantiles from approximately 0.1 to 0.9 and Figure 525.2A-18c considers one-sided error for 10% REA error percentage and the 0.90 FOT quantile. A two-sided tolerance specification might, for example, require not more than 10% (0.10 FOT quantile) of test time to lie outside the REA amplitude percentage bounds of -5% and 5%. Tolerance is in terms of what percentage of time is the error allowed to be larger than a certain percentage of REA as a reference amplitude.

In Figure 525.2A-19, a composite of the ASD estimates for r and c is provided. The ASD estimates between r and c are essentially equivalent. For time trace s, there is non-flat spectrum that normally would not be present if the replication error were of a strong Gaussian character, i.e., s was band-limited white noise. The processing parameters are an analysis bandwidth of 5 Hz applying a Hamming window with 50 percent overlap.

Figure description:

TWR Error (Stationary Vibration with REA = 1.85 g rms

Error FOT Estimate

Figure 525.2A-18a. FOT Error Assessment – 10% REA Error Fraction-of-Time (FOT)

MIL-STD-810H METHOD 525.2 ANNEX A

Figure description:

Error FOT Estimate Analysis

Error FOT Estimate - Amplitude [-5% 5%

Error FOT Estimate - Quantile [0.05 0.95

Figure 525.2A-18b FOT Error Assessment - 5% REA FOT Error Bounds

Figure description:

Error FOT Estimate (one-sided - Amplitude [10%

Error FOT Estimate (one-sided - Quantile [0.90

Figure 525.2A-18c FOT Error Assessment - One-sided 10% REA FOT Error Bounds

MIL-STD-810H METHOD 525.2 ANNEX A

Figure description:

Stationary Vibration Analysis

Stationary Vibration ASD

Legend

• ref — color: black; symbol: Dashed line
• ctl — color: black; symbol: Solid line

Stationary Vibration ASD Difference

Figure 525.2A-19a. Composite ASD estimates for r and c.

Figure description:

Error Stationary Vibration ASD

Error Stationary Vibration ASD

|

Figure 525.2A-19b. ASD estimate for s.

From the above statistics, it might be concluded that no valid distinction can be made between r and c under the stationary model assumption even though the non-Gaussian distribution of error s is difficult to interpret. It would

525.2 ANNEX A

|

appear that tolerances for this particular segment could be established as maximum 2 dB for the ASD estimates, based on the information in Figure 525.2A-19. This concludes replication error processing and tolerance development for the stationary vibration sub-event.

|

525.2 ANNEX A

A common way of comparing shock information is through the SRS, in particular the recommended pseudo-velocity SRS estimate (Method 516.8). For the r and c time traces, a composite overlay of the pseudo-velocity SRS estimates for both shocks is useful. Figure 525.2A-23 provides this comparison in addition to a maximax acceleration SRS comparison. Since the SRS is an integration/smoothing process, it is expected that the reference and control information will be highly correlated when viewed in an SRS format. For these figures no wavelet correction was attempted for low frequency correction since such a correction applied individually may lead to a less transparent comparison.

Figure description:

Shock Response Spectrum (Pseudo velocity vs. Natural Frequency

Shock Response Spectrum

Legend

• Upper Bound — color: blue; symbol: Solid line
• Lower Bound — color: blue; symbol: Solid line

Annotations

• 100 in (diagonal_line - position: Displacement constant line (downward slope))
• 10 in (diagonal_line - position: Displacement constant line (downward slope))
• 1 in (diagonal_line - position: Displacement constant line (downward slope))
• 0.1 in (diagonal_line - position: Displacement constant line (downward slope))
• 0.01 in (diagonal_line - position: Displacement constant line (downward slope))
• 0.001 in (diagonal_line - position: Displacement constant line (downward slope))
• 0.0001 in (diagonal_line - position: Displacement constant line (downward slope))
• 0.1 g (diagonal_line - position: Acceleration constant line (upward slope))
• 1 g (diagonal_line - position: Acceleration constant line (upward slope))
• 10 g (diagonal_line - position: Acceleration constant line (upward slope))
• 100 g (diagonal_line - position: Acceleration constant line (upward slope))
• 1000 g (diagonal_line - position: Acceleration constant line (upward slope))
• 10000 g (diagonal_line - position: Acceleration constant line (upward slope))
• 100000 g (diagonal_line - position: Acceleration constant line (upward slope))

Figure 525.2A-23a Composite pseudo-velocity maximax pseudo-velocity SRS for r and c.

Figure description:

SRS : xra/xca

SRS : xra/xca

Legend

• R — color: black; symbol: Solid line
• C — color: black; symbol: Dashed line

|

Figure 525.2A-23b. Composite maximax acceleration SRS for r and c.

525.2 ANNEX A

Since ESD estimates provide a way of comparing shock type events, Figure 525.2A-24 provides a composite of r and c ESD estimates, while Figure 525.2A-25 provides the ESD estimate for "s." It is clear from both of these plots that the most substantial error is found in the low frequency region. This is not surprising since the transfer function used to compensate the entire time trace was likely not optimal for the shock.

Figure description:

Shock ESD and Shock ESD Difference Analysis

Shock ESD

Legend

• ref — color: black; symbol: Dashed line
• ctl — color: black; symbol: Solid line

Shock ESD Difference

|

Figure 525.2A-24. ESD estimates for r and c.

Figure description:

Shock ESD

Shock ESD

|

Figure 525.2A-25. ESD estimate for s.

525.2 ANNEX A

525.2 ANNEX B METHOD 525.2, ANNEX B SUMMARY OF POST-TEST ANALYSIS PROCESSING PROCEDURES AND TEST TOLERANCE SPECIFICATION

525.2 ANNEX B

place, however for averaging windows on the order of the oversample factor bias error should be minimal. Whenever practical bias errors in the estimates for the error assessment need to be minimized. If there exists questions relative to the size of normalized bias and random errors, much more detailed processing beyond the scope of this Annex may need to be employed (paragraph 6.1, reference a).

In description of the error assessment expressions, the designation “local” or “global” is made. The term “local” refers to a statistic that is useful for processing short segments of time-varying traces, while the term “global” refers to a statistic that is better suited to summarizing overall time traces. For example, the collection of STA for $S$ root-mean-square provides “local” estimates related to a potential tolerance specification. The cumulative probability density function estimate for $S$ describes error as being perhaps Gaussian or non-Gaussian. This is a “global” assessment from which a tolerance specification might be based upon the distributional form of the estimate. Generic variables

$x(t)$ $\left(x[n], n=1,2, \ldots, N\right)$, $y(t)$ $\left(y[n], n=1,2, \ldots, N\right)$ and $z(t)$ $\left(z[n], n=1,2, \ldots, N\right)$

are employed in the formulas whereby $r(t), c(t)$, and $s(t)$ may be substituted at will depending upon interpretation. In the formulas to follow $M$ will be an "index" related to the time sample interval for the time average estimate (it is a time shift parameter for averaging) and $N_{a}$ will be the number of time points averaged over. $\left.\lfloor N_{a} / 2\right.\rfloor$ is the greatest integer designation for $N_{a} / 2$. It is assumed that $M=\left.\lfloor N_{a} / 2\right.\rfloor+\left.\lfloor N_{a} / 2\right.\rfloor-1$ where generally M is an odd number to prevent any phase shift introduced in the processing.

There are three cases in which joint consideration of deterministic $R$ and stochastic $C$ may be useful. In the first case a scatterplot constructed by plotting the point $\left(r(n), c(n)\right)$ in the plane will reveal valuable information relative to a single plot of the error $s(n)$. In the second case since computation of an ASD/ESD estimate over a deterministic time trace has some meaning the comparison of the ASD/ESD estimates for $r(n)$ and $c(n)$ may provide meaningful information in relation to the ASD/ESD for $s(n)$. In particular the deterministic estimate is divided into the stochastic estimate to examine the ratio in the frequency domain. Finally, comparison of SRS estimates for $r(n)$ versus $c(n)$ along with an SRS estimate for $s(n)$ i.e., the "noise" can be useful.

For easy reference the following table is provided:

525.2 ANNEX B

525.2 ANNEX B

$C I_{\sigma ; 1-\alpha}=\left[\frac{n s^{2}}{\chi_{n ; \alpha / 2}^{2}} \leq \sigma_{x}^{2} \leq \frac{n s^{2}}{\chi_{n ; 1-\alpha / 2}^{2}}\right]$ for $n=N-1$.

For application for $B=2000 H z$ and $T=0.01$ or 0.1 seconds the normalized random error for a mean comparable to the standard deviation, root-mean-square and mean-square is $0.16,0.11,0.22$ respectively for averaging time of 0.01 seconds, and $0.05,0.04,0.07$ respectively for averaging time of 0.1 seconds. To obtain a meaningful characterization of $x(t)$, it is important the normalized random error be minimized by as long an averaging time as is consistent with nominal bias error.

E3: COVARIANCE, CORRELATION, and SCATTER-PLOT (global and local)

Generally, covariance and correlation can be viewed as meaningful in the case of regression between a deterministic and a random time trace i.e., $r(t)$ and $c(t)$ Since $s(t)=c(t)-r(t)$ no new information is added by computing the correlation or covariance between $r(t)$ and $s(t)$. Covariance and correlation should be viewed in terms of a "regression fit" of $r(n)$ to $c(n)$. This particular replication error assessment is somewhat qualitative thus not particularly useful for tolerance specification e.g., specifying a correlation coefficient for tolerance would be too gross a parameter to be meaningful. The covariance relationship between two time traces over a short interval of time (local covariance), or over the entire time trace (global covariance) is computed in the time domain as follows:

This quantity can be normalized to provide the local or global correlation coefficient that can be expressed as follows:

The time trace basis for these expressions from traditional data analysis is as follows. For two arbitrary random processes $\left\{x_{k}(t)\right\}$ and $\left\{y_{k}(t)\right\}$ whose sample functions are indexed on $k$ and for which the ensemble means are defined by $\mu_{x}(t)=E\left[x_{k}(t)\right]$ and $\mu_{y}(t)=E\left[y_{k}(t)\right]$ where expectation is over index $k$ then the cross covariance function at arbitrary fixed values of $t_{1}=t$ and $t_{2}=t+\tau$ is given by

If _ 7 = 0 then C¸„ (t,t) = E[(x (1)− µx (1))(yk (1) — µ, (1))] = C (1), and this is of the form of the covariance xy μx expression above only where the expected value is not over an ensemble indexed on k, but over a finite time interval of length Nat. The expression for xy is merely a “normalized” version of the expression for cov(x,y) defined above. When the kth sample functions x (i^t) and y/ (i^t) for i = 1, 2,..., N are plotted on the x and y axes, respectively, the resulting plot is termed a "scatter-plot." The "scatter-plot" depicts the degree of covariance or correlation between two time traces. For xy in the neighborhood of zero there tends to be no correlation between time traces, and the "scatter-plot" reveals an ellipse with major and minor axes approximately equal. For a distribution of rxy close to either -1 or +1, there is substantial correlation between the time traces, and the "scatter-plot" provides an ellipse with a very small minor axis. In general "scatter-plot" information at the amplitude extremes is of most interest since this defines the correspondence between time trace peaks and valleys.

525.2 ANNEX B E6: ASD/ESD/PERIODOGRAM (global)

For a deterministic time trace such as $r(t)$ a frequency domain estimate is meaningful and similar to the fitting of a Fourier series to an analytically defined function. Visual comparison between frequency domain estimates for $r(t)$ and $c(t)$ can be made and the ratio of the estimates at each frequency line provided by ratioing the computed quantities (this must never be interpreted as a “transfer function estimate” between the reference and the control time traces).It might be noted that for TWR the “transfer function estimate” is provided in the vendor software in the form of the frequency domain Fourier “drive signal compensation” function. The frequency domain estimates provide for tolerance specification that is directly related to tolerance specifications in Method 514. The basic definition of the windowed two-sided periodogram for an $N$ point digital sequence $\left\{x_{t}, t=1,2, \ldots, N\right\}$ in continuous frequency form is as follows:

Generally the two-sided periodogram is made one sided by multiplying by a factor of 2 with 0≤ f≤0.5, and the periodogram is sampled at discrete frequencies, f, for i = 0,1,2,..., N/2 with a uniform spacing of Af = 1/NAt. The ASD and ESD can be defined in terms of the sampled periodogram. An ASD estimate is typically a time average sampled periodogram estimate over a limited time interval, with an applied window to reduce spectrum leakage. For stationary time traces the ASD represents a powerful means of comparison between r(t) and c(t), and a display of the frequency content in s(t). Paragraph 6.1, reference a provides information on ASD processing of stationary time traces including normalized random and bias error estimates. For analysis filter bandwidth B in Hz, and averaging time T in seconds, the normalized random error for the ASD estimate is given by

$\varepsilon_{r}\left[\hat{G}_{x x}(f)\right] \approx \frac{1}{\sqrt{B_{e} T}}$

(17)

while the normalized bias error is given by

$\left[\hat{G}_{x x}\left(f_{r}\right)\right]=\frac{B_{r}}{B_{e}} \tan ^{-1}\left(\frac{B_{e}}{B_{r}}\right)-1$

(18)

where

$B_{r} \approx 2 \zeta f_{r}$

is an estimate of the half-power bandwidth of a resonant peak.

An ESD estimate is typically a scaled periodogram, scaled by multiplying the periodogram by the duration of the time trace $N \Delta t$, over a very short transient time trace that cannot be characterized by an ASD estimate. A uniform or end tapered uniform time window is generally placed over the significant portion of the time trace. For transient TWR time traces, ESD estimates are useful for comparing $r(t)$ and $c(t)$ in addition to examining the character of $s(t)$.

E7: SRS – Shock Response Spectra (global)

As in the case of the frequency domain estimates in E6 a comparison between SRS estimates for deterministic r(t) and stochastic c(t) can be made. The SRS estimate for the error time trace s(t) is related to an SRS estimate for pre-shock and post-shock considered to be random in nature (see Method 516).The SRS may be expressed as a time domain convolution of an impulse response function that has the character of the response to base-input of the mass of a single- degree-of-freedom mechanical system, with a certain percentage of critical damping. The SRS estimate is a function of the output of the mass displacement, velocity, and acceleration. If the maximum absolute acceleration (positive or negative) is selected over the time interval of excitation, and plotted versus the undamped natural frequency of the single-degree-of-freedom system, the resulting plot over a selected set of frequencies is referred to as a maximax

525.2 ANNEX B

shock response spectrum. It is becoming increasingly evident that for most cases of mechanical shock the pseudo- velocity SRS estimate is a more indicative measure of potential for mechanical damage (because mechanical damage is related to mechanical stress that, in turn, is proportional to relative velocity of a mass-spring-damper system). Various references provide the details of SRS computation. For transient time trace TWR comparison, the SRS of r(t) and c(t) is useful and demonstrates the faithfulness of shock reproduction under TWR. Computing the SRS for s(t) is less useful and difficult to interpret since random variable S should represent a noise source but not Normal distributed. The mathematics for the SRS computation over a transient ( ) for 0 x t ≤≤ t T r is given as follows:

SRS (ƒ„,5) = I[y(t, ƒ„‚5)]=: I (t-t)x(t)dr | for 0≤T, ≤T for

where,

SRS (f.) the magnitude of the SRS at natural frequency fr

I a nonlinear functional operating on the resulting convolution y(t,ƒ„‚5)

h(t) impulse function response for a damped single-degree-of-freedom system with base input and undamped natural frequency for having damping ratio 5.

x(7) finite input record 0≤t≤T

T time of response assessment where generally T, <T

Natural frequency, nf , can extend beyond the sampling frequency of x(t). The SRS estimate is computed through filtering a transient time record, and does not have a clear random error or bias error criterion. Numerically, the time trace sample rate should be ten times the bandwidth of the time trace in order to provide an acceptable error in the estimates (approximately 5 percent error).

• 1 SCOPE
• 2 TAILORING GUIDANCE
• 3 INFORMATION REQUIRED
• 4 TEST PROCESS
• 5 ANALYSIS OF RESULTS
• 6 REFERENCE/RELATED DOCUMENTS
• 7 TPP STATIONARY VIBRATION
• 8 TPP SHOCK
• 9 POST-TEST PROCESSING FOR STA