• The Equivalent Response Method (ERM) is an alternate method to developing environment specifications.• The method can also be useful for comparing specifications.
• The basic idea behind the ERM is the same as NASA’s use of the Shock Response Spectrum (SRS): if one environment can produce a similar response as another environment, then the two environments can be treated as equivalent in severity (similar caveats apply to the ERM as with the SRS).
• Full details of the method are provided in Christian Lalanne’s Mechanical Vibration and Shock Analysis handbook.
ERM Goal• Develop a test specification, ytest, that produces an equivalent SDOF
response, xtest=xenvironment, as the predicted mission environment, yenvironment , over a range/spectrum of SDOF natural frequencies.• Hence, equivalent response.
• For a vibration environment (sine, random vib, etc.), the Equivalent Response Method uses two response spectrums:• The Fatigue Damage Spectrum (FDS)• The Extreme Response Spectrum (ERS)
• These two response spectrums are intended to capture the severity of a vibration environment in terms of,• The exceeding of an instantaneous stress limit (ERS).• The damage by fatigue due to the application of a large number of cycles (FDS).
• The FDS can also be useful for comparing the effects of two different environments of differing duration on a particular component.• E.g., a heritage component subjected to a new QUAL specification.
• Lalanne, C., Mechanical Vibration and Shock Analysis Vol. V: Specification Development. 2002.• Irvine, T., A Fatigue Damage Spectrum Method for Comparing Power Spectral Density Base Input
• Mathematically identical to the Shock Response Spectrum.• The distinction is that the peak response of an SDOF to a shock input can occur after the duration of the shock and
therefore the residual SDOF response should be taken into account.• The peak response of an SDOF to a vibration input typically occurs during the duration of the input vibration.
• The FDS has some unique properties that offer some useful advantages:
• The FDS is not sensitive to nonstationary signals.• This is due to the fact that Rainflow Cycle Counting accounts for all the cycles of a signal; i.e., can be performed on
any oscillating signal.
• Under the assumption of signal ergodicity, the FDS of a measured signal of a sample duration, ts, can be scaled to
account for the full event duration, T, using the following proportional relationship: 𝐹𝐷𝑆𝑒𝑣𝑒𝑛𝑡 =𝑇
𝑡𝑠𝐹𝐷𝑆𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑_𝑠𝑎𝑚𝑝𝑙𝑒.
• Depending on the nature of the environments predicted during a components operation life, multiple FDS can be combined in two different ways to develop on overall FDS• For events that occur in parallel, take the envelope of the FDSs.• For events that occur in series, sum the FDSs.
• IMPORTANT!!!: While mathematically, the ERM for one environment type could be used to develop the specification utilizing another environment type (e.g. swept sine spec developed to represent a random vibration environment), a key assumption of fatigue equivalency is that the environments produce a similar stress response. Therefore, the specification developed should match the expected environment (NASA-HDBK-7005 warns about this).
• Two Sample Scenarios1. Only environment profiles (heritage PSD, Swept Sine Parameters, etc.) available; no measured data.2. Measured test data of samples of the environment available; e.g., 1 minute of development test data for a 1 hr
expected environment duration.Scenario 11. FDSs and ERSs can be calculated directly from PSDs, Swept Sine Profiles, etc. and vice versa.
• Tom Irvine has many scripts for this.• Script inputs: PSD (or other profile), spectrum frequency range, duration, damping/DAF (Q), fatigue exponent (b)
2. To develop overall FDS environment,1. Sum FDSs of events that occur in series (e.g., 10hrs truck transport followed by 2hr train)2. Take envelope of FDSs for parallel events (e.g. two possible launch vehicles)3. If needed, scale FDS to full duration of expected operation life.
3. To develop overall ERS, take max envelope of the ERSs of expected events.4. Now that an FDS and ERS are developed that represent overall expected life events, these reference response spectrums
can be used to derive test spectrums.• FDSs can be converted directly into PSD, or enveloping PSD (or swept sine for sine events) can be created whose FDS
closely envelopes the reference FDS.5. Once test spectrum is developed, generate ERS of test spectrum to compare to reference ERS; adjust spectrum accordingly
so that reference FDS and ERS are closely matched by test spectrum; this may require adjusting the test duration.
Scenario 2: Measured Data Available• FDS and ERS can be calculated directly from the measured time history data.
• Alternatively, the PSD of the TH can be calculated and the FDS can be calculated from the FDS.• For Gaussian random vibration, it can be shown that the PSD calculated from the TH and the PSD developed from the FDS are identical.
• Create overall representative response spectrums following a similar procedure as Scenario 1.• NOTE: Since the FDS has a duration/time component, it is important to keep track of the durations used to develop the FDS.
• If developing an overall FDS from multiple FDSs, it may be necessary to scale the FDSs to the same duration prior to summing or enveloping.
E.g.: Two time histories of the
following spectrums are measured for
different durations.
Spec 1 would appear to be the dominating case; however, the differing measurement durations need to be taken into account. Spec 1 needs to be scaled down or Spec 2 needs to be scaled up to make an adequate comparison.
• Waivers and technical authority approval are often required when deviating from NASA standards and methodologies.
• The purpose of this presentation is not to propose this method as a replacement for current NASA practices, but to highlight an alternative method that may be useful for atypical scenarios.• One example is the M2020 percussion drill operating environments; use of the ERM for this project will be
demonstrated in later charts.
• One limitation is that the ERM is intended for base drive environments; i.e. not applicable to acoustic environments.
• Reminder: The specification developed should be of a similar nature to the expected environment (develop random vib spec for random vib environment, etc.); I.e., Test like you fly.
• The figures below show that the FDSs from the ERM can create the same results by increasing the duration of the most severe level (road segment A) until its FDS is equal to the sum of the FDSs of each road segment.
• Since there is a direct proportionality for time with an FDS, the increased duration for road segment A can be calculated from the ratio of the total FDS and segment A’s FDS.
• E.g., @ 40Hz: 9.029
5.94= 1.52, therefore, Segment A’s
new duration would be 1.5*1hr which is the same conclusion drawn in 7005.
• While this example may be trivial, this same concept can be applied to vibration environments with difference spectral content as well as the overall levels.• In this case, the ratio would result in a different
equivalent stationary duration at each frequency. To be conservative, the longest duration would be chosen.
• Evaluation of M2020 Percussion Drill Operation• Percussion impact force controlled by increasing the percussion frequency: Increased Frequency -> Larger Impact Force• Two operation modes: Coring & Abrading
• Data Processing• Initially calculated power spectral densities:
• PSDs showed narrowband spikes at percussion frequency & first harmonic.
• First attempt to envelope PSD led to very conservative levels.• Review of filtered time history data showed narrow bands
Equivalent Response Method• The FDSs are calculated from the measured test
data, the P95/50 limit is calculated per percussion frequency then the P95/50 limits are summed and scaled to the full environment duration (36hrs) resulting in the solid black curve.
• The figure shows the breakdown of the time history signal of a 40Hz percussion drill operation.• The bandpass filtered signals and histograms show the sinusoidal nature of the signal at the percussion frequency and the
first harmonic.• This led to the decision to represent the lower frequency range (~20-100Hz) of the drill environment as a swept sine.
Histograms of the bandpass filtered signals indicate sinusoidal signals rather than random signals.
• Swept sine parameters are chosen such that the swept sine FDS & ERS closely matches the reference environment FDS and, for swept sine, can be calculated by the shown equations.
• A waterfall FFT plot shows how the frequency content of a signal changes with time.• Note that the drill signal percussion and harmonic frequencies (40Hz & 80Hz) are constant with time (left figure) and occur simultaneously.
• However, a swept sine signal’s single frequency changes with time according to the following function: 𝑓(𝑡) = 𝑓𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑒
𝑡∗ln(𝑓𝑓𝑖𝑛𝑎𝑙𝑓𝑖𝑛𝑖𝑡𝑖𝑎𝑙
)
𝑇 .• Note how this is observable in the waterfall FFT of the swept sine signal (right figure).
• The equivalent response method offers an alternative approach to the methods provided in NASA-HDBK-7005 for atypical environments.• The method utilizes of response spectrums• Response spectrums can be calculated from PSDs, swept sine specs, and measure time histories.• Response spectrums are more robust for non-stationary environments.
• The method can also be used to implement the procedures in NASA-HDBK-7005 for test level and duration development.
• For more information:• Lalanne, C., Mechanical Vibration and Shock Analysis Vol. V: Specification Development.• Tom Irvine, https://vibrationdata.wordpress.com/
