# SRS Analysis

December 30, 2022

Back to: Shock Response Spectrum (SRS)

A time-domain study of shock pulses does not provide sufficient information about complex waveforms. By moving data to the frequency domain, engineers can better understand the energy distribution across a frequency range and the product’s response to excitation within the defined range.

Two methods of moving data from the time domain to the frequency domain are the fast Fourier transform (FFT) and the shock response spectrum (SRS). They are both mathematical tools that sequentially apply filters of increasing frequency to time-domain data. Both analyze the characteristics of the filtered waveform and generate a plot in the frequency domain.

### Difference: FFT & SRS

The FFT calculation analyzes continuous, repetitive waveforms. It produces a PSD that displays the average acceleration at each frequency bin. For shock testing, however, test engineers are interested in the maximum acceleration at each frequency bin rather than the average.

The SRS displays the peak response of a series of theoretical SDOF mass-damper-spring oscillators as an amplitude (Gpk) versus frequency (Hz) plot. It provides the peak response to a time-domain input at each frequency bin. Unlike the FFT, the SRS does not apply averaging.

Most significantly, the FFT preserves a data set’s magnitude and phase information, whereas the SRS only displays the magnitude of the peak response. The original pulse can be re-created from the Fourier transform data but not the SRS data. Many different pulses can result from the same SRS or similar spectra.

Still, engineers can better visualize the effects of a shock on a physical system using the SRS. They can determine the maximum dynamic load of a component or the entire system as a function of frequency. Then, they can correlate this information to the damage potential based on an input response. Although an SRS response cannot generate the original pulse, the engineer can determine if it is similar to the transient event.

### Supplemental SRS Analysis

SRS transform is many to one, meaning a range of waveforms can produce the same SRS in terms of amplitude, duration, and frequency content. A supplemental analysis is needed to ensure the pulses represent the original event. This analysis includes:

- Coherence
- Correlation
- Stationarity
- Measurement of the ZPA
- Peak stress cycle counting

#### Coherence

Earthquake motion is statistically independent when measured at a distance from the epicenter. Artificially generated time histories should possess a similar amount of statistical independence. Coherence and correlation functions verify statistical independence.

Coherence examines the relationship between two signals. The coherence graph confirms if the response signal is related to the drive signal across the frequency domain. Values for coherence will be between 0 and 1. Perfectly independent signals yield a value of zero, and perfectly dependent signals a value of +1.0.

The coherence defined in IEEE-344-2004 Annex E requires a value of less than 0.5. IEEE STD-344 defines the procedures that demonstrate that Class 1E equipment can perform during a seismic event.

#### Stationarity

IEEE-344-2004 defines stationarity in Annex B. Stationarity maintains a waveform’s frequency content and a consistent excitement of all vibration modes (uniform presence of energy).

Test engineers use frequency content and stationarity to create a waveform representative of a seismic event. The frequency content is in the form of a power spectral density (PSD) function. The input waveform must be statistically constant over time, which a waterfall PSD can achieve. The engineer evaluates the waterfall PSDs for uniformity and compares them to the recorded event.

#### Measuring the Zone of Polarizing Activity (ZPA)

IEEE-344-2004 defines the ZPA in Annex A. Zero-period acceleration is the acceleration level of the high-frequency, non-amplified portion of the response spectrum. It corresponds to the maximum peak acceleration of the time-history waveform from which the response spectrum is derived. Zero-period acceleration is also known as peak acceleration.

The ZPA measurement confirms if the test response spectrum (TRS) has enveloped the required response spectrum (RRS). The ZPA of the TRS should only consider the frequencies within the RRS-amplified region.

If the TRS indicates the presence of high frequencies, the engineer can implement low-pass filtering above the RRS cutoff frequency to show the true ZPA value.

All shaker tables introduce waveform distortions. The true zero-period acceleration can only be measured through filtering at the cutoff frequency. Filters support accurate and relevant time-domain data acquisition. They ensure that the data limiting the mechanical motion of the shaker is correct.

#### Peak Stress Cycle Counting

IEEE-344-2004 defines the peak stress cycle in Annex D. The peak stress cycle counts the equivalent full-amplitude cycles and ensures that the waveform motion is consistent with frequency content and duration.

Test duration alone may not be sufficient to ensure an adequate equipment response relative to the low-cycle fatigue capability. A conservative seismic test requires the input to generate a certain amount of strong-motion-response cycles in the equipment. A stress cycle counting function verifies this.