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Shock Response Spectrum (SRS)

Siemens Valued Contributor

02-14-2019
07:54 PM

A Shock Response Spectrum (SRS) is a frequency based function that is used to indicate the magnitude of vibration due to a shock or transient event. A SRS can quantify transient vibration from a myriad of different events: from earthquakes to pyrotechnic events to ballistic shocks (*Figure 1*).

Take the example of a battleship which experiences transient vibration from the shock of a nearby explosion. A Shock Response Spectrum can be measured and captured on key components of the battleship during the shock event.

Using the Shock Response Spectrum, laboratory tests can then be devised to recreate the transient shock vibration experienced by the components to ensure they would survive the event.

This article covers:

1. What is a Shock Response Spectrum?

2. How is a SRS measured?

3. Why Use a SRS?

* 3.1 Time Signal Uniqueness*

* 3.2 Amplitude Differences*

4. SRS Calculation parameters:

* 4.1 Q-factor*

* 4.2 Points per Octave and Frequency Spacing*

* 4.3 Instances and Amplitudes*

* 4.4 Correction Methods*

* 4.5 Dimensions*

5. Calculating a SRS in Simcenter Testlab

6. History of the Shock Response Spectrum

**1. What is a Shock Response Spectrum?**

The amplitude of a Shock Response Spectrum is the peak vibration response versus frequency of a series of mass-spring-damper systems created during a shock or transient event as shown in *Figure 2*.

Historically, actual mass-spring-damper systems were used in the measurements. Today, the response of the mass-spring-damper systems is calculated virtually with computer software from measured or predicted acceleration data.

The steps in a SRS calculation are as follows:

- Input: A system experiences an acceleration which is measured. The force that created the acceleration is not measured or known in the SRS calculation.
- System: A series of mass-spring-damper systems respond to the transient acceleration. The natural frequency (f
_{n}) and damping (Q-factor) of these mass-spring-damper systems are selected to cover the frequency range of interest. In the*Figure 3*, the number of mass-spring-damper systems are numbered 1 to*i*, where*i*is the number of systems.

- Responses: The acceleration versus time response of each mass-spring-system is recorded. The peak levels of the signal are determined. There are different methods for calculating the peak level which are covered later in the article.
- Output: The peak level response of each mass-spring-damper system is plotted as a function versus the corresponding natural frequencies of the systems. This is the Shock Response Spectrum (
*Figure 4*).

For the Shock Response Spectrum calculation, vibration is measured only at one location. A base with mass-spring-damper systems distributed across it is only used to illustrate how the calculation works.

The mass-spring-damper transient analysis is performed on the acceleration time history output of the accelerometer using a software algorithm on a computer. The vibration response can be plotted in acceleration, velocity, or displacement.

**2. How is a SRS measured?**

An SRS is normally measured so that it captures the input into a component that is experiencing a transient event.

Imagine a product that consists of a base with a component mounted to it. The SRS measurement is not made directly on the component, but on the base adjacent to component attachment location as shown in *Figure 5.*

By measuring the SRS at the base, the input into the component is captured during field usage. Then, if a similar transient event needs to be recreated in a laboratory setting, it can be reproduced by placing the control accelerometer of a closed loop vibration control shaker test in a similar location on the shaker head.

If a new or different component will be mounted to the same base location, they can be tested using the same SRS profile as shown in *Figure 6*.

Simcenter Testlab Shock Control can use a SRS function as the target for a shock test.

**3. Why Use a SRS?**

There are a few reasons why a Shock Response Spectrum (SRS) might be used rather than a Fourier Transform to capture transient vibration events.

*3.1 Time Signal Uniqueness*

For a transient vibration signal:

- The Fourier Transform of a time domain transient yields both magnitude and phase information.
- A Shock Response Spectrum is an envelope (magnitude) only, and no phase information is retained from the input signal.

The original input time signal cannot be regenerated from an SRS.

Because there is no unique relationship between a SRS and the time signal (unlike a Fourier Transform), there can be an infinite number of transient time signals that can generate the same SRS (*Figure 7*).

This can be an advantage when recreating a shock transient event on a shaker table. For example, a time signal for a given SRS can be selected that is within the performance constraints (maximum displacement, maximum acceleration, etc) of the shaker system, so the test can be run without issues.

*3.2 Amplitude Differences*

Another advantage of using a SRS over the Fourier Transform is that the amplitude is not dependent on the time window (assuming the whole event is captured) or the frequency resolution used in the Fourier analysis.

Because the transient event is so short, the Fourier Transform has difficulty getting a consistent amplitude due to the tradeoff between frequency resolution and the time analysis window as shown in *Figure 8*. This is true whether the Fourier Transfrom is done with a Fast Fourier Transform or a Power Spectral Density (PSD).

Functions like a Power Spectral Density (PSD) are better suited for quantifying random vibration.

The SRS processing bypasses these issues and provides a consistent amplitude for the transient event. The consistency of the SRS amplitude will be illustrated further in the upcoming *points per octave* section.

**4. SRS Calculations**

There are several parameters that can be set by the user during the calculation of a SRS:

**4.1 SRS Calculations: Q-Factor**

The *Q-factor* parameter sets the damping in the mass-spring-damper systems. When calculating a SRS, the Q-factor for the mass-spring-damper systems is set by the user.

The Q-factor determines with how much amplitude the mass-spring-damper system responds to a given input. By increasing the Q-factor, a mass-spring-damper system responds with higher amplitude to the same input as shown in *Figure 9*.

The Q-factor is calculated by measuring the width of the response of the mass-spring-system as shown in *Figure 10*.

The higher the Q-factor, the lower the damping. The lower the damping, the sharper the peak of the system response, and the higher the amplitude of the system response.

Besides Q-factor, the damping of a mass-spring-damper system can also be expressed as percent critical (%Cr) damping as shown in *Equation 1*.

The Q-factor, from a numerical point of view, is inversely related to damping. Percent critical damping is directly porportional to damping.

Depending on the software implementation, damping may be entered in either format. A Q-factor of 10, which corresponds to a critical damping of 5%, is often used in Shock Response Spectrum calculations.

Because the Q-factor affects the amplitude of the mass-spring-damper response, it is important to compare Shock Response Spectrums (SRS) functions that use the same Q-factor.

For more information about Q-factor, see the Knowledge Base article ‘Determining Damping from a FRF’.

**4.2 SRS Calculation: Points per Octave and Frequency Spacing**

The *Points per Octave* parameter determines the number and natural frequencies of mass-spring-damper systems used in the calculation of the SRS. The higher the points per octave, the:

- Higher the number of mass-spring-damper systems that are used in the SRS calculation.
- Finer or narrower the frequency spacing between points in the SRS.

Together with the Q-factor, the points per octave should be selected so the response of the mass-spring-damper systems have an overlap as shown in *Figure 11*.

This overlap should ensure the entire frequency range of interest for the Shock Response Spectrum measurement (*Figure 11*) is covered, and no energy of the shock event is missed.

Like octave bands, fewer data points are calculated at higher frequencies. The spacing between the natural frequencies of the mass-spring-damper systems gets wider at higher frequencies.

Unlike the Q-factor, the points per octave does not affect the calculated amplitude for a given mass-spring-damper system. A comparison of 6, 12, and 24 points per octave on the same transient event are shown in *Figure 12*.

With more points, the SRS can contain more details as a function of frequency, but the amplitudes of any common natural frequencies will be the same. In Simcenter Testlab, possible values of the points per octave range from 1 to 48.

In a typical SRS calculation, the values are:

- Q-factor: A value of 10, which corresponds to 5% critical damping.
- Points per Octave: A value of 6 points per octave.

There are other considerations for how the amplitude of the shock event is calculated.

**4.3 SRS Calculation: Instances and Amplitudes**

When calculating a SRS, the transient response of the mass-spring-damper system is not treated as a single event. It is composed of two different *instances*.

The amplitude of the system response can be calculated from either the primary instance, the residual instance, or over both instances combined (*Figure 13*):

- The
*primary*instance is the response while the transient excitation is applied to the system. - The
*residual*instance is response after the excitation is no longer being applied.

When determining the amplitude of the response from the primary instance:

- Absolute (1): The highest absolute amplitude of the primary instance of the time waveform is recorded.
- Positive (2): The highest positive amplitude value of the primary instance time waveform.
- Negative (3): The highest negative amplitude value of the primary instance time waveform.

When determining the amplitude of the response from the residual instance:

- Absolute (4): The highest absolute amplitude of the residual instance of the time waveform is recorded.
- Positive (5): The highest positive amplitude value of the residual instance time waveform.
- Negative (6): The highest negative amplitude value of the residual instance time waveform.

These values are calculated for the time history of each mass-spring-damper system as shown in *Figure 14*.

In *Figure 15*, all six methods of calculating the amplitude of the Shock Response Spectrum from the same transient event are overlaid. In addition, the *Maximax SRS* is also shown.

The Maximax SRS is the absolute value over both the primary and residual instance. The Maximax SRS is commonly used. Some reasons why:

- When running a shock vibration control test, the exact time that the excitation is applied is known so the primary and residual instance are easily identified. For field data, the excitation is not always measured, so it is not easy to determine the primary and residual parts of the response.
- The absolute maximum value could occur either in the primary or residual response (especially possible for very low frequencies that take time to ramp up). The absolute maximum value may be of greatest interest, regardless where it occurs.

In Simcenter Testlab, the Maximax spectrum is calculated when the Instance is set to ‘Maximum’ and the Amplitude is set to ‘Absolute’ as shown in *Figure 16*.

At the end of the article, there is a more detailed explanation of how to calculate SRS functions in Simcenter Testlab.

**4.4 SRS Calculation: Correction Methods**

If the time signal being used in the SRS calculation is not centered about zero, but instead has offsets, the resulting amplitudes of the SRS function can be higher than expected. This can be the case in *measured* acceleration transient signals.

Offsets can be created in unexpected ways. For example, playing back data from a tape might introduce an offset, or an offset can be created by external signal conditioning equipment used with the accelerometers.

Even though the DC offset is at zero Hertz, the amplitude of the SRS calculation at every natural frequency can be affected. A correction is needed to compensate for this increased amplitude as shown in *Figure 17*.

Offsets on data can be constant or vary slowly over time. Depending on the type of offset, there are different types of corrections available in the Simcenter Testlab software:

- DC Offset – Removes a constant acceleration offset from the time data.
- Velocity – Removes a linear 1
^{st}order velocity trend from the time data. - Displacement – Removes a linear 1
^{st}order displacement trend from the time data.

For more information about recording acceleration data with offsets see the knowledge base article ‘AC versus DC coupling’.

**4.5 SRS Calculation: Dimension**

The SRS can be calculated with units (i.e., dimensions) of acceleration, velocity, or displacement. Sometimes there are advantages to viewing the SRS in different formats, based on the application.

*Acceleration*

A SRS can be shown in units of acceleration as shown in* Figure 18*.

In Simcenter Testlab, there are two ways of calculating the acceleration: *absolute* and *equivalent static*.

- Absolute acceleration is derived from the maximum response acceleration of the SDOF mass relative to the base acceleration.
- Equivalent static acceleration is calculated by multiplying the relative displacement by the natural frequency twice.

Acceleration based SRS functions are commonly used in the aerospace industry.

*Velocity*

SRS can be calculated in units of velocity. In Simcenter Testlab, both *relative velocity* and *pseudo velocity *can be calculated (*Figure 19*).

A *pseudo velocity shock spectrum* (pvss) has some useful properties. The shock response spectrum shape will have a hill like shape for a zero mean acceleration transient event:

- Maximum Displacement: There will be a left asymptote which is governed by the maximum displacement in the transient event.
- Maximum Velocity: The plateau of the hill is the maximum velocity reached. It is proportional to the damage on the object.
- Maximum Acceleration: The right asymptote is governed by the peak acceleration in the test.

It is helpful if this is plotted on what is called four co-ordinate paper (4cp). The acceleration, velocity, and displacements can be read directly from a 4cp graph.

The hill shape of pseudo velocity SRS is only possible if the acceleration signal has a zero mean. Depending on the segment of time that is being analyzed, even from the same acceleration recording, the SRS could change due to the mean of the segment selected as shown in *Figure 20*.

For a zero mean acceleration signal, the velocity versus time must be zero at the beginning and end of the recorded time signal. If a segment of time is analyzed where the velocity is not zero at the beginning and end, then the classic hill shape of the pseudo velocity does not occur.

The pseudo velocity SRS is often used in Naval applications.The pseudo velocity is obtained by multiplying the relative displacement by the natural frequency.

*Displacement*

The *relative displacement* model for shock response spectrum calculation is used when internal stresses or movements in the structure are of more interest (*Figure 21*).

Stress is proportional to strain which in turn is related to the relative deflection (displacement) of the response mass. The relative displacement spectrum is derived therefore from the maximum relative displacement of the SDOF mass relative to the base. The input to the base is the acceleration provided by the time pulse to be analyzed.

The pseudo velocity SRS is derived directly from the relative displacement SRS by multiplying each value by its corresponding natural frequency.

Looking at a relative displacement SRS can be useful to indicate if internal parts of the test component could hit or interfere with each other. By knowing the clearances between parts in the component, the chances of interference can be assessed. The damping value, or Q-factor, must be indicative of material of the object under test.

**5. Calculating a SRS in Simcenter Testlab**

A SRS spectrum can be calculated either interactively in the Simcenter Testlab Desktop, or can be calculated in batch mode in Simcenter Testlab Signature Throughput Processing.

*Simcenter Testlab Desktop*

In the Simcenter Testlab Desktop, in the Navigator worksheet, there is a SRS calculation icon as part of the conditioning toolbar (*Figure 22*).

Pressing on the SRS calculation button while a time history selected in the display yields a menu where SRS calculation parameters can be set (*Figure 23*).

After selecting the desired settings and pressing OK, the SRS spectrum will be placed in folder called “Conditioning” in the current section of the open project.

*Simcenter Testlab Throughput Processing*

In Signature Throughput processing, a SRS can be calculated on multiple channels and multiple files at one time as shown in *Figure 24.*

For more details on processing SRS functions in Simcenter Testlab, see the forum post on processing a Shock Response Spectrum in Simcenter Testlab.

**6. History of the Shock Response Spectrum**

1933 – Maurice Anthony Biot (a graduate of the Catholic University of Leuven, Belgium) describes the Shock Response Spectrum in his Ph.D. thesis from the California Institute of Technology. This is the first published reference to the Shock Response Spectrum. It was initially developed to understand earthquake vibrations.

1962 - The first edition of MIL-STD-810 for environmental test tailoring is published by the United States Department of Defense. Dr. Irwin Vigness of the U.S. Navy Research Lab did much to establish the method as an indicator of mechanical shock severity.

Questions? Post a reply or email john.hiatt@siemens.com.

**Durability Links**

- History of Fatigue
- Stress and Strain
- Calculating Damage with Miner's Rule
- What is a SN-Curve?
- Rainflow Counting
- Difference between 'Range-Mean' and 'From-To' Counting
- Power Spectral Density
- How to Calculate a Shock Response Spectrum with Testlab?
- Some Thoughts on Accelerated Durability Testing
- Goodman-Haigh Diagram for Infinite Life
- Measuring strain gauges in Simcenter Testlab
- Rosette Strain Gauges
- Calculating Damage in Simcenter Tecware Process Builder
- Strain Gauges: Selecting an Excitation Voltage
- Simcenter Testlab SCADAS and Long Strain Cables

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