**Introduction**

Shock Response Spectrum (SRS) analysis is used to calculate the peak response of a physical system subjected to shock loads in the frequency domain. This is a very useful, inexpensive tool that provides qualitative comparisons between preliminary designs. It is often essential in the fields of engineering, material science, and electronics where rapid transient forces are common. Some of the examples of impulsive loads are earthquakes, avionic loads, transportation loads, blast and impact loads.

The main goal of this type of analysis is to evaluate if there is permanent damage in the physical system or any decline of performance of the system during or after the shock. This method is not suitable if the excitation is very severe and produces highly nonlinear effects in the that need to be accounted for. In this case, a dynamic analysis has to be performed by inputting the variation of the base motion in the time domain. However, this analysis might be computationally very expensive as the time increment has to be very small to properly account for the sudden variations in the base excitations.

**What Is A Shock Wave?**

The term ‘shock’ refers to a very sudden change of a force, position, velocity or acceleration resulting in a temporary change of state of a physical system. This modification usually occurs within a very short time period compared to the system’s natural frequency. To measure this shock, we have to trace the value of shock input parameters such as displacement, velocity or acceleration over time which is termed as ‘time history’. The figure below shows an example of time history of shock characterized by acceleration.

**What Is A Shock Response Spectrum?**

First, let’s understand how to calculate the response of a single degree of freedom system (SDOF) when subjected to shock excitation. The figure below shows a SDOF with input acceleration of ẍ. Because of this input excitation, the system will have a response acceleration of y. In the figure, m, k and c denote mass, spring stiffness and viscous damping coefficient of the oscillator respectively.

The equation of motion for this SDOF system when subjected to external excitation is obtained as,

Where, z(t) is the relative displacement of the mass which is calculated as z(t) = y(t) – x(t). Q is the quality factor and wn is the natural angular frequency in radians.

The solution for this above equation of motion in terms of relative and absolute accelerations of the system are expressed as,

Where, w_{d} is the damped angular natural frequency, which is defined as,

This absolute acceleration response y(t) of a SDOF oscillator is primarily used to compute the shock response spectrum. This response can be computed using discrete time methods like the recursive filtering method or the Prefilter-Smallwood method.

To calculate the SRS, the physical system is represented as several SDOF systems with increasing order of their natural frequencies, and their absolute peak response is calculated when subjected to a transient input excitation. This peak acceleration is plotted against natural frequency in a graph as shown in the figure below, which is termed as Shock Response Spectrum (SRS). It describes the frequency characteristics of the transient shock input signal. This SRS can be applied to the physical system and its maximum linear dynamic response can be evaluated

**Finite Element Analysis Using SRS In Abaqus**

The response spectrum analysis is performed in 2 steps.

**Performing Modal Analysis**

The first step in evaluating shock response of a system is to perform a modal analysis to obtain natural frequencies and corresponding mode shapes of the physical system. This is a linear perturbation procedure in which stiffness and mass matrices are calculated for the system. These matrices are used to extract the eigenvalues and mode shapes. Instabilities and rigid body modes can lead to negative or zero eigenvalues. It should be noted that the number of eigenvalues extracted in this step must be greater than or equal to the maximum frequency on the SRS curve. For more information on performing modal analysis, please refer to our previous blog here.

**Performing Shock Response Spectrum Analysis**

In this step, we generate a response spectrum and specify the SRS loading curve and the direction of the loading. For each mode, we can define the damping ratios as required. This approach assumes that the response of the system is linear.

**Results**

This is a linear analysis, so the natural modes can be used to analyze the response in the frequency domain. It is used to determine the maximum dynamic response of mechanical systems. The output variables available are stress, strain and displacement of the structure. The magnitudes computed for these variables do not correspond to any particular frequency but, instead, represent the peak magnitude over the entire frequency domain in SRS.

**Final Thoughts**

Hopefully this blog has summarized the calculation of SRS and outlined the process of evaluating shock response using finite element analysis. Response spectrum analysis is essential in ensuring the safe performance of the structures in industries like electronics, aerospace, automotive, manufacturing and civil engineering. SRS is used to describe the transient shock wave characteristics in frequency domain.

If you would like to learn more about using response spectrum analysis for you design or any other types of analysis in general, **get in touch with our expert team today**!!