Shock Response Spectrum Measurements

Use the shock response spectrum (SRS) to characterize a dynamic mechanical environment and to estimate the damage potential of a specific shock to a component.

The following figure shows a single-degree-of-freedom (SDOF) mechanical system.

Figure 43. Single-Degree-of-Freedom Mechanical System

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An SDOF mechanical system consists of the following components:

  • Mass, whose value is represented with the variable m
  • Spring, whose stiffness is represented with the variable k
  • Damper, whose damping coefficient is represented with the variable c

x represents the motion of the base and y represents the motion of the mass.

The input to the SDOF system is the base acceleration ẍ(t). The response of the SDOF system is the absolute acceleration of the mass ÿ(t). SRS measurements often use this absolute acceleration model. The following equation represents the transfer function in the Laplace domain for the SDOF system:

H ( s ) = n s 2 ζ ω + n 2 ω s 2 + 2 ζ ω n s + ω n 2

Another common model for SRS measurements is the relative displacement model. You use the relative displacement model when the damage potential is correlated with relative displacement between the component and the base. The relative displacement model uses an additional coordinate z, where z(t) = y(t) - x(t). Use the following equation to calculate the transfer function.

H ( s ) = 1 s 2 + 2 ζ ω n s + ω n 2

To keep the stimulus signal and response signal in the same physical units, you also can represent the relative displacement in terms of the equivalent static acceleration, as defined in the following equation:

Y ¨ e q = z ω n 2

You then calculate the transfer function using the following equation:

H ( s ) = ω n 2 s 2 + 2 ζ ω n s + ω n 2

The resonance frequency, fn, and the critical damping factor, ζ, characterize an SDOF system, where

f n = 2 x ω n = 1 2 x k m
ζ = c 2 k m

For light damping where ζ is less than or equal to 0.05, the peak value of the frequency response occurs in the immediate vicinity of fn and is given by the following equation, where Q is the resonant gain:

Q ∼ 1 2 ζ = k m c

Obtain SRS by applying the acquired shock pulse to a series of SDOF mechanical systems. Plot the system maximum response as a function of the resonance frequency of the system.

The following figure illustrates the response of an SDOF system to a half-sine pulse with a 10 g acceleration amplitude and 10 ms duration.

Figure 44. SDOF System Response to 10 ms, 10g Amplitude Half-Sine Pulse"

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The top graph shows the time-domain acceleration. The middle graph is the SDOF system response with a 50 Hz resonance frequency. The bottom graph is the SDOF system response with a 150 Hz resonance frequency. In both cases, ζ is 0.05.

Use the signals shown in the previous figure to construct the SRS. For example, the maximax, the absolute maximum response of the calculated shock response signal over the entire signal duration, uses the absolute maximum system response as a function of the system natural frequency. The following figure illustrates the maximax SRS for the same half-sine pulse.

Figure 45. Maximax SRS of SDOF System from 10 ms, 10 g Amplitude Half-Sine Pulse

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Note Each computed SRS is specific to the pulse used to perform the measurement.

You can use other types of shock spectra depending on the application. These spectra include the initial shock response from the system response over the pulse duration or from the residual shock spectrum of the system response after the pulse. You can use the positive maximum, the negative maximum, or the absolute maximum response signal value. The following figure shows different types of SRS measurements.

Figure 46. Different Types of SRS Measurements

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