Shock Testing & Analysis
By Tom Irvine
Shock Isolator Photo Gallery
Figure 1.1. Titan II Missile Silo, Launch Control Room

The control room is mounted underground via huge isolation springs. A typical spring is shown in the background. The purpose is to isolate the control room from mechanical shock and vibration in the event of a nuclear strike above the launch site. The springs allow 18 inches of relative displacement. The control room could thus carry out a retaliatory strike, as ordered by the U.S. president. This site is located south of Tucson, Arizona. It has been decommissioned and is now a museum.

Simple Drop Shock
Figure 2.1. iPhone 6 Accidental Drop

The first iPhone 6 was purchased in Perth, Australia on September 19, 2014. The event was covered by a live TV report. The buyer mishandled the phone as he unboxed it. The phone survived the drop onto the ground but may have had some fatigue or fracture damage.

Portable electronic devices (PEDs) are expected to survive multiple drops. Most original equipment suppliers specifying between 30 and 50 drops. Recommended test methods are given in Reference [28].

Figure 2.2. Drop Shock Analytical Model

The accidental drop shock of a component onto a hard surface is difficult to model accurately. The item may undergo rigid-body translation and rotation in each of its six degrees-of-freedom during freefall. The item may have a nonlinear response with plastic deformation during impact. It may or may not bounce. Furthermore, a box-shaped object may strike the ground on any of its corners, edges or faces.

A very simple method, as a first approximation, is to assume that the object is a linear, undamped, single-degree-of-freedom subjected to initial velocity excitation as it strikes the ground and remains attached to it via its spring. The object then undergoes free vibration in his configuration. The initial velocity is calculated using a familiar physics formula where the change in kinetic energy is equal to the change in potential energy due to gravity.

Assume that the object is dropped from rest. The initial velocity as it strikes the ground is


The equation of motion is


Or equivalently


The peak displacement is


The peak velocity is equal to the initial velocity in equation (1.1).

The peak acceleration is


An example is shown in the following table for three natural frequency cases.

Table 2.1. Peak Response Values for 36 inch Drop Height
Natural Freq (Hz) Displacement (in) Velocity (in/sec) Acceleration (G)
200 0.133 167 543
600 0.044 167 1630
1000 0.027 167 2710
Classical Shock
Figure 3.1. Idealized Classical Pulse Examples

Classical pulses are the simplest base excitation pulses. They are deterministic and can be represented by simple mathematical functions. They are typically one-sided. An SDOF system’s response to a classical pulse can be solved for exactly using Laplace transforms.

Four classical pulse types are shown in Figure 18.4. Other types include rectangular and trapezoidal pulses. These pulses do not necessarily represent real field environments, but they are still used throughout industry to test equipment ruggedness for convenience.

Shock tests are performed on military equipment [26] to:

  • a. Provide a degree of confidence that materiel can physically and functionally withstand the relatively infrequent, non-repetitive shocks encountered in handling, transportation, and service environments. This may include an assessment of the overall materiel system integrity for safety purposes in any one or of the handling, transportation, and service environments.
  • b. Determine the materiel's fragility level, in order that packaging may be designed to protect the materiel's physical and functional integrity.
  • c. Test the strength of devices that attach materiel to platforms that can crash.

Potential equipment failure modes due to shock excitation include:

  • a. Materiel failure resulting from increased or decreased friction between parts, or general interference between parts.
  • b. Changes in materiel dielectric strength, loss of insulation resistance, variations in magnetic and electrostatic field strength.
  • c. Materiel electronic circuit card malfunction, electronic circuit card damage, and electronic connector failure. (On occasion, circuit card contaminants having the potential to cause short circuit may be dislodged under materiel response to shock.)
  • d. Permanent mechanical deformation of the materiel resulting from overstress of materiel structural and nonstructural members.
  • e. Collapse of mechanical elements of the materiel resulting from the ultimate strength of the component being exceeded.
  • f. Accelerated fatiguing of materials (low cycle fatigue).
  • g. Potential piezoelectric activity of materials, and materiel failure resulting from cracks in fracturing crystals, ceramics, epoxies, or glass envelopes.
Figure 3.2. Drop Shock Test Machine, Initial Velocity Excitation
(Courtesy of Lansmont)

Classical pulse shock testing has traditionally been performed on a drop tower. The component is mounted on a platform which is raised to a certain height. The platform is then released and travels downward to the base, which has pneumatic pistons to control the impact of the platform against the base. In addition, the platform and base both have cushions for the model shown. The pulse type, amplitude, and duration are determined by the initial height, cushions, and the pressure in the pistons. This is a textbook example of case where the initial potential energy of the raised platform and test item are converted to kinetic energy. The final velocity of the freefall becomes the initial velocity of the shock excitation.

Figure 3.3. A 50 G, 11 msec, Terminal Sawtooth Pulse for Shaker Shock Test

Classical pulse shock testing can sometimes be performed on shaker tables, but there some constraints. The net velocity and net displacement must each be zero. Also, the acceleration, velocity and displacement peaks must each be within the shaker table stroke limits. Pre and post pulses are added to classical pulses to meet these requirements. A hypothetical terminal sawtooth pulse suitable for shaker shock testing is shown in Figure 1.6.

Half-Sine Shock Example

Consider a single-of-freedom is subjected to a 50 G, 11 msec half-sine pulse applied as base excitation per Figure 12.12. Set the amplification factor to Q=10. Allow the natural frequency to be an independent variable. Solve for the absolute response acceleration. The equation of motion is for the relative displacement z is


The primary response occurs during the half-sine pulse input. The residual response occurs during the quiet period thereafter. The total response is the combination of primary and residual. The exact response for a given time can be calculated via a Laplace transform. Note that the quiet period solution is free vibration with initial velocity and displacement excitation.

Figure 4.1. Fourier Transform Half-Sine Shock

A common misunderstanding is to regard the half-sine shock pulse as having a discrete frequency which would be the case if it were extended to a full-sine pulse. The Fourier transform in Figure 18.7 shows that the half-sine pulse has a continuum of spectral content beginning at zero and then rolling-off as the frequency increases. There are also certain frequencies where the magnitude drops to zero. The magnitude represents the acceleration, but the absolute magnitude depends on the total duration include the quiet period after the pulse is finished. The post-pulse duration was 10 seconds in the above example, for a total of 10.011 seconds.

Figure 4.2. SDOF System Response to Half-Sine Shock, 10 Hz

The response in the above figure has an absolute peak value less than the peak input. This is an isolation case. The response positive and negative peaks occur after the base input pulse is over.

Figure 4.3. SDOF System Response to Half-Sine Shock, 75 Hz

The response in Figure 1.9 has an absolute peak value that is 1.65 times the peak input. This is resonant amplification case. The absolute peak response occurs during the base input pulse.

Figure 4.4. SDOF System Response to Half-Sine Shock, 400 Hz

The response converges to the base input as the natural frequency becomes increasingly high. This becomes a unity gain case. The system is considered as hard-mounted.

The peak results from the three cases are shown in Table 1.2. The calculation can be repeated for a family of natural frequencies. The peak acceleration results can then be plotted as a shock response spectrum (SRS) as shown in Figure 1.11. The peak relative displacement values can likewise be plotted as an SRS as shown in Figure 1.12.

Figure 4.5. Acceleration SRS, 50 G, 11 msec Half-Sine Pulse

The two curves in Figure 1.11 contain the coordinates in Table 1.2.

The initial slope of each SRS curve is 6 dB/octave indicating a constant velocity line. The curves also indicate that the peak response can be lowered by decreasing the natural frequency. A low natural frequency could be achieved for a piece of equipment by mounting it via soft, elastomeric isolator bushings or grommets. But a lower natural frequency leads to a higher relative displacement as shown in Figure 1.12.

Table 4.1. Summary of Peak Response, 50 G, 11 msec, Half-Sine Base Input
Natural Frequency (Hz) Peak Positive (G) Absolute Value of Peak Negative (G)
10 20.3 17.3
75 82.5 65
400 51.8 4.38
Figure 4.6. Relative Displacement SRS, 50 G, 11 msec Half-Sine Pulse

The curves in the above figure could be used for designing isolator mounts for a component. The isolators must be able to take up the relative displacement without bottoming or topping out. There must also be enough clearance and sway space around the component.

Figure 4.7. Acceleration SRS, 50 G, 11 msec Terminal Sawtooth Pulse

The positive and negative SRS curves are reasonably close for the terminal sawtooth pulse shown in Figure 1.13. In contrast, the positive and negative SRS curves for the half-sine pulse in Figure 1.11 diverge as the natural frequency increases above 80 Hz. The terminal sawtooth pulse is thus usually preferred over the half-sine pulse for classical shock testing.

Another means of visualizing the SRS concept is given in Figure 1.14.

Figure 4.8. Half-Sine Shock Applied as Base Input to Independent SDOF Systems

The systems are arranged in order of ascending natural frequency from left to right and subjected to a common half-sine base input. The Soft-mounted system on the left has high spring relative deflection, but its mass remains nearly stationary. The Hard-mounted system on the right has low spring relative deflection, and its mass tracks the input with near unity gain. The Middle system ultimately has high deflection for both its mass and spring. The peak positive and negative responses of each system are plotted as a function of natural frequency in the shock response spectrum.

Response to Arbitrary Excitation

Recall from Section 12.3 that the response of a single-degree-of-freedom system to base excitation can be expressed in terms of a second order, ordinary differential equation for the relative displacement for a base acceleration ÿ.


Equation (1.7) can be solved via Laplace transforms if the base acceleration is deterministic such as a half-sine pulse. A convolution integral is needed if the excitation varies arbitrary with time. The convolution integral is computationally inefficient, however. An alternative is to use the Smallwood ramp invariant digital recursive filtering relationship [17], [19].

Again, the recursive filtering algorithm is fast and is the numerical engine used in almost all shock response spectrum software. It is also accurate assuming that the data has a sufficiently high sample rate and is free from aliasing. One limitation is that it requires a constant time step.

The equation for the absolute acceleration is


The damped natural frequency is


The digital recursive filtering relationship for relative displacement is omitted for brevity but is available in Reference [19].

The relationship in equation (1.8) is recursive because the response at the current time depends on the two previous responses, which are the first two terms on the righthand side of the equation. This is a feedback loop in terms of control theory. The relationship is filtering because the energy at and near the natural frequency is amplified whereas higher frequency energy above √2 times the natural frequency is attenuated. See Figure 12.14.

Shock Response Spectrum Test Specification Objective
Figure 5.1. Mid-Field Pyrotechnic Shock Time History

Consider the measured mid-field shock time history in the above figure as taken from Reference [29]. The time history would be essentially impossible to reproduce in a test lab given that it is a high-frequency, high-amplitude complex, oscillation pulse. The aerospace practice instead is to derive an SRS to represent the damage potential of the shock event. The test conductor may then use an alternate pulse to satisfy the SRS specification within reasonable tolerance bands. This is an indirect method of achieving the test goal. There are some limitations to this approach. One is that the test item is assumed to be linear. Another is that it behaves as a single-degree-of-freedom system. Nevertheless, this method is used in aerospace, military and earthquake engineering fields, for both analysis and testing.

Figure 5.2. Mid-Field Shock Response Spectrum & Envelope

The customary approach is to draw a conservative envelope over the measured SRS. The ramp-plateau format is the most common, although there are variations. The enveloping process shown in the above vibration is very conservative in the mid frequency domain. An additional dB uncertainty factor may be needed to develop the envelope into a test specification, given that the SRS envelope is derived from a single time history. Industry standards, such as Reference [30], give guidelines for the dB factor.

Seismic Shock
Seismic Waveforms
Figure 6.1. Mother Earth

The Earth experiences seismic vibration. The fundamental natural frequency of the Earth is 309.286 micro Hertz, equivalent to a period of approximately 54 minutes [1]. The structure of Earth's deep interior cannot be studied directly. But geologists use seismic waves to determine the depths of layers of molten and semi-molten material within Earth.

Figure 6.2. P-Wave

The primary wave, or P-wave, is a body wave that can propagate through the Earth’s core. This wave can also travel through water. The P-wave is also a sound wave. It thus has longitudinal motion. Note that the P-wave is the fastest of the four waveforms.

Figure 6.3. S-Wave

The secondary wave, or S-wave, is a shear wave. It is a type of body wave. The S-wave produces an amplitude disturbance that is at right angles to the direction of propagation. Note that water cannot withstand a shear force. S-waves thus do not propagate in water.

Figure 6.4. Love Wave

Love waves are shearing horizontal waves. The motion of a Love wave is similar to the motion of a secondary wave except that Love wave only travel along the surface of the Earth. Love waves do not propagate in water.

Figure 6.5. Rayleigh Wave, Retrograde

Rayleigh waves produce retrograde elliptical motion. The ground motion is thus both horizontal and vertical. The motion of Rayleigh waves is similar to the motion of ocean waves in Figure 1.22 except that ocean waves are prograde. Rayleigh waves resulting from airborne acoustical sources may either be prograde or retrograde per Reference [31]. In some cases, the motion may begin as prograde and then switch to retrograde. Airborne acoustical sources include above ground explosions and rocket liftoff events.

Figure 6.6. Ocean Surface Wave Particle Motion, Prograde

The Love and Rayleigh waves are both surface waves. These are the two seismic waveforms which can cause the most damage to building, bridges and other structures.

As an aside, seismic and volcanic activity at the ocean floor generates a water-borne longitudinal wave called a T-wave, or tertiary wave. These waves propagate in the ocean’s SOFAR channel, which is centered on the depth where the cumulative effect of temperature and water pressure combine to create the region of minimum sound speed in the water column. SOFAR is short for “Sound Fixing and Ranging channel.” These T-waves may be converted to ground-borne P or S-waves when they reach the shore. Acoustic waves travel at 1500 m/s in the ocean whereas seismic P and S-waves travel at velocities from 2000 to 7000 m/s in the crust.

Seismic Response Spectrum Method

Professors Theodore von Kármán and Maurice Biot were very active in the early 1930s in the theoretical dynamics aspects of what would later become known as the response spectrum method in earthquake engineering. Biot proposed that rather than being concerned with the shape of the input time history, engineers should instead use a method describing the response of systems to those shock pulses. The emphasis should instead be on the effect, as represented by the response of a series of single-degree-of-freedom oscillators, similar to that previously shown for the case of a half-sine input in Figure 18.14.

Practical use of the response spectrum method had to wait until the 1970s due to the intricacy of the response calculation for complex, oscillating pulses which required digital computers. Time was also needed to establish and publicize databases of strong motion acceleration time histories.

The response spectrum method was adopted for pyrotechnic shock in the aerospace industry and renamed as shock response spectrum.

El Centro Earthquake
Figure 6.7. El Centro, Imperial Valley Earthquake Damage

Nine people were killed by the May 1940 Imperial Valley earthquake. At Imperial, 80 percent of the buildings were damaged to some degree. In the business district of Brawley, all structures were damaged, and about 50 percent had to be condemned. The shock caused 40 miles of surface faulting on the Imperial Fault, part of the San Andreas system in southern California. Total damage has been estimated at about $6 million. The magnitude was 7.1. The was the first major earthquake for which strong motion acceleration data was obtained that could be used for engineering purposes.

Figure 6.8. El Centro Earthquake, Triaxial Time History
The highest excitation is in the North-South axis, parallel to the ground.
Figure 6.9. El Centro Earthquake North-South SRS, Three Damping Cases

The acceleration levels reached 1.5 G for the 1% damping curve. Recall that large civil engineering structures can have nonlinear damping. The damping values tend to increase as the base excitation levels increase as shown for the Transamerica Title Building in Section 10.7.

Figure 6.10. El Centro Earthquake Tripartite SRS

Seismic SRS curves are often plotted in tripartite format which displays relative displacement, pseudo velocity and acceleration responses all on the same graph. The pseudo velocity is calculated from the relative displacement as


Stress can be calculated from pseudo velocity using the methods in Section 19. The acceleration curve might be the most important design metric for equipment mounted inside a building. The relative displacement might be the most important concern for analyzing the foundational strength. The curves also show design tradeoffs. Lowering the building’s natural frequency below 1 Hz reduces the acceleration response but increases the relative displacement. Note that Q=10 is the same as 5% damping.

Golden Gate Bridge
Figure 6.11. Golden Gate Suspension Bridge, San Francisco, California

In addition to traffic loading, the Golden Gate Bridge must withstand the following environments:

  • 1. Earthquakes, primarily originating on the San Andreas and Hayward faults
  • 2. Winds of up to 70 miles per hour
  • 3. Strong ocean currents

The Golden Gate Bridge has performed well in all earthquakes to date, including the 1989 Loma Prieta Earthquake. Several phases of seismic retrofitting have been performed since the initial construction. The bridge’s fundamental mode is a transverse mode with a natural frequency of 0.055 Hz, with a period of 18.2 seconds

Note that California Department of Transportation (CALTRANS) standards require bridges to withstand an equivalent static earthquake force (EQ) of 2.0 G. This level is plausibly derived as a conservative envelope of the El Centro SRS curves in Figure 1.25.

Vandenberg AFB
Figure 6.12. Rocket Launch from Vandenberg AFB, California

The Vandenberg launch site is near the San Andreas fault system. The vehicle is mounted on the pad as a tall cantilever beam. The vehicle must be analyzed to verify that it can withstand a major seismic event. The vehicle may be mounted to the pad for only two weeks prior to launch. The odds of an earthquake occurring to that time window are miniscule. But the launch vehicle and payload together may cost well over $1 billion. The risk thus necessitates the analysis. Areas of concern are the loads imparted at the launch vehicle’s joints and to the payload.

Figure 6.13. NASA SRS Curves for Launch Vehicles at Vandenberg

SRS curves are given for three damping cases. The curves are taken from Reference [29]. The vehicle would typically be analyzed as a multi-degree-of-freedom system via a finite element model. Each SRS curve could be applied to the model using a modal combination method. An alternative is to synthesize a time history to satisfy a selected SRS curve. The time history could then be applied to the model via a modal transient analysis.

Seismic Testing
Figure 6.14. Electrical Power Generator Seismic Shock Test

The diesel generator is mounted onto a platform at the top of a shaker table which is located below the ground floor. This could be an emergency power generator for a hospital in an active seismic zone. A video clip of the test is available on YouTube at:

Pyrotechnic Shock
Flight Events
Figure 7.1. Stage Separation Ground Test, Linear Shaped Charge

The plasma jet cuts the metal inducing severe mechanical shock energy, but the smoke and fire would not occur in the near-vacuum of space.

Launch vehicle avionics components must be designed and tested to withstand pyrotechnic shock from various stage, fairing and payload separation events that are initiated by explosive devices. Solid rocket motor ignition is another source of pyrotechnic shock. The source shock energy can reach thousands of acceleration Gs at frequencies up to and beyond 100 kHz. The corresponding velocities can reach a few hundred in/sec, well above the severity thresholds in the empirical rules-of-thumb in Section 19.4.

Empirical source shock levels for a variety of devices are given in References [29], [32], [33]. These levels are intended only as preliminary estimates for new launch vehicle designs. The estimates should be replaced by ground test data once the launch vehicle hardware is built and tested.

Pyrotechnic Device Photo Gallery
Figure 7.2. Metal Clad Linear Shaped Charge

The chevron focuses a pyrotechnic plasma jet at the launch vehicle’s separation plane material. Severe shock levels are generated as a byproduct.

Figure 7.3. Frangible Joint

A frangible joint may be used for stage or fairing separation. The key components of a frangible joint:

  • Mild Detonating Fuse (MDF)
  • Explosive confinement tube
  • Separable structural element
  • Initiation manifolds
  • Attachment hardware

The hot gas pressure from the MDF detonation cause the internal tube to expand and fracture the joint.

Figure 7.4. Frangible Nuts, Hold Down Posts

The purpose of the nuts was to hold the SRBs in place against wind and ground-borne excitation. The nuts were separated just before liftoff.

Figure 7.5. Clamp Band with Pyrotechnic Bolt-Cutters
(image courtesy of European space agency)

Clamp bands are often used for payload separation from launch vehicle adapters. They are also commonly used for stage separation in suborbital launch vehicles, similar to the one in Figure 4.1. A pyrotechnic bolt-cutter uses an explosive charge to drive a chisel blade to cut the band segments’ connecting bolt. The cutters produce some shock energy, but much of the shock is due to the sudden release of strain energy in the preloaded clamp band. This action can excite a ring mode in the radial axis. Recall Section 7.3.

The total clamp band release shock is significantly less than linear shaped charge and frangible joint. Note that an analysis must be performed to verify that no gapping will occur in the between the band and the joint as the vehicle undergoes bending mode vibration during powered flight.

Pyrotechnic Shock Data
Initial SRS Slopes
Figure 8.1. Expected Pyrotechnic SRS Initial Slope Limits

Near-field pyrotechnic shock can be difficult to measure accurately. The accelerometer data may have a baseline shift or spurious low frequency transient. This error could be a result of the accelerometer’s own natural frequency being excited or to some other problem. Aerospace pyrotechnic shock SRS specifications usually begin at 100 Hz due to the difficult in accurately measuring low-amplitude, low-frequency shock, while simultaneously measuring high-amplitude, high-frequency shock.

There are several methods for checking whether the data is acceptable. One is to the check the initial slopes of both the positive and negative SRS curves. Each should have an overall trend of 6 to 12 dB/octave. The actual slopes may have local peaks and dips due to low frequency resonances. A 12 dB/octave slope represents constant displacement and zero net velocity change. A 6 dB/octave slope indicates constant velocity. Recall the slope formulas in Section 16.1.2.

A second method for checking data accuracy is to verify that the positive and negative SRS curves are within about 3 dB of each other across the entire natural frequency domain. A third method is to integrate the acceleration time history to velocity. The velocity time history should oscillate in a stable manner about the zero baselines.

These verification goals are challenging to meet with near-field shock measurements of high-energy source shock, such as that from linear shaped charge and frangible joints. In practice, some high-pass filtering or spurious trend removal may be necessary. There is no one right way to perform this “data surgery.” It is a matter of engineering judgment.

Re-entry Vehicle Separation, Flight Data
Figure 8.2. Re-Entry Vehicle, Separation Shock, Near-Field Measurement, Time History

The source device was linear shaped charge.

Figure 8.3. Re-Entry Vehicle, Separation Shock, Near-Field Measurement, SRS

The SRS reached 20,040 G at 2400 Hz, which is an extreme level per the rule-of-thumb Section 19.4.

Water Impact Shock
Figure 9.1. Solid Rocket Booster Water Impact Shock

Each of the two Space Shuttle Boosters was recovered and refurbished after every flight. Each booster contained sensitive avionics components which underwent shock testing for the water impact event.

Figure 9.2. Solid Rocket Booster Water Impact Shock, Time History

The data is from the STS-6 Mission. The accelerometer was mounted at the forward end of the booster adjacent to a large IEA avionics box. This was the worst-case shock event for this component.

Figure 9.3. Solid Rocket Booster Water Impact Shock, Tripartite SRS

The maximum acceleration response is 257 G at 85 Hz. The maximum pseudo velocity response is 201 in/sec at 76 Hz, which is severe per the rule-of-thumb Section 1.4.

Shock Response Spectrum Synthesis
Synthesis Objectives

Consider an SRS specification for a component, subsystem or a large structure. The article should be tested, if possible, to verify that it can withstand the shock environment. Certain shock tests can be performed on a shaker table, like the generator test in Figure 18.30. This requires synthesizing an acceleration time history to satisfy the SRS. The net velocity and net displacement must each be zero for this test. These requirements can be met with a certain type of wavelet series, where the individual wavelets may be nonorthogonal. The resulting wavelet time history should meet the SRS within reasonable tolerance bands, but it may not “resemble” the expected time history which is a limitation of this method. Another requirement is that the peak acceleration, velocity and displacement values must be within the shaker table’s capabilities.

An innovative method for meeting the SRS with a wavelet series that resembles one or more measured shock time histories is given in Reference [34].

A synthesized time history can also be used for modal transient analysis. This could be done for articles which are too large or heavy for shaker tables. This analysis can also be done on small components prior to shock testing to determine whether they will pass the test. Or the analysis could be done in support of isolator mounting design. There is again a need for the synthesized time history to have net velocity and net displacement which are each zero, to maintain numerical stability in stress calculations from relative displacement values.

Damped-sines can be used for modal transient analysis where the goal is to meet the SRS with a time history that plausibly resembles the expected field shock event. But damped-sines do not meet the desired zero net velocity and displacement goals. The workaround is to first synthesize a damped-sine series to meet the SRS and then reconstruct it via a wavelet series, in Rube Goldberg fashion.

Wavelet and damped-sine synthesis are shown in the following examples.

Wavelet Synthesis
Wavelet Equation

A wavelet is a sine function modulated by another sine function. The equation for an individual wavelet is



    = acceleration of wavelet at time

    = wavelet acceleration amplitude

    = wavelet frequency

    = number of half-sines

    = wavelet time delay

Note that must be an odd integer greater than or equal to 3. This is required so that the net velocity and net displacement will each be zero.

The total acceleration at time for a set of wavelets is

Figure 10.1. Sample Wavelet

A sample, individual wavelet is shown in the above figure. This wavelet was a component of a previous analysis for an aerospace project.

Wavelet Synthesis Example

Consider the specification: MIL-STD-810E, Method 516.4, Crash Hazard for Ground Equipment.

Table 10.1. SRS Q=10, Crash Hazard Specification
Natural Frequency (Hz) Peak Accel (G)
10 9.4
80 75
2000 75

Synthesize a series of wavelets as a base input time history for a shaker shock test to meet the Crash Hazard SRS. The goals are:

  • Satisfy the SRS specification
  • Minimize the displacement, velocity and acceleration of the base input

The synthesis steps are shown in the following table.

Table 10.2. Wavelet Synthesis Steps
Step Description
1 Generate a random amplitude, delay, and half-sine number for each wavelet. Constrain the half-sine number to be odd. These parameters form a wavelet table.
2 Synthesize an acceleration time history from the wavelet table.
3 Calculate the shock response spectrum of the synthesis.
4 Compare the shock response spectrum of the synthesis to the specification. Form a scale factor for each frequency.
5 Scale the wavelet amplitudes.
6 Generate a revised acceleration time history.
7 Repeat steps 3 through 6 until the SRS error is minimized or an iteration limit is reached.
8 Calculate the final shock response spectrum error.
Also calculate the peak acceleration values.
Integrate the signal to obtain velocity, and then again to obtain displacement. Calculate the peak velocity and displacement values.
9 Repeat steps 1 through 8 many times.
10 Choose the waveform which gives the lowest combination of SRS error, acceleration, velocity and displacement.

The resulting time history and SRS are shown in Figure 1.43 and Figure 1.44, respectively.

Figure 10.2. Crash Hazard Time History Synthesis

The acceleration time history has a reverse sine sweep character. It is an efficient and optimized waveform for a shaker shock test, and it satisfies the SRS as shown in the next figure. A drawback is that it does not resemble an actual crash shock time history.

Figure 10.3. Crash Hazard SRS

The positive and negative curves are from the synthesized waveform. The tolerance bands are set at +3 dB.

Damped-Sine Synthesis
Damped-Sine Equation

The equation for an individual damped sinusoid is



    = acceleration of damped sinusoid at time

    = acceleration amplitude

    = angular frequency

    = damping value

    = time delay

The total acceleration at time for a set of damped sinusoids is

Damped-Sine Example

Consider the following specification which could represent a stage separation shock level as some location in a launch vehicle. A modal transient finite element analysis is to be performed on a component to verify that the component will pass its eventual shock test. The immediate task is to synthesize a time history to satisfy the SRS. The time history should “resemble” an actual pyrotechnic shock pulse.

Note that pyrotechnic SRS specifications typically begin at 100 Hz. The author’s rule-of-thumb is to extrapolate the specification down to 10 Hz in case there are any component modes between 10 and 100 Hz. This guideline is also to approximate the actual shock event which should have an initial ramp somewhere between 6 and 12 dB/octave.

Table 10.3. SRS Q=10, Stage Separation Shock
Natural Frequency (Hz) Peak Accel (G)
10 10
2000 2000
10,000 2000

The specification has an initial slope of 6 dB/octave. The synthesis steps are shown in the following table.

Table 10.4. Damped-Sine Synthesis Steps
Step Description
1 Generate random values for the following for each damped sinusoid: amplitude, damping ratio and delay. The natural frequencies are taken in one-twelfth octave steps.
2 Synthesize an acceleration time history from the randomly generated parameters.
3 Calculate the shock response spectrum of the synthesis.
4 Compare the shock response spectrum of the synthesis to the specification. Form a scale factor for each frequency.
5 Scale the amplitudes of the damped sine components.
6 Generate a revised acceleration time history.
7 Repeat steps 3 through 6 as the inner loop until the SRS error diverges.
8 Repeat steps 1 through 7 as the outer loop until an iteration limit is reached.
9 Choose the waveform which meets the specified SRS with the least error.
10 Perform wavelet reconstruction of the acceleration time history so that velocity and displacement will each have net values of zero.

The resulting time history and SRS are shown in Figure 1.45 and Figure 1.46, respectively.

Figure 10.4. Stage Separation Time History Synthesis

The acceleration time history somewhat resembles a mid or far-field pyrotechnic shock. The velocity and displacement time histories each have a stable oscillation about their respective baselines.

Figure 10.5. Stage Separation SRS

The positive and negative curves are from the damped-sine synthesis.

Modal Transient Finite Element Analysis for Uniform Base Excitation

Consider a rectangular plate mounted to a base at each of its four corners. The plate is to be subject to uniform seismic excitation. There are two methods to apply the base excitation in a finite element analysis, as shown in Figure 1.47 and Figure 1.48.

Figure 10.6. Direct Enforce Acceleration Method

The direct enforcement method is computationally intensive, requiring matrix transformations and a matrix inversion as shown in Reference [35].

Figure 10.7. Seismic Mass Method

The seismic mass is chosen to be several orders of magnitude higher in mass than the plate. An equivalent force is calculated and apply to the seismic mass to excite the desired acceleration at each of the plate’s corners. This method adds a degree-of-freedom to the plate system resulting in a rigid-body mode at zero frequency. But the remaining natural frequencies and mode shapes should be the same as if the plate were mounted normally to its joining structure. The author’s experience is that the seismic mass method is faster and more accurate and reliable than the direct enforced method. This method was introduced in Section 12.3.3.

Shock Fields

The near-field environment is dominated by direct stress wave propagation from the source. Peak accelerations in excess of 5000 G occur in the time domain with a frequency content extending beyond 100 kHz. The near-field usually includes structural locations within approximately 15 cm of the source for severe devices such as linear shaped charge and frangible joint. No shock-sensitive hardware should be mounted where it would be exposed to a near-field environment.


The mid-field environment is characterized by a combination of wave propagation and structural resonances. Peak accelerations may occur between 1000 and 5000 G, with substantial spectral content above 10 kHz. The mid-field for intense sources usually includes structural locations between approximately 15 and 60 cm of the source, unless there are intervening structural discontinuities.


The far-field environment is dominated by structural resonances. The peak accelerations tend to fall below 2000 G, and most of the spectral content below 10 kHz. The far-field distances occur outside the mid-field. The typical far-field SRS has a knee frequency corresponding to the dominant frequency response. The knee frequency is the frequency at which the initial ramp reaches the plateau in the log-log SRS plot.

Joint & Distance Attenuation

The source shock energy is attenuated by intervening material and joints as it propagates from the near-field to the far-field. Empirical distance and joint attenuation factors for the SRS reduction are given in References [29], [32], [33]. A typical attenuation curve from [32] is given in Figure 18.49, which assumes an input source shock SRS consisting of a ramp and plateau in log-log format. Such curves should be used with caution given that the attenuation is highly dependent on damping and structural details. The curves can be used for preliminary estimates for new launch vehicle designs, but ground separation tests are still needed for the actual launch vehicle hardware. These tests are needed to measure the source shock as well as the levels at key component mounting locations.

Figure 10.8. Shock Response Spectrum Versus Distance from Pyrotechnic Shock Source
Shock Mitigation
Figure 10.9. Sensor Electronics, Wire Rope Isolators
(image courtesy of NASA/JPL)

The source shock energy is attenuated as it propagates to avionics mounting locations through the launch vehicle’s material and joints. The input shock to a component can be mitigated by mounting the component as far away from the source device as possible. Another effective attenuation technique is to mount the component via elastomeric bushings or wire rope isolators. The NASA Mars Science Laboratory Sensor Support Electronics unit is mounted on vibration isolators in Figure 1.50.

Figure 10.10. SCUD-B Avionics Component Isolation

The bushings are made from some type of rubber or elastomeric compound. The bushings provide damping, but their main benefit is to lower the natural frequency of the system. The isolators thus attenuate the shock and vibration energy which flows from the instrument shelf into the avionics component.

Pyrotechnic Shock Testing Methods
Figure 10.11. Near-Field Shock Simulation using a Plate

The test component is mounted on other side of plate. The source device is a textile explosive cord with a core load of 50 gr/ft (PETN explosive). Up to 50 ft of Detonating Cord has been used for some high G tests. The maximum frequency of shock energy is unknown so analog anti-aliasing filters are needed for the accelerometer measurements per the guidelines in Section 14.

Note that a component may be mounted in the mid-field shock region of a launch vehicle. The SRS test level derivation for this zone may include a significant dB factor for uncertainty or as a qualification margin. This conservatism may require a near-field-type shock test for a component that is actually located in a mid-field zone. This situation can also occur for components mounted in far-fields.

Figure 10.12. NASA JPL Tunable Shock Beam

The NASA/JPL Environmental Test Laboratory developed and built a tunable beam shock test bench based on a design from Sandia National Laboratory many years ago. The excitation is provided by a projectile driven by gas pressure. The beam is used to achieve SRS specifications, typically consisting of a ramp and a plateau in log-log format. The intersection between these two lines is referred to as the “knee frequency.” The beam span can be varied to meet a given knee frequency. The high frequency shock response is controlled by damping material.

Shock Failure Modes
Figure 10.13. Sensitive Electronic Parts

Pyrotechnic shock can cause crystal oscillators to shatter. Large components such as DC-DC converters can detached from circuit boards. In addition, mechanical relays can experience chatter or transfer.

Figure 10.14. Shock Test Case History, Adhesive & Solder Joint Failure

The image shows adhesive failure and rupture of solder joint after a stringent shock test. A large deflection of the PCB resulting from an insufficient support/reinforcement of the PCB combined with high shock loads can lead to these failures. Staking is needed for parts weighing more than 5 grams.

Figure 10.15. Shock Test Case History, Lead Wire Failure

The image shows a sheared lead between solder joint and winding of coil. The lacing cord was insufficient by itself. The lacing should be augmented by staking with an adhesive.

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