Calculation of response spectra from incomplete shock time histories

Missiles and space launch vehicles are subjected to extreme dynamic loads in the form of short duration vibration transients, or shocks. Information on the severity of these shocks comes primarily from accelerometers mounted in various locations on the airframe structure. However, phenomena related to the shock events also interrupts the radio telemetry that transmits the data to the ground. This paper explores mitigation techniques for the calculation of response spectra from these incomplete shock time histories. Additionally, an empirical study of the error from the Shock Response Spectra (SRSs) calculated from incomplete time histories is presented. It is concluded that any time history that drops out before 20% of its action time is practically unusable. For particular families of shock events, increasing the margin above that typically added can account for any residual error.

CALCULATION OF RESPONSE SPECTRA FROM INCOMPLETE SHOCK TIME HISTORIES by Scott Lawrence Stebbins A thesis submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Master of Science Department of Mechanical Engineering The University of Utah December 2016 Copyright © Scott Lawrence Stebbins 2016 All Rights Reserved The University of Utah Graduate School STATEMENT OF THESIS APPROVAL Scott Lawrence Stebbins The thesis of has been approved by the following supervisory committee members: James Edward Guilkey , Chair 11 August 2016 Meredith M. Metzger , Member 11 August 2016 Sanford G. Meek , Member 11 August 2016 and by Timothy A. Ameel the Department/College/School of and by David B. Kieda, Dean of The Graduate School. Date Approved Date Approved Date Approved , Chair/Dean of Mechanical Engineering ABSTRACT Missiles and space launch vehicles are subjected to extreme dynamic loads in the form of short duration vibration transients, or shocks. Information on the severity of these shocks comes primarily from accelerometers mounted in various locations on the airframe structure. However, phenomena related to the shock events also interrupts the radio telemetry that transmits the data to the ground. This paper explores mitigation techniques for the calculation of response spectra from these incomplete shock time histories. Additionally, an empirical study of the error from the Shock Response Spectra (SRSs) calculated from incomplete time histories is presented. It is concluded that any time history that drops out before 20% of its action time is practically unusable. For particular families of shock events, increasing the margin above that typically added can account for any residual error. iii I dedicate this thesis to my wife, Amanda. Without her support, encouragement, and feedback, this would not have been possible. iv "We can lick gravity, but sometimes the paperwork is overwhelming." — Wernher von Braun, Chicago Sun Times, July 10, 1958 v TABLE OF CONTENTS ABSTRACT....................................................................................................................... iii ACKNOWLEDGEMENTS ............................................................................................. viii Chapters 1. INTRODUCTION .........................................................................................................1 2. THE SHOCK RESPONSE SPECTRUM ......................................................................7 2.1 History................................................................................................................9 2.2 Algorithm ...........................................................................................................9 2.3 Comparison to Other Methods of Shock Quantification .................................11 2.4 Standard Practices ............................................................................................12 3. EFFECT OF SIMULATED DROPS IN ACCELERATION TIME HISTORIES ......21 3.1 Definitions........................................................................................................21 3.2 Effect of Simulated Drops on Classic Shock Pulses ........................................22 3.3.1 Haversine Pulse .....................................................................................22 3.3.2 Half-Sine Pulse ......................................................................................23 3.3.3 Sawtooth Pulse ..................................................................................... 24 3.3 Effect of Drops on Simulated Shocks to Multi-Degree-of-Freedom Systems 24 3.4 Effect of Drops on Shock Data ........................................................................24 3.4.1 Complex Shock Environments ..............................................................24 3.4.2 Reentry Vehicle Separation Shock ........................................................26 3.4.3 Solid Rocket Motor Ignition Shock...................................................... 26 3.4.4 1000g Shock ..........................................................................................27 3.4.5 2000g Shock ..........................................................................................27 3.4.6 T Flight Shock ...................................................................................... 28 3.5 Effect of Drops on Simulated Shocks ..............................................................28 3.5.1 Multi-Degree-of-Freedom Systems .......................................................29 3.5.2 Commercial Finite Element Model .......................................................30 vi 4. ADVANCED COMPENSATION STRATEGIES ......................................................59 4.1 Decomposing Shock Events ............................................................................59 4.2 Time of Maximum Amplitude SRS .................................................................61 4.3 Recommendations ............................................................................................62 4.3.1 Conditions to Discard Shock Data ....................................................... 64 5. CONCLUSION ............................................................................................................90 REFERENCES ..................................................................................................................92 vii ACKNOWLEDGEMENTS I would like to acknowledge the patience and guidance displayed by my advisor, Dr. James Guilkey, and the rest of my committee. I am grateful that Dr. Guilkey showed enough interest to encourage me to pursue a thesis even after I expressed skepticism about my abilities. Additionally, I would like to thank my managers, Jennifer Milburn and Michael Cohen, for their understanding and encouragement. Finally, I would like to thank my mentor, Kim Dahl, for taking the time to educate me on the fundamentals of shock and vibration testing, as well as giving me the problem to tackle. viii 1 CHAPTER 1 INTRODUCTION Missiles, launch vehicles, and spacecraft experience dramatic loads during the liftoff and staging environment. These loads come from a variety of sources including combustion instability, resonance burn, and pyrotechnical shocks due to stage separation or spacecraft deployment [1]. A stage separation of the Saturn V rocket is shown in Figure 1. These loads are of interest for structural and test engineers. It is critical to know whether the components that make up the rocket have been qualified to survive and function to sufficient levels, and if not, what risk there is that a component will malfunction. It is important to know if the components have been over-tested by exposing them to environmental loads greater than those found during flight. In this case the engineer can use this information to lower costs or reduce weight. Short duration high g transient vibrations called shock, can be especially damaging to electronic components, relays, and valves. After the initial development, ongoing testing is an integral part of any complex system and rocketry is no exception. Even long after the system has matured, agingsurveillance requires that systems, subsystems, and individual components be tested to evaluate age-related degradation. Additionally, as components age-out or are consumed in 2 launch new parts must be procured, qualified, and accepted. Comparing the severity of dynamic environments is not as simple as finding the maximum acceleration amplitude in the time history. Along with magnitude there are also frequency considerations as well as the duration. Two of the techniques commonly used are the shock response spectrum (SRS) and the power spectral density (PSD). These two tools are similar in that they take an acceleration time history and represent the data in the frequency domain in a way that can be compared to other dynamic environments with different, unique time histories. Power spectral density curves, also called auto-spectral densities, are usually used to describe stationary random vibrations [3,4]. The shock response spectrum is more commonly used to describe short duration mechanical vibration transients [5], and will be the focus of this paper. For many aerospace launch systems, the acceleration time history is collected by accelerometers and data acquisition systems on the spacecraft and then transmitted via radio telemetry to ground stations. Sometimes the effects of the shock event cause the telemetry system downlink to “drop” or cut out. This momentary loss of data creates a hole in an otherwise uniformly sampled time history. As launches are rare events the loss of a single measured flight environment means an increase in risk. Therefore, it is desirable to know the effect and error that these “drops” cause and what mitigation strategies engineers can use when processing shock data. Figure 2 shows a typical mechanical shock transient. It has a magnitude in the hundreds to thousands of g, and a short duration. The decay of the peaks follows an exponential decay curve. Figure 3 shows a telemetry dropout during the shock event, indicating a loss of data for the duration of the drop. The dropouts always happen after the 3 start of the shock, and the telemetry stream does not resume until the after the transient decends below the noise floor. The missing data call into question the use of the Maximum Predicted Enviroment (MPE) derived from the shock. 4 Figure 1: Artist’s rendering of a Saturn V stage separation [2]. Stage separations produce severe shocks. 5 Figure 2: A mechanical shock acceleration time history, caused by the the seperation of a reentry vehicle. The transeint is less than 10 ms and is in thousands of g [1,6] 6 Figure 3: A mechanical shock acceleration time history with telemetry dropout. 7 CHAPTER 2 THE SHOCK RESPONSE SPECTRUM The SRS is used to describe the acceleration amplitude of a shock event. Imagine a series of mass-spring-damper systems mounted to a base which is, in turn, excited to acceleration time history 𝑦𝑦̈ (t). The equation for each single-degree-of-freedom (SDoF) system with base excitation is [7]: 𝑚𝑚𝑥𝑥̈ + 𝑐𝑐(𝑥𝑥̇ − 𝑦𝑦̇ ) + 𝑘𝑘(𝑥𝑥 − 𝑦𝑦) = 0 (1) Systems react more vigorously to inputs that have sinusoids at the same frequency as the natural frequency of the system. The natural frequency of a system is defined as: 𝜔𝜔𝑛𝑛 = �𝑘𝑘�𝑚𝑚 (2) The damping ratio is defined as: 𝜁𝜁 = 1 𝑐𝑐 = 2√𝑘𝑘𝑘𝑘 2𝑄𝑄 (3) 8 The frequency ratio r is the ratio between the forcing function frequency ω and the natural frequency of the system: 𝑟𝑟 = 𝜔𝜔�𝜔𝜔𝑛𝑛 (4) Figure 4 shows the nondimensional response of a single-degree-of-freedom system to different steady-state harmonic inputs. The amplitude of the displacement is X, and the forcing function amplitude is f0. The equation for the steady state response to a harmonic excitation is [7]: 2 𝑋𝑋𝜔𝜔𝑛𝑛 𝑓𝑓0 = 1 �((1−𝑟𝑟 2 )2 +(2𝜁𝜁𝜁𝜁)2 )2 (5) As evident in Figure 4, the maximum amplitude reached depends heavily on both the frequency ratio and the damping ratio. Note that Equation (5) does not define the number of cycles of f0 are needed before the SDoF system reaches the steady state response. Each Single-Degree-of-Freedom (SDoF) system that is simulated has a natural frequency higher than the last, and is assumed not to affect the base excitation, as shown in Figure 5. Equation (1) can be used to calculate the response time history for a succession of systems, shown in Figure 6. In Figure 6 the responses of several systems are shown. The red stars indicate the maximum acceleration of each system response. The same points are shown on Figure 7. Plotting the maximum acceleration amplitude of each system as a function of natural frequency creates a shock response spectrum. If the natural frequencies are chosen on close 9 intervals then they can be plotted as a continuous function as shown in Figure 7. Calculating the SRS on close intervals allows for the direct comparison of two shock events. Typically, the label on the x-axis is Fn and the label on the y-axis is acceleration, for brevity. 2.1 History The shock response spectrum was originally used by civil engineers who were faced with the problem of designing structures to withstand earthquakes. This approach was originated by M. A. Biot in 1932. The earthquake displacement time history was reduced to a series of single-degree-of-freedom responses plotted by their natural period. A civil engineer evaluating a building for resistance to seismic activity need only to calculate the natural period of the structure and read off the maximum displacement. If the structure has multiple modes then either simply summing them, or finding the root-sumsquare of the displacements will yield the displacements for all the natural periods. [8]. Later Biot applied the technique to aeronautical problems such as the design of landing gear [9]. 2.2 Algorithm The calculation of a shock response spectrum uses a z-transform to convert the base input into the response time history of each SDoF system [10]. The z-transform is the discrete counterpart to the continuous Laplace transform, and is defined as: −𝑘𝑘 Ƶ{𝑓𝑓(𝑘𝑘)} = 𝐹𝐹(𝑧𝑧) = ∑∞ 𝑘𝑘=0 𝑓𝑓(𝑘𝑘)𝑧𝑧 (6) 10 where f(k) is the discrete sampling of some function f(t) [11]. The algorithm used is a digital filter described by the transfer function in Equation (7), which takes the base excitation, 𝑦𝑦̈ (𝑡𝑡) and converts it to the response acceleration, 𝑥𝑥̈ 𝑛𝑛 (𝑡𝑡). Ƶ{𝑥𝑥̈ 𝑛𝑛 (𝑘𝑘)} Ƶ{𝑦𝑦̈ (𝑘𝑘)} = 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂(𝑧𝑧) 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼(𝑧𝑧) = 𝐻𝐻(𝑧𝑧) = 𝛽𝛽0 +𝛽𝛽1 𝑧𝑧 −1 +𝛽𝛽2 𝑧𝑧 −2 1+𝛼𝛼1 𝑧𝑧 −1 +𝛼𝛼2 𝑧𝑧 −2 (7) where the coefficients are: 𝜁𝜁 = 𝐴𝐴 = 1 𝑐𝑐 = 2√𝑘𝑘𝑘𝑘 2𝑄𝑄 𝜔𝜔𝑛𝑛 𝑇𝑇 2𝑄𝑄 𝐵𝐵 = 𝜔𝜔𝑛𝑛 𝑇𝑇�1 − 1�4𝑄𝑄 2 𝛼𝛼1 = 2𝑒𝑒 −𝐴𝐴 cos 𝐵𝐵 𝛼𝛼2 = 𝑒𝑒 −2𝐴𝐴 𝛽𝛽0 = 1 − 𝑒𝑒 −𝐴𝐴 ∗ sin 𝐵𝐵�𝐵𝐵 𝛽𝛽1 = 2𝑒𝑒 −𝐴𝐴 ∗ �sin 𝐵𝐵�𝐵𝐵 − cos 𝐵𝐵� 𝛽𝛽2 = 𝑒𝑒 −2𝐴𝐴 − 𝑒𝑒 −𝐴𝐴 ∗ �sin 𝐵𝐵�𝐵𝐵 � (8) (9) (10) (11) (12) (13) (14) (15) Equation (7) corresponds to a difference equation that calculates the response of 𝑥𝑥̈ at the nth data point, given an input 𝑦𝑦̈ : 𝑥𝑥̈ 𝑛𝑛 = 𝛽𝛽0 𝑦𝑦̈𝑛𝑛 + 𝛽𝛽1 𝑦𝑦̈ 𝑛𝑛−1 + 𝛽𝛽2 𝑦𝑦̈ 𝑛𝑛−2 − 𝛼𝛼1 𝑥𝑥̈ 𝑛𝑛−1 + 𝛼𝛼2 𝑥𝑥̈ 𝑛𝑛−2 (16) 11 The Equation (7) creates a time history of a response for each system. These calculations can be used to find the maximum absolute response for each response time history. From this data, the maximum acceleration of the response that would occur if the system was exposed to the base excitation. The damping ratio for each system is usually assumed to be 1% (Q = 50), or 5% (Q = 10), based on Equation (8). These damping ratios envelope the typical ranges for metal structures [12,13]. 2.3 Comparison to Other Methods of Shock Quantification There are a few other ways that engineers describe shock events, but these are less common. Ultimately the shock response spectrum is used in this paper due to its wide use in military specifications. Military Standard 810G, Department of Defense Test Method Standard [3], which describes data reduction techniques for mechanical shocks, only mentions the response spectrum. The discrete Fourier Transform (DFT) describes a signal in terms of coefficients in a series expansion. The DFT is the digital counterpart to the Fourier transform done to continuous signals [14]. Figure 8 shows the DFT of a shock event. It is not preferred for short duration transients because the sample window is small. Additionally, it provides no way to estimate the maximum response of a component when it is subjected to the shock. Pseudo-Velocity Spectrum is very similar to the shock response spectrum, except that it plots the maximum displacement amplitudes and maximum velocity amplitudes along with the acceleration amplitudes and natural frequencies [15]. It includes all of the information contained on a SRS, but is described only briefly in MIL-STD-810G [3]. This method was advocated by H. Gaberson, who suggested that damage potential was directly 12 proportional to the maximum velocity amplitude [15]. For a given SDoF system, the maximum acceleration, velocity, and displacement can be directly read off the chart. If a system with a natural frequency of 2000 Hz and a damping Q = 10 were subjected to the base excitation that produced Figure 9 it would have experienced a displacement of 0.04 inches and an acceleration amplitude of 20,000g. 2.4 Standard Practices The collected accelerometer data often has a steady-state offset due to instrument errors. These are removed from the time history before the SRS algorithm is run by subtracting the mean [8,16,17]. Figure 10 shows the differences in the RV Separation SRS when the time history is preprocessed in this manner. There are large steady-state errors in the raw data from systems with natural frequencies below 100 Hz. The shock response spectrums are used to define the maximum predict environment (MPE). The MPE is usually simplified by enveloping the SRSs as shown in Figure 11. This simplification introduces some conservative error above the actual measured environments. The process is not standardized, and results will vary between engineers. Ideally several shocks measured in the same location are available to establish an MPE that envelopes the 95th percentile of all shock environments. If not, the MPE has between 4.5 and 6 dB of extra margin is added [3,18]. The MIL-STD-810G requires that the spectrum is defined one octave lower than the lowest modal frequency of the unit under test (UUT). In practice this is often no lower than 5 Hz [3]. 13 Figure 4: Response of a single-degree-of-freedom to harmonic excitations. The greatest response is for a system with very little damping which is driven at resonance. When the forcing function has a frequency much less than the natural frequency of the system the normalized response is unity. 14 Figure 5: The mechanical analogy to the shock response spectrum. The maximum, x(t), is found for each SDoF system exposed to the base excitation, (t) [5]. The base excitation is usually the movement of the spacecraft structure measured by mounted accelerometers. 15 Figure 6: Simulated responses of several single-degree-of-freedom systems to a reentry vehicle (RV) separation [1,6]. The stars mark the maximum acceleration amplitude reached during the shock event. A SRS is created when the maximum acceleration is plotted against the natural frequency of the system. 16 Figure 7: Shock response spectrum of the reentry vehicle separation when Q = 10. The red stars indicate the maximum responses of the SDoF system when subjected to shock. 17 Figure 8: Discrete Fourier transform of RV separation. This gives a description of the frequency content of the shock event. 18 Figure 9: A pseudo-velocity spectrum of RV separation shock event. For a given SDoF system, the natural frequency, maximum acceleration, maximum velocity, and maximum displacement could be read off the plot. 19 Figure 10: RV separation SRS with and without preprocessing. The shock response spectra were found using the algorithm described in section 2.2. The time history used to calculate the “Mean Removed” SRS has the average acceleration subtracted from every point before SRS calculation. 20 Figure 11: The MPE for RV separation based on the SRS. When writing requirements for testing it is customary to simplify the response spectrum. The dashed line adds 6 dB, or 200%, for margin of safety. 21 CHAPTER 3 EFFECT OF SIMULATED DROPS IN ACCELERATION TIME HISTORIES In this chapter, the effects of dropping the acceleration time histories will be shown as well as some simple mitigation strategies. First, the effect of dropping classical shock impulse shapes will be shown. Then the effect on real and simulated shock events will be discussed, as well as some processing techniques. 3.1 Definitions The action time of a shock event, TE, is defined in MIL-STD-810G [3] as the period between the start of the shock event to where it has decayed to one-third of its peak acceleration value. In practice, the response for many systems has not yet reached the peak acceleration amplitude. Including time until 3*TE would envelope all maximum responses. TE on a shock time history is shown in Figure 12. A damping value of Q = 10 was used to calculate the response spectrums in this paper. 22 3.2 Effect of Simulated Drops on Classic Shock Pulses As an alternative to complex shock inputs there are several shock pulses that test engineers use to simulate impacts or shock events. In this section, the effect of drops on these classical shock pulses is examined. The SRS is plotted with the complete time history in black, and the times at which the data drops occur are indicated by the vertical colored lines. These drops are at TE times 0.2, 0.4, 0.6, 0.8, and 1. The SRSs are plotted with insets of the time history with the same lines showing when the simulated telemetry drops occur. 3.2.1 Haversine Pulse The haversine function is equal to sin2(𝑥𝑥�2). It produces a SRS that is similar to the SRS of a sawtooth pulse or a half-sine pulse except that there is no rippling in the high frequency range. Figure 13 shows an acceleration time history of a haversine impulse with an amplitude of 1g and a duration of 0.5 sec. The vertical lines in blue, red, green, orange, and purple indicate where simulated telemetry drops happen at 0.2, 0.4, 0.6, 0.8, and 1 times TE. The same colors represent the SRS calculated from the dropped time histories. Figure 14 shows the shock response spectra from the original time history and the dropped time history. Only the drops that occurred after the peak acceleration have a worst case error less than 20%. The systems with a natural frequency much higher than the fundamental frequency of the input show a maximum acceleration amplitude equal to the maximum acceleration of the base excitation because they are much stiffer. The time histories that dropped out before the peak generate SRS very similar to the response spectra from a sawtooth input. 23 3.2.2 Half-Sine Pulse Figure 15 is the acceleration time history from a half-sine impulse. It has a magnitude of 1g, and a duration of 0.5 sec. It is called a half-sine because it is one half of a sine wave cycle, and it is an impulse because it is a relatively short application of acceleration that results in a change in velocity. Figure 16 shows the shock response spectra from the original time history and the dropped time history of the half-sine impulse. The SRS generated from a half-sine shock pulse is similar to the haversine pulse SRS, except that the high frequency responses vary more around the maximum input acceleration. The system with the greatest acceleration has a resonance with the input frequency. 3.2.3 Sawtooth Pulse Figure 17 is the acceleration time history from a sawtooth impulse. It is so-called because the acceleration time history resembles the tooth in a sawblade. It has a magnitude of 1g, and a duration of 0.5 sec. Figure 18 is the SRS of the sawtooth shock pulse. The systems with natural frequencies between 10 and 11 Hz, were not amplified much over the highest acceleration of the base input. The SRS of the time histories with drops have the same fundamental shape, but are shifted to higher frequencies and lower magnitudes. The dropped time histories have the same shape as a quicker, lower acceleration sawtooth impulse. 24 3.3 Effect of Drops on Simulated Shocks to Multi-Degree-of-Freedom Systems The SRSs are calculated from the dropped and original time histories taken during aerospace tests. These time histories were measured using accelerometers mounted to aerospace vehicles. The SRSs from the original time histories will show examples of actual vibration environments encountered during flight. The SRSs of time histories with simulated data drops will be shown to compare the relative responses. 3.4 Effect of Drops on Shock Data This section will focus on the differences in shock response spectrums calculated from pristine time histories and those calculated from time histories with simulated data drops. These shocks were measured from aeronautical and astronautical tests including the mechanical separation of a weapon from a launch vehicle, stage separations, and rocket motor ignition transients. 3.4.1 Complex Shock Environments Figure 19 shows the first shock event investigated, and is referred to as “2000g” shock. All of the shock time histories from tests are from Tom Irvine’s public pages [6]. It is typical of a pyrotechnically induced shock event. The drops occur at TE times 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, and 1. Several different strategies were employed to mitigate the effect of drops. In this first strategy, the data from both ends of the telemetry drop is joined; no effort is made to replace the missing frames. This technique simply “concatenates” the data on both ends. 25 The data points on both sides of the telemetry drop are joined by a straight line. This required a small change in the standard algorithm for calculating SRS. In most codes, the sampling rate T is found by taking the average of the differences between time steps. However, if there are missing samples then the time step used in equations (9) and (10) will be longer than the actual sample rate. This leads to a shift in the SRS curve to lower frequency responses. A more accurate method for finding the time step T is to find the mode of the differences between samples. Figure 20 shows that concatenating the time history around the gap under-predicts the shock levels. This is particularly evident for the range of systems from 10-100 Hz. The next strategy is to fill the missing data with samples of zero acceleration. The missing time steps are filled in with the same sampling rate as the rest of the time history, at an acceleration level of zero. Figure 21 shows the SRS generated using this technique. A comparison of the SRSs from Figure 20 and Figure 21 shows no discernable difference. The last strategy is to fill the missing data with samples of the last acceleration level recorded. Figure 22 shows the SRS calculated from a time history with missing data filled in with the last sample. The differences in the SRSs in Figure 22 show a considerable amount of error in all responses below 300 Hz. This is a severe over-prediction of the acceleration levels for these systems. If the response spectrum is calculated from a dropped time history and it overpredicts then components tested to this environment will fail in an invalid manner. All of the response spectra for dropped time history will be calculated by concatenating the data unless otherwise stated. 26 3.4.2 Reentry Vehicle Separation Shock The mechanical separation of a reentry vehicle from the launch vehicle is the source of significant structural shock. In Figure 23 the transient from such a procedure can be seen with data drops at TE times 0.2, 0.4, 0.6, 0.8, and 1.0. The SRS in Figure 24 shows little error in the responses that are 100 Hz and higher. The responses below 200 Hz are slightly underpredicted. It is clear the responses in the lower frequency range are driven by more cycles at a lower acceleration level and take longer than TE to build. To reach the maximum responses the systems with lower natural frequencies needed to be driven at resonance for more cycles. 3.4.3 Solid Rocket Motor Ignition Shock Rockets also experience shock during ignition. Figure 25 shows the transient from a S-19 solid rocket motor [6]. This time history contains very little high frequency data when compared to the previous shock events. Figure 26 shows the SRS of the ignition. The data that dropped out before 0.4*TE shows significant underpredictions for most systems. The systems with a natural frequency from 50 to 60 Hz shows the greatest responses, and data that was dropped before 0.6*TE shows diminished acceleration. The higher frequency systems have a maximum acceleration response of about 20g. This is the highest acceleration of the base input. 3.4.4 1000g Shock Figure 27 shows the acceleration time history of the “1000g” shock [6]. Figure 28 shows SRS of the “1000g” shock time history and the dropped time histories. The drop at 27 20 percent of TE is before the peak acceleration of the shock, and the large difference between the 0.2 response spectrum and the pristine spectrum is evident. The dropped spectrums all under predict the responses from 5 to 200 Hz suggesting the low frequency systems reach their maximum acceleration well after TE. The results are sometimes counterintuitive. In Figure 28, the drop at 0.8 TE has better predictions in the 2-90 Hz range than the drop at TE. The SRS calculated from time histories that have telemetry drops are sensitive to the exact moment the data stream drops out. 3.4.5 2000g Shock Figure 29 shows the acceleration time history of the “2000g” shock [6]. The “2000g” shock is the same example used in section 3.4.1. Figure 30 shows SRS of the “2000g” shock time history, and the dropped time histories. Again, the drop at 20% of TE is before the peak acceleration of the shock, and there is a large difference between the 0.2 response spectrum and the pristine spectrum. All of the dropped spectrums under-predict the responses from 3 Hz to 200 Hz. The systems with the highest responses were those with natural frequencies from 2000 to 10000 Hz. The time histories that dropped out before 0.6*TE under predict the responses for these systems. 3.4.6 T Flight Shock Figure 31 shows the acceleration time history of the “T Flight” shock [19]. Figure 32 shows SRSs of the “T Flight” shock time history and the dropped time histories. The drop at 20 % of TE is after the peak acceleration of the shock. The systems with the highest 28 responses were those with natural frequencies around 300 Hz. The time histories that dropped out before 0.6*TE underpredict the responses for these systems. The SRSs calculated from the time histories that dropped at 0.2, 0.6, and 0.8 time TE overpredict the low frequency responses from 0.5 to 70 Hz. This is an unexpected result that warrants further investigation. Figure 33 shows the response of a SDoF system with a 10 Hz natural frequency to the “T Flight” shock with and without the mean removed. Additionally, the response to the same shock with a simulated telemetry drop at 0.2*TE is shown. The responses to the dropped time histories suggest that the removal of the mean from dropped time histories is not sufficient to remove all instrumentation error. Because the drop time history only has a zero offset for a short time, the mean acceleration level is less. A high pass digital filter may be a better choice for removing accelerometer offsets. 3.5 Effect of Drops on Simulated Shocks To provide further examples this section will show what effect dropped data has on simulated shocks. Shock events were simulated on systems built two different ways. The first example is a discretized 12 degree system. In the second example a finite element model of a pipe impact will be simulated. 3.5.1 Multi-Degree-of-Freedom Systems Consider a missile or rocket as a beam that can be approximated as a series of discretized mass-spring-dampers. An impact is simulated by giving a mass an initial condition with no displacement but a high velocity. The initial conditions as such would 29 be: 𝑉𝑉 0 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = � ⋮ � , 𝑥𝑥̇ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = � ⋮ � 0 0 (17) The entire system is simulated. If a node on the opposite end of the system is assumed to contain a shock sensitive component of interest, and that component itself is considered in the lumped mass of that node, then the acceleration of the node during the transient can be used to calculate a shock response spectrum. Figure 34 shows such a system. The shock shown in Figure 35 is from a 12-DoF system. It has a TE of about 30 ms. It is a linear system, and the low acceleration levels are an artifact from the initial velocity. Table 1 lists the natural frequencies of the 12-DoF system. Figure 36 shows the SRSs of the simulated shock time history and the dropped time histories. All of the SRSs calculated form dropped time histories show underprediction for the low frequency responses from 0.1 to 8 Hz. With the exception of the SRS that dropped at 0.2 TE all of the other dropped time histories correctly predict the responses for these higher frequency systems. None of the other dropped time histories lost the maximum acceleration from the shock event. The maximum responses of systems with higher natural frequencies are driven by the high frequency base inputs that are present early in the shock event. The responses of systems with lower natural frequencies are driven by frequency components in the base input which need longer shock time histories than the telemetry drops allow. 30 3.5.2 Commercial Finite Element Model Many FEM software suites can be used to model impacts. If there is a location on the model that has sensitive components the acceleration time history can be used to calculate the shock response spectrum. The Abaqus FEA software package was used to simulate the impact of two steel pipes, shown in Figure 37. The node presented is on the far end of the top pipe along the x-coordinate. This pipe was given an initial rotational velocity, and therefore the pipe only had rotational accelerations. This was converted into a translational acceleration time history as shown in Figure 38. Figure 39 shows SRS of the pipe impact and the dropped time SRSs. All SRSs calculated from dropped time histories show under-prediction for the low frequency responses from 0.1 to 300 Hz. The systems with the highest responses were those with natural frequencies from 20,000 to 50,000 Hz. Only the drops that occur after 0.4*TE correctly predict the responses for these high frequency systems. The correct prediction of this peak is essential for establishing the MPE. This trend suggests that any telemetry drops that occur before 0.4*TE time will render the time history unusable for the calculation of a SRS. 31 Figure 12: The shock transient action time, TE. It is defined as the period between the start of the event to when the transient decays to 1/3 of its peak value. It does not include the entire transient. The red bars mark the duration of TE. 32 Figure 13: A 0.5 sec haversine pulse with data drops at different percentages of pulse duration. The haversine pulse is one type of classical impulse used instead of a complex shock transient. Several telemetry drops were simulated. The times of the drops is indicated by the colored vertical lines. 33 Figure 14: Shock response spectrum of a 0.5 sec, 1g haversine pulse. The SRSs calculated from the time histories with simulated telemetry drops are shown in the same colors used on the time history plot. 34 Figure 15: The acceleration time history of a 0.5 sec duration half-sine pulse. It is a more commonly used shock impulse. 35 Figure 16: SRS of a 0.5 sec duration half-sine pulse. The drops that occur at later times produce a SRS closer to the original time history. All of the drops shifted the frequency of the systems showing the greatest response. 36 Figure 17: Acceleration Time History of a 0.5 sec duration, 1g sawtooth pulse. 37 Figure 18: SRS of a 0.5 sec duration unit sawtooth impulse. The time histories with simulated drops shift the SRS curve to a higher frequency and a lower peak. 38 Figure 19: Acceleration time history of the “2000g” shock event showing data drops at different times. It has a TE of about 10 ms. 39 Figure 20: Shock response spectrum with drops that are concatenated. The time history on both sides of the telemetry drop are connected with no attempt to smooth the transition. 40 Figure 21: SRS with drops filled in with zeros. The missing time steps from the telemetry drop are resampled, and the accelerometer values for these steps are zero. 41 Figure 22: SRS with drops filled in at the acceleration level of last sample. It produces a significant amount of error when compared to the concatenate and zero acceleration methods. 42 Figure 23: Shock time history of a reentry vehicle separation with drop times. The telemetry drops were simulated at TE times 0.2, 0.4, 0.6, 0.8, and 1.0. This shock has a TE of about 3 ms. 43 Figure 24: SRS of an RV separation with drops. Significant underpredictions occur for all dropped time histories for systems with Fn between 3-90 Hz. 44 Figure 25: Solid rocket motor ignition acceleration time history. It has a TE of about 60 ms. 45 Figure 26: SRS of a solid rocket motor ignition with drops. All of the dropped time SRSs are in good agreement with the original time SRS except the drops at TE time 0.2 and 0.4 which underpredict the resonant response. 46 Figure 27: Acceleration time history of the “1000g” Shock. TE is approximately 18 ms. 47 Figure 28: SRS of the “1000g” Shock. Significant under-predictions occur for all dropped time histories. This is especially evident in systems with Fn between 5-200 Hz. 48 Figure 29: Acceleration time history of the “2000g” shock with drops. TE is about 10 ms, and the maximum acceleration of the base input is about 900g. 49 Figure 30: The SRS of the “2000g” Shock with drops. All of the telemetry drops underpredict the responses with natural frequencies between 3 and 300 Hz. 50 Figure 31: Acceleration time history of the “T Flight” shock with drops. It has a TE of about 10 ms. 51 Figure 32: The SRS of the “T Flight” shock with drops. Interestingly the responses at 0.2, 0.6, 0.8 time TE overpredict the responses that have Fn under 60 Hz. 52 Figure 33: The responses of a SDoF system with Fn = 10 Hz to the “T Flight” shock, and SDoF responses to a drop at 0.2*TE. The large amount of high-frequency energy is evident. The overprediction of the dropped time history only occurs when the zero-shift from the original signal is removed. Subtracting the mean from the dropped time history of the base input does little to remove the error associated with the zero-shift. Figure 34: A discretized, dynamic model of a rocket. One mass is given an initial velocity, as if a linear charge had been applied. The acceleration of a mass on the far end is collected, and the time history is used to calculate a SRS, as if a component were mounted to it. 53 Figure 35: A simulated shock to a 12-DoF system, as measured on node 12. This shock has a TE of about 30 ms. 54 Table 1: The natural frequencies of the 12-DoF System. Mode ωn (radians/sec) Fn (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 561.6 1675.9 2763.9 3808.2 4792.5 5701.2 6520.0 7236.0 7837.8 8316.1 8663.2 8873.6 89.3 266.7 439.8 606.1 762.7 907.3 1037.6 1151.6 1247.4 1323.5 1378.7 1412.2 55 Figure 36: The SRS of the 12-DoF Shock with drops. The original time SRS has peaks at the natural frequencies of the parent structure. The dropped time SRS under-predicts the responses with natural frequencies below 8 Hz. The drop that occurred at 0.2*TE also under-predicts the response at 90 Hz. 56 Figure 37: Finite element simulation of two pipes impacting. The pipe aligned with the xaxis was given an initial rotational velocity about the z-axis. The acceleration was measured on the end opposite of the rotational constraint. Symmetry was used to reduce the computational resources used for the simulation. 57 Figure 38: Pipe impact shock time history with drops. The shock produced by the impact has a TE of about 2 ms. 58 Figure 39: The SRS of the pipe impact with drops. The drops cause a significant underprediction for systems with a natural frequency below 300 Hz. The drops at 0.2 and 0.4 times TE also underpredict the higher frequency systems. 59 CHAPTER 4 ADVANCED COMPENSATION STRATEGIES Strategies that can be employed to reduce the errors associated with missing data include rebuilding the shock time history from Fourier coefficients, studying similar shock events, and looking at the SRS from a time history perspective. As discussed in section 3.4.1, concatenating the data on both sides of the telemetry drop is the simplest method of reducing the error in a SRS when using dropped data. In the next section, a few of the more sophisticated techniques are explored. 4.1 Decomposing Shock Events Telemetry drops usually do not lose all of the data associated with a shock. The first few ms are preserved. If enough of the shock event is captured, the frequency content of the shock can be estimated using frequency domain techniques, and the damping can be found from similar events [20]. Figure 40 shows the DFT of the “1000g” shock along with the dropped time histories. The DFTs are shown in the same colors as drop lines in the time history. The DFT is a description of the frequency content during the shock event. An equivalent shock time history can be built using the Fourier coefficients even if 60 the DFT was found for a dropped time history. Assuming an exponential decay, the formula for a vibration transient is [7]: 𝑢𝑢̈ (𝑡𝑡) = 𝐴𝐴𝑒𝑒 −𝜁𝜁𝜁𝜁 sin(𝜔𝜔𝜔𝜔) (18) Using the vector of Fourier coefficients, A(𝜔𝜔), an equivalent shock time history can be built: 𝑛𝑛 𝑦𝑦̈ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ℎ (𝑡𝑡) = �𝑖𝑖=1�𝐴𝐴(𝜔𝜔𝑖𝑖 )𝑒𝑒 −𝜁𝜁𝜁𝜁 sin(𝜔𝜔𝑖𝑖 𝑡𝑡)� (19) A conservative value for the damping ratio is assumed as one percent for this case. To scale the rebuilt time history the ratio between max(𝑦𝑦̈ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ) and max(𝑦𝑦̈ 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 ) is found, and the time history produced from Equation (19) is multiplied by this ratio. Figure 41 is such a time history found from the DFT of the “1000g” time history that is dropped at 0.4*TE. The synthesized shock was not padded with preshock acceleration levels. In the interest of maintaining conservatism the maximum of the dropped time SRS and the rebuilt time SRS was used. Figure 42 shows this SRS, compared to the original time SRS. The error between the two is shown in Figure 43 and Figure 44. Equation (20) shows the how the error between the original time SRS and the dropped time SRS is calculated. The underprediction of the response for systems with natural frequencies between 1 and 70 Hz is the most concerning. If a component MPE was based on the rebuilt shock it would be under tested compared to the flight environment. More than 12 dB would need to be added to the SRS to account for the errors introduced by the telemetry drops. 61 Figure 45 shows the original time history of the RV separation shock, and the time history rebuilt from a drop at 0.4*TE. Figure 46 shows the SRS calculated from the original time history and the rebuilt time history. Figure 47 and Figure 48 show the error from using the rebuilt time history from the RV shock to calculate the SRS. The rebuilt SRS shows a large over-prediction for systems with natural frequencies between 90 and 1000 Hz. 4.2 Time of Maximum Amplitude SRS Figure 49 is a variation of the standard shock response that also plots the time that the maximum acceleration amplitude was reached for each SDoF system on the x-axis. Plotting the SRS in such a way has several advantages. First it shows what frequencies would be affected by a drop at a certain time. This is not useful if there are only shocks that have telemetry drops, but if the data available contains similar shocks this can help the cognizant engineer decide if a particular shock has enough data to be used. An additional use for this plot is to help the test engineer decide if two shock events have the same damage potential. Real systems have several modal frequencies. If two shock events have the same potential to damage a system then the time for each mode to reach its maximum response should be the same during each shock event. Using the SRS in Figure 50 as an example if a component has modal frequencies at 5 Hz and 40 Hz then each of these modes would reach their maximum response at about 91.48 sec. If the test shock event does not excite these two modes simultaneously then the two modal responses will not occur simultaneously, and the test shock event will not have been as severe as the in-flight environment. 62 4.3 Recommendations The errors between the dropped time SRS and the original time history SRS are calculated. The errors were calculated in percent error and in decibel. The formula for percent error is: 𝑆𝑆𝑆𝑆𝑆𝑆𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 −𝑆𝑆𝑆𝑆𝑆𝑆𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 � 𝑆𝑆𝑆𝑆𝑆𝑆𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 � ∗ 100% = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 (20) The formula for error in decibel is: 𝑆𝑆𝑆𝑆𝑆𝑆𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 20 log10 �𝑆𝑆𝑆𝑆𝑆𝑆 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 � = 𝑑𝑑𝑑𝑑 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 (21) Figure 51 and Figure 52 show the error between the dropped time SRS and the original SRS for the reentry vehicle shock. The drop at 0.2*TE under predicts for all frequencies except for those 200-600 Hz. The drop at TE is the next worst, but would be corrected if a margin of 15 dB was added. The drops at 0.4 and 0.8*TE have the least error and could be corrected with a margin of 12 dB. Figure 53 and Figure 54 show the error between the dropped time SRS and the original SRS for the solid rocket motor ignition. The dropped data over-predicts the responses for systems with natural frequencies below 10 Hz. The drop at 0.2*TE under predicts responses above 200 Hz. The remaining dropped SRSs could be rendered conservative with the addition of 6 dB or less. Figure 55 and Figure 56 show the error between the dropped time SRS and the original SRS for the “1000g” shock. The errors for the SRSs based on the 0.2, 0.4, 0.6, and 63 1 times TE renders them unusable. The drop at 0.8*TE would need a margin of more than 15 dB to account for the error. Figure 42 is the SRS of the original and rebuilt “1000g” shock. The rebuilt time history is based on a drop at 0.4*TE. The compensation technique reduced the error from 32 dB to 12 dB. Figure 57 and Figure 58 show the error between the dropped time SRS and the original SRS for the “2000g” shock. The errors for the SRSs based on the all of the dropped time SRSs renders them unusable. Figure 59 and Figure 60 show the error between the dropped time SRS and the original SRS for the “T Flight” shock. The dropped time histories generally over predict the responses. A maximum of 5 dB of margin would be needed depending on when the drops occur. Figure 61 and Figure 62 show the error between the dropped time SRS and the original SRS for the 12-DoF shock. All of the drops underpredict responses for systems with natural frequencies below 10 Hz. If the SRS did not include responses at these frequencies then it would not be necessary to add 6 dB to drops that happened before 0.4*TE, and 2 dB to drop that occur after 0.4*TE. Figure 63 and Figure 64 show the error between the dropped time SRS and the original SRS for the pipe impact. The drops that happened before 0.6*TE had significant underpredictions, but any drops that occurred later than that could be rendered useable with the addition of 6 dB of margin. 64 4.3.1 Conditions to Discard Shock Data Inspections of Figure 51 through Figure 64 do not reveal many universal rules to aid in the decision of when to discard corrupted shock data. Shock time histories where the data dropped before 0.4*TE occurs are not trustworthy. The availability of similar shock time histories without drops would be needed for further conclusions about the characteristics concerning the necessary margins as well as the frequency range of interest. 65 Figure 40: A discrete Fourier transform of the “1000g” shock with drops. 66 Figure 41: A rebuilt time history using the DFT coefficients from “1000g” and the original time history. The rebuilt time history had to be divided by about four to scale it correctly. 67 Figure 42: The SRSs from the original undropped “1000g” and the rebuilt time history. The rebuilt time history is the sum of a series of decaying sinusoids. The relative amplitudes of the decaying sines are based on the DFT of the dropped shock event. 68 Figure 43: The percent error between the original SRS and the rebuilt SRS from the “1000g” shock. 69 Figure 44: The decibels error between the original SRS and the rebuilt SRS from the “1000g” shock. More than 12 dB of margin are needed to compensate for the errors in the rebuilt time history. 70 Figure 45: Shock of the RV separation and the rebuilt time history from a DFT calculated from the 0.4*TE drop. Although there is very little visual similarity the SRS from the rebuilt time history is close. 71 Figure 46: The SRSs from the original un-dropped RV separation and the rebuilt time history. The rebuilt time history is the sum of a series of decaying sinusoids. The maximum of the dropped SRS and the rebuilt SRS is used to increase conservatism. 72 Figure 47: The percent error between the original SRS and the rebuilt SRS from the RV separation shock. The overprediction for systems with natural frequencies between 100 and 1000 Hz is significant. 73 Figure 48: The decibels error between the original SRS and the rebuilt SRS from the RV separation shock. Although there are no under-predictions below 6 dB the SRS would cause over-testing of components that have natural frequencies between 80 and 1500 Hz. 74 Figure 49: SRS of the “1000g” shock with the time of maximum amplitude. The top typical SRS. The bottom curve shows when the maximum response was reached. The time scale is linear. 75 Figure 50: SRS of the RV separation shock with the time of the maximum amplitude. For systems with natural frequencies above 100 Hz the maximum response is reached in the first 10 ms. 76 Figure 51: Percent error for reentry vehicle shock drops. 77 Figure 52: Decibel error for reentry vehicle shock drops. MPE’s based on the dropped SRS would under-test systems with natural frequencies below 100 Hz. The drop at 0.2*TE has a significant under prediction for systems with frequencies above 60 Hz. 78 Figure 53: Percent error for rocket ignition drops. The dropped SRSs overpredict the response of systems below 10 Hz. Depending on the frequency range of interest this may not be of concern. 79 Figure 54: The decibel error for rocket ignition drops. If the first frequency of interest is 10 Hz then an additional 6 dB of margin would account for the under-predictions with the exception of the drop at 0.2*TE. 80 Figure 55: Percent error for “1000g” shock drops 81 Figure 56: Decibel error for “1000g” shock drops. The significant under-prediction for systems with natural frequencies between 0.1 and 200 Hz would likely render any dropped time histories unusable. 82 Figure 57: Percent error for “2000g” shock drops 83 Figure 58: Decibel error for “2000g” shock drops. The significant underprediction for systems with natural frequencies between 0.2 and 100 Hz would likely render any dropped time histories unusable. 84 Figure 59: Percent error for “T Flight” shock drops. 85 Figure 60: Decibel error for “T Flight” shock drop. An additional 6 dB of margin would compensate for the nonconservative errors. This would increase the overprediction for systems with natural frequencies below 80 Hz. 86 Figure 61: Percent error for 12-DoF shock drops. There is significant overprediction of systems with natural frequencies below 10 Hz. This is not necessarily within the frequency range of interest. 87 Figure 62: Decibel error for 12-DoF shock drops. If the lowest frequency of interest is 10 Hz then the addition of a 6 dB margin would cover any drops after 0.2 time TE. 88 Figure 63: Percent error for pipe impact shock drops. The drops at 0.4*TE and beyond overpredict the responses below 40 Hz. All drops under-predict the response of systems near 100 Hz. 89 Figure 64: Decibel error for pipe impact shock drops. The drops at 0.2*TE and 0.4*TE cause wide-band underpredictions. For drops that occur after that time an additional 6 dB of margin would correct for the errors caused. 90 CHAPTER 5 CONCLUSION Incomplete acceleration time histories present a problem in spacecraft shock analysis. The missing data has the potential to cause over or underpredictions when calculating the SRS which can lead to components which are not properly qualified for flight. Through an empirical study of the error associated with incomplete time histories it is possible to develop strategies for minimizing the error. The overreaching conclusion from the findings presented is that the design of vibration data collection and telemetry systems should carefully consider how to avoid corrupting the data during both collection and transmittal. A simple one second delay before broadcasting to ground stations would have prevented the most key shock data from being lost. Failing this, loads engineers should study available uncorrupted shock time histories to estimate the action time, TE, frequency distribution, and time of maximum acceleration for each frequency range. Any time history that dropped out before 0.4 of TE should be discarded, and at least 6 dB should be added to the response spectrum after calculation to conservatively account for error. The use of rebuilt time histories would need to be evaluated against the particular characteristics of the shocks that require rebuilding. Since it is the maximum of the dropped 91 SRS and the rebuilt-time SRS the use of this technique can only decrease the negative margin. 92 REFERENCES [1] Irvine, T., “Shock & Vibration: 28. 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