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Six-Degree-of-Freedom Accelerations: Linear Arrays Compared with Angular Rate Sensors in Impact Events.

INTRODUCTION

Both linear and angular metrics are important to analyses of head injury mechanics. Linear and angular accelerations have been correlated with brain injury, and linear acceleration has been correlated with skull fracture [1, 2, 3, 4, 5, 6]. Many novel head injury metrics, such as BrIC, GAMBIT, HIP, PRHIC, Principle Component Score (PCS), RIC, RVCI, and [DELTA][[omega].sub.peak] require knowledge of angular terms (velocities and accelerations) [7-8]. Angular terms are also influential in determining linear acceleration injury metrics; as six-degree-of-freedom (6DOF) data is required to transfer data collected on the perimeter of an object (usually the head) to center of mass (COM) linear accelerations [9, 10, 11]. 6DOF data is most important when studying brain injury mechanisms and human tolerance to acceleration in situations with complicated kinematics where the activity under study is not amenable to high-speed video analysis [5, 7], such as helmet impacts.

To understand brain injury tolerances, analysis of dynamic loadings in real-world and testing environments must be investigated. Studies performed with human subjects often use angular rate sensors as part of 6DOF systems due to their smaller geometric size requirement, reduced channel count, and stability in long-duration measurements [12, 13, 14, 15, 16, 17, 18, 19]. It is increasingly common for researchers to perform ATD testing with angular rate sensors mounted in a headform either in replacement of a linear accelerometer array [20-21], or supplementing one [22, 23, 24, 25]. However, while data processing and filtering methods are standardized and straightforward when using a linear accelerometer array [26], how to handle angular rate sensor data is less established, especially for impact or other short-duration events [26-27].

Measuring Angular Velocity and Acceleration

There are a variety of ways to either measure or calculate angular velocity and angular acceleration. Theoretically, only six linear accelerometers are needed to solve the three linear and three angular terms, although in practice nine accelerometers are required because it provides improved error-correction and an algebraic means of calculating angular acceleration [27-28]. Two-dimensional (planar) solutions based on linear accelerometers have existed since the late 1960s and the three-dimensional solution since 1975 (Figure 1). also known as the nine-accelerometer-package (NAP) [10].

3-2-2-2 Linear Accelerometer Array (NAP)

Padgaonkar et al. described a nine-accelerometer array of linear accelerometers in a 3-2-2-2 configuration that offered matched pairs of accelerometers to cancel Coriolis components of angular accelerations [28]. This yielded angular acceleration Eqs (1-1) independent of angular velocity terms. In these equations, A represents linear acceleration, [rho] represents the offset distance from the center of mass, and [omega] represents angular velocity. The subscripts describe the axis and sensor position. Angular velocities were calculated as the numerical time integral of the angular accelerometer data. DiMasi provided an exhaustive analysis of the NAP implementation in standard Hybrid-III headforms, including how to handle non-coplanar sensors, alignment errors, and non-centroidal seismic mass locations for "COM" accelerometers [10].

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

The advantages of the 3-2-2-2 configuration were numerous versus the six-channel configurations, at the cost of increased channel count and size. The NAP configuration, due to the required numerical integration step to calculate angular velocity, is prone to cumulative integration error from bias, drift, misalignment, or vibration. Any measurement error will be propagated through all later times [27, 29].

The 3-2-1 or 3-1-1-1 Array

Although theoretically equivalent to the 3-2-1 configuration, in practice the 3-1-1-1 is not favored, as it offers no advantages over the 3-2-1 design in terms of channel count or mathematical rigor, but takes up more space by its three-dimensional nature. The 3-2-1 array is generally presented as a configuration consisting of a triaxial linear acceleration unit at the origin with a pair of y-axis and z-axis oriented accelerometers on an arm along the x-axis with a single z-axis oriented accelerometer located on an arm along the y-axis; the equations for angular acceleration for this configuration consist of Eqs 4-6 [27, 30, 31, 32].

[mathematical expression not reproducible] (4)

[mathematical expression not reproducible] (5)

[mathematical expression not reproducible] (6)

Eqs 4-6 demonstrate that the angular acceleration components require knowledge of the angular velocity terms. Although these differential equations can be solved numerically, the solution is unstable and because of errors in the experimentally measured accelerations, stepwise integration results in an accumulation of error in the angular accelerations that in turn propagate to the angular velocity terms [31-32]. The result is angular terms that rapidly lose accuracy, regardless of the how the six linear accelerometer channels are analyzed [27, 30, 31, 32].

Mixed Arrays of Linear Accelerometers and Angular Rate Sensors

Studies have also used arrays with more exotic combinations of linear accelerometers [33-34] or 6DOF systems using combinations of triaxial linear accelerometers and triaxial angular rate sensors. Such arrays have been favorably compared to the angular velocity and angular acceleration calculations of linear accelerometer arrays [23, 35, 36, 37]. They offer direct measurement of angular velocity terms and the advantage of reduced size and channel count, which is useful for studies involving sensors worn for more than a brief period of time. The primary difficulty with the use of angular rate sensors is the numerical differentiation required to calculate angular accelerations. Noise components are preferentially increased by numerical differentiation, and over-filtering is usually employed to compensate for this [38]. The error caused by differentiation is transient and applies only to each individual time step, as opposed to the persistent nature of integration errors.

The 3a[omega] Array

An increasingly common mixed array is a combination of three linear accelerometers and three angular rate sensors, often implemented as two triaxial arrays or one cluster with the three angular rate sensor channels mounted to their corresponding channel of a triaxial linear accelerometer. This array type has been shown to be effective for determining angular kinematics, including during impacts [35, 36, 37,39]. A six-channel (3a[omega]) combination of linear accelerometers and angular rate sensors is theoretically advantageous in that no such calculations are needed to directly measure six terms, and angular accelerations can be independently calculated as the time derivatives of the angular velocity terms [24, 27, 36, 40]. Because multiple sensor locations are not required, direct measurement of angular velocities is less susceptible to error from bias, drift, and misalignment.

The primary limitation of 3a[omega]-based arrays is that a numerical differentiation step is required to calculate the angular accelerations.

Theoretically, differentiation error does not accumulate over time as integration error does, and is less sensitive to drift and bias [27]. However, differentiation preferentially amplifies high-frequency noise and can lead to grossly inaccurate angular acceleration results [24, 27, 38]. In practice, numerical differentiation of inherently noisy experimental data is problematic, and requires care in the selection of filtering methods. Unfortunately, there is little standardization in how data should be filtered prior to differentiation, and methods in the literature vary considerably [13, 14, 19, 21, 36, 40].

The 6a[omega] Array

One means of avoiding both the numerical integration step required by the 3-2-2-2 linear array and the numerical differentiation step required by the 3a[omega] array is to construct a 6a[omega] array, which can take the form of a 3-2-1 array or a 3-1-1-1 array with an additional triaxial array of angular rate sensors. Kang et al. provided an analysis of a 3-1-1-1 array with the angular rate sensors mounted to each non-centroidal arm [25]. They found that the 6a[omega] array yielded angular acceleration results that were closer to that of the NAP than a 3a[omega] array, transformed peripheral linear accelerations to the COM better than the 3a[omega] array and at least as well as the NAP, yielded angular velocity data more accurate than the NAP, and provided angular velocity data which could be integrated into angular displacement data at higher accuracy than the NAP [24-25]. With sufficiently capable angular rate sensors, all channels of the 6a[omega] array can be filtered to CFC1000, which preserves time-integrity for transformation purposes [24].

The limitation of 6a[omega]-based arrays is that they are just as channel count and geometrically demanding as the 3-2-2-2 NAP, and the individual sensors are often slightly heavier than the linear accelerometers they replace. This can make the 6aco array difficult to accommodate in applications with limited space, and it is a more costly solution.

Angular Accelerometers

There is a paucity of data on the use of sensors which directly measure angular acceleration of the head. Angular accelerometers have mostly been used for modal analyses [42], and little is available regarding their efficacy in an automotive or biomechanical context. These sensors tend to be larger and with a lower frequency response than the linear accelerometers and angular rate sensors typically used. A CFC1000-capable angular accelerometer does exist (K-Shear[R] 8838/8840, Kistler Instrument Corporation, Amherst, NY), however it is approximately four times larger and six times more massive on a per-channel basis than a comparable angular rate sensor. A 3-3 array of linear and angular accelerometers (3a[alpha]) may offer no advantages over a 3-3 array of linear accelerometers and angular rate sensors (3a[omega]), because it merely trades the problems of integration for differentiation, and the time-history for integration accumulates error. A 3-3-3 array of linear accelerometers, angular rate sensors, and angular accelerometers is viable and would directly measure all nine quantities. However, this array offers no means of redundantly calculating linear accelerations, unlike the 3-2-2-2 or 6a[omega] arrays.

Filtering

For short-duration impact analyses, the numerical integration required to calculate angular velocity by arrays of linear accelerometers is generally preferred to the numerical differentiation required to calculate angular accelerations from angular velocities, as numerical integration acts as a low-pass filter to small errors in angular acceleration calculations. However, these integrations are stable for only a short period of time (on the order of seconds), before cumulative error from sources such as inaccurate zero-bias, sensor drift, and small mismatches between sets of accelerometers overwhelm the signal under analysis [27, 28, 29, 30, 32]. This is an obstacle to impact measurement in scenarios where there may be a series of discrete impacts, where there is significant non-negligible pre-impact motion, or where initial angular velocity is non-zero and of relevance. Examples of these scenarios would be occupant kinematics in rollovers, bicycle-automobile collisions, or forklift tipover or off-dock scenarios.

Short-duration (0-200 ms) Events

For short-duration events, linear arrays of accelerometers are customary and have been in wide use since the mid-1970s. Filtering for linear accelerometers is straightforward and defined by SAE J211 as a CFC1000 (1650 Hz cutoff) low-pass filter for head accelerations [26]. The state of the art is less defined for angular rate sensors. SAE J211 specifies no appropriate channel frequency class for differentiation of angular rate sensor data, and the footnote referenced defines a means of analyzing long-duration events [27].

Many authors have analyzed power spectral densities on a case-by-case basis, or simply resorted to an ad hoc cutoff of 300 Hz (sometimes described as CFC 180) [13, 14, 19, 21, 24]. Occasionally, higher cutoff frequencies are chosen [36 40-41]. Laughlin compared the angular accelerations from differentiated 1000 Hz magnetohydrodynamic (MHD) angular velocity data to calculated angular accelerations from a 3-2-2-2 linear accelerometer array and found the MHD data to be equal to or superior to the linear array data [40]. Martin et al. investigated MHD angular rate sensors versus a rotary potentiometer and 10000 fps high-speed film, and found 600 Hz to be an appropriate low-pass filter cutoff [36]. Marshall and Guenther compared multiple types of rate sensors to potentiometer data, a 3-2-2-2 linear accelerometer array, and high-speed video data, and found MHDs had acceptable results at CFC1000 for pendulum head-form drops with and without impact [35]. It is noteworthy that the ARS-1 MHD used in that study only had a frequency response up to 1000 Hz; a 1650 Hz low-pass filter would have been of limited use to that rate sensor. Voo et al. similarly compared an ATA DynaCube 3 MHD to high-speed video and a 3-2-2-2 linear accelerometer array, with all electronic data filtered at CFC1000. However, the authors found that the angular velocity data was limited to 100 Hz and the linear acceleration data was limited to 300 Hz, so lower cutoff frequencies could have been used [37], Kang et al. in an analysis of the 6a[omega] array, also processed the data as a 3a[omega] array. They found using low-pass filters of CFC60 (100 Hz) appropriate for low-severity impacts and CFC180 (300 Hz) appropriate for high-severity tests. CFC60 was found to overly reduce peak angular acceleration for high-severity impacts. In general, they found that the 3a[omega] array was either excessively noisy or excessively reduced in peak magnitude [24]. Funk et al., in a mouthpiece-based study, low-pass filtered all head kinematic data at CFC180. For differentiation into angular acceleration data, they filtered both prior to and after numerical differentiation [14], Camarillo et al. in a comparison to a mouthpiece accelerometer system, low-pass filtered both angular rate sensor and linear accelerometer data to CFC180 prior to transformation [13]. Suderman et al. analyzed impacts to an ATD wearing a hardhat, and, based on a power spectral density analysis, determined that a 100 Hz low-pass filter was appropriate for all tests in that series [21]. Lloyd and Conidi indicated they filtered COM linear accelerations to CFC1000 and COM angular rate sensor data via a phaseless 8th-order Butterworth low-pass filter with a 500 Hz cutoff. The data were numerically differentiated using a 5-point least-squares quartic equation [41]. For purposes of comparison to the NAP, Siegmund et al., filtered mouthpiece angular rate sensor data to 300 Hz, using a phaseless 4-pole Butterworth filter [19].

Long-Duration (>200 ms) Events

Bussone et al. found per-event appropriate low-pass filter cutoff frequencies for differentiating the angular rate sensor data into angular accelerations by using a residual analysis to find individual cutoff frequencies for the three center of mass (COM) linear accelerometer channels and the three angular rate sensor channels and taking the arithmetic mean of the six cutoffs. This was effective because residual curves were reasonably insensitive to small variations in the cutoff frequency in the band between linear and rotational cutoff frequencies. The angular rate sensors provided more accurate rotational velocities than integrated angular accelerations calculated from a 3-2-2-2 NAP, especially for events lasting longer than 200 ms [27].

Optimal Filtering

Filtering is a process that seeks to balance signal preservation and noise reduction. Any analysis of a real signal requires a compromise between the amount of distortion of true signal accepted and the amount of noise rejected. By necessity, as one reduces noise passed one also increases signal lost. In theory, a given angular velocity signal contains a maximum signal frequency and an optimal cutoff frequency which yields a subjectively noise-free velocity and acceleration which would allow a "true differentiation" of the whole signal [38]. A method used in kinematics analyses and employed in prior analyses of differentiation of angular velocities of the head is to self-analyze a given channel's frequency content and accept the point at which the lost signal and lost noise are equal, or an analysis of residuals [43]. Winter's method quantifies the difference between the filtered and unfiltered signals as a function of the filter cutoff frequency across a wide range of possible filter frequencies. The calculation of residuals is shown in Eq 7.

[mathematical expression not reproducible] (7)

where [X.sub.i] is raw data at the [i.sup.th] sample and [X.sub.i] is filtered data at the [i.sup.th] sample.

A theoretical signal containing only random noise would exhibit a residual plot consisting of a straight line decreasing from an intercept at 0 Hz to an intercept on the abscissa at the Nyquist frequency (half the sample rate). The magnitude of the intercept value would represent the root-mean-square (rms) value of the noise. When a signal consists of a true signal combined with noise, the residual will rise above the straight line as the cutoff frequency is reduced. This represents the signal distortion caused as the cutoff is reduced. By setting equal the amount of signal distorted with the amount of noise passed, a cutoff frequency can be chosen. This is found by projecting a horizontal line from the noise intercept until it intercepts the residual line. The index of the intersection is an optimal cutoff frequency. This is depicted in Figure 2. This method assumes noise is randomly distributed and white, and the true signal of interest has a maximum frequency below the Nyquist frequency. This method has been found to be effective for long-duration head accelerations generated by inertial head motions [27] because it tends to arrive at a solution range consistent with least-squares matches to reference solutions. It has not yet been validated for impact-based head kinematics.

A complication of filtering three-dimensional kinematics data is that all three axes channels must be filtered to the same cutoff frequency. Because low-pass filters shift data in the time domain as a function of order and frequency, to avoid inconsistent time-shifting of filtered data, all axes must be filtered to the same cutoff. Accordingly, choosing a universal cutoff requires balancing the filtering needs of each component channel. This is also a complication of integrating angular rate sensor data with linear accelerometer data, especially when the frequencies responses and filtering requirements are not equal.

Arrays of linear accelerometers, specifically the 3-2-2-2 configuration, are time-proven, standards-compliant, and validated for analysis of impact events. However, it is increasingly common for researchers to use angular rate sensors to capture 6DOF kinematics at reduced cost or with reduced channel counts. Less settled is an equivalent appropriate filtration for angular rate sensor data into angular acceleration data by numerical differentiation for impact events. The purpose of this study was to identify an SAE J211-compliant, a priori filtering method for angular velocity data in short-duration (impact) events that most closely matched angular acceleration data produced by linear arrays of accelerometers, and then to generalize that method to determine appropriate filters for angular rate sensor data in impact events. This was performed via a combination of physical testing and multibody simulation. Physical testing provided performance data in real-world environment that allowed the analyses of rate sensor data to be compared to an array of linear accelerometers. The multibody simulation provided data and kinematic results which were objectively correct and not dependent on an intermediate comparison to a reference standard, which while validated and widely-accepted, is not without limitation.

METHODS

ATD Testing

The head of a 50th percentile male Hybrid-Ill ATD (1846-D head, Denton ATD, Milan, Ohio) was instrumented with nine linear accelerometers (7264-2000, Endevco, San Juan Capistrano, California) in a 3-2-2-2 configuration in accordance with DiMasi [10]. The full scale range of the accelerometers was +/- 2000 Gs with a resolution of 0.061 Gs/bit (16-bit system). The ATD head was also instrumented with triaxial angular rate sensors (ARS-1500, Diversified Technical Systems, Seal Beach, California). Channel orientations were in accordance with the SAE J211-2014 standardized dummy head coordinate system: x-axis positive forward, y-axis positive right, z-axis positive down [26]. The full scale range of the angular rate sensors was +/- 1500 deg/sec, with a resolution of 0.732 deg/sec/bit (12-bit system). To record the data, the linear accelerometers were connected to a 16-bit data acquisition system (PicoDAS, EME Corporation, Arnold, Maryland). The rate sensors were connected to a 12-bit data acquisition system (NanoDAS, EME Corporation, Arnold, Maryland), as less resolution was necessary for the rate sensors due to the acceleration events using more of their dynamic range. The data acquisition systems were sampled at frequencies up to 20 kHz per channel.

The ATD was exposed to direct impact head motions with a rigid rubber-covered mallet about each axis for linear and angular motion by the following mechanisms:

1. The ATD was struck in the face at approximately the level of the COM.

2. The ATD was struck in the head from the right side at approximately the level of the COM.

3. The ATD was struck in the head from the right side more forcefully at approximately the level of the COM.

4. The ATD was struck near the vertex of the head from the right side.

5. The ATD was struck upwards at the chin.

6. The ATD was struck in the face from the right side at the level of the nose.

All ATD data processing was completed using GNU Octave (4.0.1). Digital filtering was done using phaseless, four-pole Butterworth low-pass filters, in accordance with SAE J211-2014 [26, 44-45]. Data were untrimmed and feature separate impacts with interval periods without motion. Linear accelerometer data were filtered to CFC600 to match the frequency response characteristics of the ARS-1500 rate sensor, in order to make a like-for-like comparison.

Differentiation of filtered angular velocity data into angular acceleration data was performed using three different numeric differentiation equations:

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

In these equations, f represents the value of the angular velocity data at a specific point in time, x represents the specific point in time, and h represents the difference in quantity between two adjacent points in time.

Angular rate sensor data was analyzed by sweeping cutoff frequencies from 1-1000 Hz to determine a single cutoff frequency that best matched the resultant on a sum-of-the-square basis and which best matched the peak value of the resultant. Data were also analyzed to determine cutoff frequencies that yielded least-square or peak value matches within [+ or -]5% of the optimum value.

NAP angular acceleration data was integrated into angular velocity data numerically using the trapezoidal method with unit spacing.

Data from the angular rate sensors were validated against the 3-2-2-2 NAP results. NAP angular accelerations were filtered to a CFC600 (1000 Hz) cutoff and numerically integrated and qualitatively and quantitatively compared to the measured angular velocities from the angular rate sensors.

Multibody Simulation

An ellipsoid model standing 50th percentile male Hybrid-Ill ATD in the reference position was struck in the head by an effectively rigid sphere with a diameter of approximately 7.4 cm using MADYMO (R7.5, TASS BV, Rijswijk, The Netherlands). Linear and angular kinematics for the COM of the head were tracked using kinematic time-history outputs. Linear acceleration outputs were also generated at locations corresponding to non-COM NAP accelerometer locations. The head was struck from the front and from the left side using two different mass-velocity combinations. These two mass-velocity combinations had the same initial kinetic energies, but the momenta were different. The four simulations were as follows:

1. The ellipsoid model was struck in the head from the front by a ball of mass 0.145 kg and a velocity of 26.8 m/s.

2. The ellipsoid model was struck in the head from the front by a ball of mass 1.308 kg and a velocity of 8.9 m/s.

3. The ellipsoid model was struck in the head from the left side by a ball of mass 0.145 kg and a velocity of 26.8 m/s.

4. The ellipsoid model was struck in the head from the left side by a ball of mass 1.308 kg and a velocity of 8.9 m/s.

Simulations were run with a multi-body integration time step of 0.01 ms. Output were written with a time interval equivalent to a sample rate of 20 kHz per channel.

All kinematic data processing was completed using GNU Octave (4.0.1). Digital filtering was done using phaseless, four-pole Butterworth low-pass filters, in accordance with SAE J211-2014 [26, 44-45]. Unmodified data was unprocessed and treated as the "true" solution. White noise was added numerically to linear accelerometer channel data in post-processing within the allowances of SAE J211-2014. White noise was added numerically to angular velocity channel data in post-processing consistent with the specifications for the ARS PRO-1500 (Diversified Technical Systems, Seal Beach, California) or ARS PRO-18K (Diversified Technical Systems, Seal Beach, California).

Linear accelerometer data were filtered to CFC1000. Angular rate sensor data was analyzed using residual analyses from 1-1650 Hz to determine a single cutoff frequency that best matched the resultant on a least-square basis and which best matched the peak value of the resultant. Differentiation of filtered angular velocity data into angular acceleration data was performed using two-point differentiation, Eq 8. Data were also analyzed to determine cutoff frequencies that yielded least-square or peak value matches within [+ or -]5% of the optimum value.

RESULTS

ATD Testing

For the six ATD tests, there existed no single cutoff frequency for the angular rate sensor data that matched NAP-calculated angular accelerations on both least-squares and peak-matching bases. Only for test 1 (ATD struck in the face at approximately the level of the COM) and test 4 (ATD struck near the vertex of the head from the right side) were the least-square and peak-match optimum cutoff frequency solutions comparable. An example of least-square versus peak-match cutoff performance is shown in Figure 3. Cutoff frequencies are described in Table 1. Maximum resultant linear accelerations and FI[C.sub.15] values are reported in the Appendix, in Table A1.

The specific equation used for numeric differentiation had less than a 1% effect on optimum cutoff frequency selection for five of the six tests, and less than a 3% effect for test 4. The difference in cutoff frequency selection was smaller for the choice of numeric differentiation equation than it was for least-square versus peak-match.

By contrast, the selection of cutoff frequency and cutoff frequency criterion had a large effect on the resulting angular accelerations (Tables 2a-b). By definition, the match-peak cutoff selection matched the NAP peak resultant magnitude, and for most tests reasonably matched the peak magnitude of the primary component axis. The least-square criterion poorly matched the peak resultant magnitude and the component axes magnitudes. Most often the result was an underestimation of the peak resultant magnitude and an overestimation of the component axes magnitudes. For tests 2 and 4, where resultant estimation was most accurate, the differences in performance between the least-square and match-peak criteria were not large. In short, in situations where the least-square criterion performed best, the match-peak criterion performed better. This performance was only better on a relative basis, however; even the best matches yielded component axis errors of almost 100% in magnitude.

The 1000 Hz cutoff frequency generated angular acceleration results that were essentially unusable. When angular rate sensor data was low-pass filtered to this cutoff and then numerically differentiated, it yielded resultant magnitudes that were two to three times larger than the NAP values and per-axis magnitudes that were an overestimate by two to three orders of magnitude.

The 300 Hz cutoff was an improvement over the 1000 Hz cutoff, but resembled a slightly less accurate version of the least-square criterion. Generally, the 300 Hz cutoff underestimated the peak resultant by a factor of two or less and overestimated non-primary component axes by a similar factor. However, the 300 Hz cutoff was very inaccurate for test 5, where it underestimated the magnitude of the resultant and of the dominant x-axis by nearly a factor of five.

Neither of the a priori cutoff selections for pre-filtering prior to numerical differentiation yielded acceptable results. Of the a posteriori methods, a peak-match criterion was more useful than a least-square criterion and yielded resultant and primary axis peak values that were essentially correct. Neither method yielded non-primary components of the correct magnitude. It is noteworthy that even peak match badly tracked the time-history of the angular accelerations found by the NAP, and while it matched the peak values adequately, it did not necessarily identify those values from the correct impact (Figure 4). While peak match is the best current method, how to find its cutoffs is non-trivial and not presently obvious. The Winter method used in prior studies [27] has not been found to arrive at peak-match cutoffs.

A related question is how one best filters the angular velocity data measured by an angular rate sensor. SAE J211-2014 [26] recommends CFC180 (300 Hz) for filtering of linear acceleration or angular velocity data prior to integration. It also recommends CFC180 for processing of vehicle control module angular rates. Thus, 300 Hz is a reasonable starting point for the analysis of filtering of angular velocity data. It is difficult to determine, without prior knowledge of a solution, whether a 300 Hz cutoff, a 1000 Hz cutoff, or some other cutoff is most effective for filtering directly measured angular velocity data (Table 3) In part, this is because drift substantially influences integrated NAP angular acceleration data. By the end of a 6-second recording, drift is the primary component of naive calculations. A simple detrend procedure, which eliminated the average slope of the directly integrated NAP data was able to recover the angular velocity values with varying success. More elaborate techniques are feasible, including baselining and detrending solely in the time window of the signal of interest, but these make assumptions about the nature of the angular velocity immediately preceding the impacts, which--for events such as automobile rollovers or off-dock events--may be non-negligible, unknown, or both. This difficulty in normalizing integrated angular acceleration data makes it difficult to provide a reference gold-standard solution by which to evaluate various selections of filter cutoffs for angular rate sensor data. In general, a 300 Hz filter of angular rate sensor data performed most closely to the integration of (de-biased and detrended) NAP data and exhibited less oscillation about impacts than 1000 Hz data (Figure 5).

While a 300 Hz cutoff frequency for filtering angular rate sensor data appeared subjectively better, this selection is not necessarily the best one. There may exist compelling reasons to select a 1000 Hz cutoff or some other frequency based on data processing needs, signal frequency analysis, or adherence to some other standard. A 300 Hz cutoff has become accepted [26] and commonplace, but at present, this selection remains arbitrary.

Multibody Simulation

Similarly to the AID tests, for the four multibody simulation tests, there existed no single cutoff frequency for the angular rate sensor data that matched noiseless MADYMO output angular accelerations on both least-squares and peak-matching bases. In a general sense, the least-square cutoffs were similar to CFC60 and the peak-matching cutoffs were similar to CFC600. An example of least-square versus peak-match cutoff performance is shown in Figure 6. Cutoff frequencies are described in Table 4. As in the ATD tests, the match-peak cutoff selection matched the MADYMO peak resultant magnitude, and for most tests reasonably matched the peak magnitude of the primary component axis. The least-square criterion poorly matched the peak resultant magnitude and the component axes magnitudes.

There was not a large difference in angular velocity traces regardless of filter cutoff chosen. The unfiltered NAP array, the NAP array filtered to CFCIOOO, the ARS with a 1500 rad/s noise amplitude filtered to 300 Hz, and the ARS with a 1500 rad/s noise amplitude filtered to CFC1000 performed similarly. Only the ARS with an 18,000 rad/s noise amplitude filtered to 300 Hz noticeably performed poorly. As noise was defined as a function of sensor range, the ARS array with the larger range also exhibited larger amounts of noise. For the ARS systems, sensor range had a larger effect than cutoff frequency. These are demonstrated in Table 5 and Figure 7.

As in the ATD tests, the selection of cutoff frequency and cutoff frequency criterion had a large effect on the resulting angular accelerations (Table 6, Figure 8).

Most often the result was a reasonably accurate estimate of overall kinematics, but an underestimation of the peak resultant magnitude and component axes magnitudes. Filtering to CFC1000 often reasonably estimated the magnitude of peaks, but at significant retained noise and a loss of correct tracking of the overall kinematics. The data corresponding to an 18,000 deg/s sensor was substantially noisier than that corresponding to a 1500 deg/s sensor. It is noteworthy that the peak resultant angular velocity for test 4--24.4 rad/s resultant and 22.7 rad/s in the x-axis--nearly saturated the 26.2 rad/s range of a 1500 deg/s sensor.

The results of multibody simulation corroborate the ATD testing finding that there exists no one cutoff frequency which simultaneously replicates both overall kinematics and peak kinematics. The 300 Hz filter often used for ARS data yielded accurate angular velocity data but angular accelerations which underestimated true values. A CFC1000 cutoff frequency also reasonably estimated angular velocities and, additionally, peak angular acceleration values, but retained significant noise and poorly tracked non-peak values. It is also noteworthy that the NAP array reasonably estimated the correct angular kinematics even without filtering. Maximum resultant linear accelerations and HI[C.sub.15] values are described in the Appendix, in Table A2.

DISCUSSION

Previous work on the problem of filtering angular velocity data prior to numeric differentiation into angular acceleration data for inertial head motions found that a residual analysis technique was effective in an a priori context - when the solution was not already known. However, that method assumes that an optimum solution does exist. The problem encountered in the current study is that at least for some impact scenarios, an optimum solution may not exist. An a priori solution method cannot be determined because no effective a posteriori solution exists. In the six impacts analyzed via ATD, there was no particular cutoff frequency that was effective for all events. There also existed no particular cutoff frequency that was effective for simultaneously preserving peak magnitudes and average signal magnitude. Even when a cutoff could be found that would preserve the correct value of the peak resultant magnitude, it did not accurately track the time-history of the angular acceleration data and in some cases, matched the correct peak value at the wrong impact event fFigure 4). Thus, at best, numeric differentiation of angular rate sensor data was only able to coincidentally match peak values of the angular accelerations as found by standard NAP techniques, regardless of empirical cutoff frequency selected.

This effect is not specific or particular to these six tests. Multibody simulation data with the controlled addition of white noise exhibits the same problem. This appears to be an inherent limitation to the application of numerical differentiation techniques in the presence of white noise in the measurement of short-duration (high-frequency) impact events. This is depicted in Figure 9, which plots an "angular velocity" signal consisting of solely white noise against the angular acceleration derived from it. While the angular velocity signal has a constant magnitude versus frequency spectrum, the magnitude of the differentiated angular acceleration signal increases with increasing frequency. This increasing magnitude as a function of increasing frequency is a consequence of numeric differentiation in general, and did not appear to be restricted to any specific equation used [38]. The net effect is that numeric differentiation techniques preferentially increase the magnitude of high frequency noise. As shown in this study, 4-pole phaseless Butterworth filters (as suggested by SAE J211-2014) are only capable of partially ameliorating this. More exotic curve-fitting techniques exist for smoothing data prior to differentiation, but were not analyzed here.

The effect of numeric differentiation is a complicated interaction between sensor range, sensor frequency response, signal content, noise content within the system, filter technique, and sample rate. Noise is increased by increased sensor range and sample rate. Thus, care must be taken when selecting a sample rate. Butterworth filters have traditionally been used for digital post-filtering for SAE J211-2014-compliance [44], as their flat pass-band and rolloff characteristics accommodated the required corridors. This has most often taken the form of the phaseless 4th-order filter, with Alem's correction to account for the effects of the double-filtration inherent to phaseless filtering on the attenuation at the corner frequency [44-45]. The 1995 version of J211 specified that post-recording digital filtration was to be performed only once; while this is no longer strictly required, filtering is still required to be performed prior to any non-algebraic operations (such as numeric differentiation or integration). Thus, in order to accommodate the CFC corridors and to satisfy the other requirements of SAE J211-2014, the most frequent solution is a single application of a 4-pole Butterworth filter prior to numeric operations. Higher-order Butterworth filters will under most conditions not satisfy the CFC corridors. While non-Butterworth solutions are allowed, the requirement to match the CFC corridors which Butterworth filters meet in turn prevents more exotic techniques to eliminate noise inserted by differentiation. In short, compliance with SAE J211-2014 means that the numeric differentiation required to convert angular rate sensor data into angular accelerations will create problematic noise.

It appears that angular rate sensor technology at present is not sufficiently noise-free to match NAP performance after numeric differentiation. This is not to say it is impossible for the state of the art of angular rate sensors to make up the gap. However, it would require angular rate sensors to exhibit baseline performance far beyond that of linear accelerometers. A review of Figure 9 demonstrates that differentiation of a 10 kHz signal introduces 30 dB of noise between 0 Hz and 1000 Hz. Therefore, an angular rate sensor designed for differentiation would have to be approximately 30 times less noisy than a comparable linear accelerometer in order to provide equivalent performance.

Situations where discrete impacts are separated by non-negligible amounts of time (more than at most a few tenths of a second) or where pre-impact kinematics are non-trivial represent worst-case scenarios for sensors arrays, for they present difficulties for both differentiation (noise) and integration (drift).

The most straightforward means of avoiding both the numerical integration step required by the 3-2-2-2 linear array (NAP) and the numerical differentiation step required by the 3a[omega] array is to construct a 6a[omega] array, which Kang et al. implemented as a 3-1-1-1 array with the angular rate sensors mounted to each non-centroidal arm [25]. This implementation yielded angular acceleration results that were closer to that of the NAP than a 3a[omega] array, transformed peripheral linear accelerations to the COM better than the 3a[omega] array and equivalently to the NAP, yielded angular velocity data more accurate than the NAP, and provided angular velocity data which could be integrated into angular displacement data at higher accuracy than the NAP [24-25]. With sufficiently capable angular rate sensors, all channels of the 6a[omega] array can be filtered to CFC1000, which preserves time-integrity for transformation purposes [24]. This is ideal when kinematic measurements are non-centroidal, such as when measuring living human beings, because data filtered to different cutoff frequencies are also time-shifted by different amounts, and transformations performed with mixed cutoff frequencies may yield temporally-inaccurate kinematics. The primary limitation of 6a[omega]-based arrays is size and channel count. It requires as many channels and as much physical space as the 3-2-2-2 NAP, and the individual sensors are often slightly heavier than the linear accelerometers they replace. This can make the 6a[omega] arrays difficult to accommodate in applications with limited space, and it is a more costly solution.

As long as the events can be analyzed as separate components and long periods of pre-impact or post-impact kinematics are not essential, a 3-2-2-2 NAP array is an acceptable solution and the manner traditionally used. Under these conditions, the increased fidelity of a 6a[omega] array may be less valuable than the decreased weight or cost of a NAP configuration.

Under most conditions, a 3a[omega] array will handle impact events poorly, due to the inherent noisiness of numerical differentiation and the present and foreseeable limitations of angular rate sensor technology. In general, for impacts, both a NAP array and a 6a[omega] array will provide a more robust solution than a 3a[omega] array. These should be used unless long-duration analyses are required and impact events will be of sufficiently low magnitude and low frequency [27].

In this study, physical testing could not be conducted with an ARS system capable of CFC1000 performance because of the type of ARS sensor used and in order to maintain consistency between data traces. Therefore, the linear accelerations had to be filtered to CFC600 for comparison purposes in this portion accordingly. However, the multibody simulation results, which were not constrained by instrumentation availability or capability, demonstrate that a CFC1000-capable ARS system would not have materially affected any of these conclusions.

SUMMARY/CONCLUSIONS

It was found that there was no specific cutoff frequency that consistently matched the angular accelerations measured by the 3-2-2-2 array, and the combination of a triaxial linear accelerometer and a triaxial rate sensor could not simultaneously match the 3-2-2-2 summed square and peak angular accelerations regardless of cutoff frequency. The differentiation step to convert angular velocity data into angular acceleration data was found to insert frequency-dependent noise which low-pass filtering could not adequately eliminate. While this may be overcome in the future if the noise floor for angular rate sensors can be constructed to many times less noise than contemporary linear accelerometers, this will continue to be a problem inherent to time differentiation. While angular rate sensors are essential for long-duration accelerometer recording and are effective for analyses requiring integration, care must be taken when they are used for impact analyses. For impact analyses, a 3-2-2-2 array (NAP) or a 6DOF array paired with an angular rate sensor, such as a 6a[omega] implementation, will generate results with lower noise, and should be used wherever possible. In general, for impacts, both a NAP array and a 6a[omega] array will provide a more robust solution than a 3a[omega] array. These should be used unless long-duration analyses are required and impact events will be of sufficiently low magnitude and low frequency.

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CONTACT INFORMATION

[Contact details for the main author should be included here. Details may include mailing address, email address, and/or telephone number (whichever is deemed appropriate).]

DEFINITIONS/ABBREVIATIONS

3a[omega] - Array of 3 linear accelerometers and 3 angular rate sensors

6a[omega] - Array of 6 linear accelerometers and 3 angular rate sensors

6DOF - Six-degrees-of-freedom

ARS - Angular rate sensor

BrIC - Brain Injury Criterion

CFC - Channel Frequency Class

COM - Center of mass

GAMBIT - Generalized Acceleration Model for Brain Injury Threshold

HIC - Head Injury Criterion

HIP - Head Injury Power

MHD - Magnetohydrodynamic

NAP - Nine accelerometer package (a specific implementation of the 3-2-2-2 array of linear accelerometers

PRHIC - Power Rotational Head Injury Criterion

PCS - Principle Component Score

RIC - Rotational Injury Criterion

RMS - Root-mean-square

RVCI - Rotational Velocity Change Index

SAE - Society of Automotive Engineers

APPENDIX

Select tables from paper body shown full-width in appendix.

Full-width figures
Table 2b. Positive and negative peak values for tests 1-6 as
determined by angular rate sensor array filtered to the listed cutoff
criteria and numerically differentiated by various means. ND methods
1-3 are specified according to Eqs 8-10.

                                   Rate sensor
                                   Least square
                           Angular acceleration (rad/[s.sup.2])
Test  ND Method     +X     +Y     +z     +R      -X      -Y      -Z  -R

          1       1643   3905   2217  12790   -2449  -12701   -2063   0
1         2       1652   3891   2193  12817   -2419  -12706   -2049   0
          3       1651   3895   2202  12796   -2421  -12686   -2043   0
          1       3719   2609   3291   5251   -2987   -1504   -4231   0
2         2       3727   2569   3289   5214   -3001   -1470   -4241   0
          3       3718   2571   3255   5178   -2986   -1472   -4241   0
          1       6735   4161   5077   9369   -6020   -2643   -9248   0
3         2       6821   4191   5039   9396   -5961   -2658   -9281   0
          3       6829   4214   5038   9378   -5977   -2636   -9263   0
          1      34484  16362  22664  36766  -32365  -14539  -23284   0
4         2      34693  16419  22567  36625  -31981  -15268  -23719   0
          3      34460  16318  22607  36406  -32147  -15289  -23717   0
          1       1205   1488    900   2019   -1281    -344   -1377   0
5         2       1197   1486    896   2016   -1271    -342   -1372   0
          3       1189   1481    890   2006   -1264    -338   -1361   0
          1       6193   2983   3148  11858   -3621   -5237   -9252   0
6         2       6153   2987   3195  11882   -3644   -5229   -9247   0
          3       6167   2984   3222  11904   -3621   -5224   -9264   0

                       Rate sensor
                       Match peak
               Angular acceleration (rad/[s.sup.1])
Test     +x     +Y     +Z     +R      -X      -Y      -Z  -R

       1900   4015   2420  13873   -2755  -13814   -2411   0
1      1907   4011   2399  13870   -2718  -13791   -2418   0
       1911   4020   2414  13871   -2730  -13793   -2418   0
      10184   9604   9584  14999   -5473   -7742   -8788   0
2      9987   9567   9573  14996   -5440   -7690   -8793   0
      10024   9618   9610  14997   -5448   -7799   -8809   0
      14228   9176  10625  18179   -8726   -5029   17704   0
3     13823   9071  10289  18159   -8956   -5163  -17256   0
      13784   9088  10260  18158   -9025   -5056  -17243   0
      37867  19159  27500  42028  -36174  -17741  -26585   0
4     38499  19047  27384  42051  -35789  -18435  -26773   0
      38319  19060  27513  42064  -36047  -18494  -26830   0
       2475   2699   3029   4742   -3905   -1993   -3617   0
5      2469   2707   3037   4750   -3899   -1981   -3643   0
       2483   2714   3045   4750   -3902   -2001   -3649   0
       9037   4567   7545  16416   -5516   -7267  -13368   0
6      9102   4573   7653  16396   -5436   -7336  -13635   0
       9127   4563   7630  16416   -5436   -7360  -13624   0

                          Rate sensor
                        300 Hz cutoff
              Angula acceleration (rad/[s.sup.2])
Test    +X  +V    +Z       +R  -X         -Y  -Z     -R

      1360  3710  1998  11449  -2109  -11330  -1660  0
1     1340  3698  1960  11423  -2080  -11282  -1656  0
      1356  3709  1988  11487  -2100  -11350  -1672  0
      4742  3587  4484   6914  -3477   -2314  -4671  0
2     4693  3491  4422   6790  -3447   -2205  -4666  0
      4770  3578  4478   6889  -3468   -2276  -4706  0
      4410  2559  3900   7072  -4618   -1455  -6954  0
3     4343  2505  3878   7056  -4607   -1436  -6952  0
      4398  2544  3909   7100  -1634   -1454  -6997  0
      8417  3442  5964   8502  -7449   -2473  -6588  0
4     8386  3405  5953   8404  -7321   -2464  -6544  0
      8500  3435  5986   8518  -7433   -2478  -6594  0
      2342  2534  2744   4347  -3518   -1686  -3257  0
5     2327  2520  2724   4310  -3466   -1649  -3251  0
      2349  2541  2755   4346  -3503   -1691  -3288  0
      5001  2348  1919   9722  -2829   -4170  -7783  0
6     4998  2348  1916   9678  -2828   -4169  -7751  0
      5029  2361  1955   9736  -2847   -4191  -7778  0

                       Rate sensor
                      1000 Hz cutoff
              Angula  acceleration (rad/[s.sup.2])
Test   +x       +V  +Z     +R     -X      -Y      -Z      -R

      14313  16173  20421  34020  -18450  -29683  -15469  0
1     12643  15444  18974  32477  -16364  -29326  -14936  0
      13780  16643  20424  33945  -17928  -30377  -15986  0
      43614  39988  34889  54605  -33475  -52555  -30913  0
2     41310  38588  33838  51052  -30718  -48202  -29767  0
      44151  39921  36401  55210  -33571  -51804  -31258  0
      33463  25178  34975  50549  -20881  -17636  -45382  0
3     31684  21274  30803  48272  -18024  -17438  -41582  0
      34654  23273  34021  53253  -19539  -19666  -45468  0
      49216  31149  46251  63979  -52166  -32024  -38740  0
4     47042  31006  43626  60104  -51401  -29278  -36049  0
      48429  33811  47788  64290  -55860  -31964  -38288  0
      40638  21364  47241  69722  -41074  -21556  -56800  0
5     40037  18393  43884  6362b  -34771  -20543  -55439  0
      44965  20311  47734  70115  -39029  -22175  -61048  0
      20013   7303  19470  33977  -13349  -13899  -23681  0
6     19048   6973  19385  30710  -12383  -12550  -22993  0
      20371   7394  20704  32996  -13515  -13344  -24818  0

Table 3. Cutoff frequencies for angular velocity data which best
matched least-square or peak-match selection criteria for

differentiation into angular accelerations. High and low correspond to
[+ or -]5% best match.

                   Rate sensor
                 1000 Hz cutoff
                 Angular velocity (rad/s)
Test    +X  +Y     +Z    +R  -X       -Y     -Z  -R

1      3.7  8.7   2.3  14.9  -2.2  -14.6   -4.9  0.0
2      5.8  7.6   5.3  13.1  -9.1   -9.2   -9.5  0.0
3      4.0  4.1   7.1   9.7  -5.6   -3.7   -9.4  0.0
4     14.9  8.8  10.4  15.6  -9.4   -6.6   -5.2  0.0
5      6.7  9.0   9.1  12.8  -4.2   -1.8   -6.4  0.0
6      8.8  4.2   6.3  15.6  -5.1   -6.6  -14.9  0.0

                       Rate sensor
                       300 Hz cutoff
                  Angular velocity (rad/s)
Test  +x   +Y   +Z     +R  -X    -Y       -Z  -R

1     1.2  8.6  1.6  13.9  -1.1  13.8   -1.6  0.0
2     2.7  1.5  3.0   8.1  -5.9  -0.9   -7.0  0.0
3     3.5  1.7  3.4   8.3  -5.6  -0.6   -7.9  0.0
4     4.6  3.1  5.6   8.4  -6.5  -1.4   -3.0  0.0
5     2.9  5.8  0.6   5.9  -1.3  -1.8   -3.6  0.0
6     6.5  4.1  6.3  13.8  -5.1  -5.3  -12.2  0.0

                   3-2-2-2 array
                   1000 Hz cutoff
                   Angular velocity (rad/s)
Test     +X    +Y    +Z     +R     -X     -Y     -Z  -R

1       0.0   4.9  19.0   25.1   -3.3  -20.3    0.0  0.0
2       0.0   0.0  15.3   17.4  -11.3   -5.4    0.0  0.0
3       7.1   1.9   0.0   25.5  -18.5   -1.7  -20.2  0.0
4     110.3   0.8  38.4  138.8  -10.5  -82.2    0.0  0.0
5       0.0   6.4   6.6    9.3   -6.4   -1.3    0.0  0.0
6      21.6  21.2   0.0   62.9    0.0    0.0  -58.6  0.0

                            3-2-2-2 array
                      1000 Hz cutoff, detrended
                       Angular velocity (rad/s)
Test    +X    +y    +Z    +R     -X     -Y     -Z  -R

1      0.6   8.7   1.5  15.3   -1.0  -15.2   -1.4  0.0
2      3.2   1.1   2.3   9.9   -6.9   -1.0   -9.5  0.0
3     11.3   1.6   2.7  12.7   -9.7   -1.9   -9.4  0.2
4     37.5  25.7  15.5  52.7  -46.0  -24.6   -5.9  2.7
5      2.8   6.0   1.1   6.4   -1.9   -1.7   -1.5  0.1
6      8.2   4.4   6.5  14.5   -5.1   -6.1  -12.1  0.0

Table 5. Positive and negative peak values for angular velocity for
tests 1-4 as determined by angular rate sensor arrays with noise added
and then filtered via the listed cutoff criteria.

                                      MADYMO output

                                 Angular velocity (rad/s)
Run  Description             +X  +Y   +Z     +R    -X     -Y    -Z  -R

1    Frontal impact         0.0  1.4  0.0   5.8   0.0   -5.8   0.0  0.0
2    Frontal impact (v39)   0.0  1.3  0.0  15.9   0.0  -15.9   0.0  0.0
3    Lateral Impact         9.7  1.2  4.5  10.5  -3.0   -0.3  -3.0  0.0
4    Lateral impact (v39)  22.7  2.2  9.3  24.4  -4.4   -0.7  -6.7  0.0

                  3-2-2-2 array
             noise added, unfiltered
             Angular velocity (rad/s)
Run    +x  +Y   +Z     +R    -X     -Y    -Z  -R

1     0.0  1.4  0.0   5.8   0.0   -5.8   0.0  0.0
2     0.0  1.3  0.0  15.9   0.0  -15.9   0.0  0.0
3     9.7  1.2  4.5  10.5  -3.0   -0.3  -3.0  0.0
4    22.7  2.2  9.3  24.3  -4.3   -0.7  -6.8  0.0

                     3-2-2-2 array
                noise added, CFC1000
                Angular velocity (rad/s)
Run    +x  +Y   +Z     +R    -X     -Y    -Z

1     0.0  1.4  0.0   5.8   0.0   -5.8   0.0
2     0.0  1.3  0.0  15.9   0.0  -15.9   0.0
3     9.7  1.2  4.5  10.5  -3.0   -0.3  -3.0
4    22.7  2.2  9.3  24.3  -4.3   -0.7  -6.8

                  Rate sensor
           1500 deg/s, 300 Hz filter
           Angular velocity (rad/s)
Run    +X  +Y   +z     +R  -X       -Y  -Z    -R

1     0.1  1.4  0.1   5.9  -0.1   -5.9  -0.1  0.0
2     0.1  1.3  0.1  16.0  -0.1  -16.0  -0.1  0.0
3     9.7  1.3  4.5  10.6  -3.1   -0.4  -3.1  0.0
4    22.8  2.2  9.4  24.4  -4.4   -0.7  -6.8  0.0

                         Rate sensor
                     18k deg/s, 300 Hz filter
                     Angular velocity (rad/s)
Run    +x  +Y     +Z    +R  -X       -Y  -Z    -R

1     0.8  2.1   0.8   6.7  -0.8   -6.6  -0.8  0.0
2     0.8  1.9   0.8  16.7  -0.8  -16.7  -0.8  0.0
3    10.4  2.0   5.0  11.5  -3.8   -1.1  -3.8  0.0
4    23.4  2.9  10.0  25.4  -5.0   -1.4  -7.5  0.1

                   Rate sensor
           1211-allowed noise, 300 Hz filter
              Angular velocity (rad/s)
Run    +X  +Y   +z     +R  -X       -Y  -Z    -R

1     0.3  1.6  0.3   6.0  -0.3   -6.0  -0.3  0.0
2     0.3  1.5  0.3  16.2  -0.3  -16.1  -0.3  0.0
3     9.9  1.4  4.7  10.9  -3.3   -0.6  -3.2  0.0
4    23.0  2.4  9.6  24.6  -4.6   -0.9  -7.0  0.0

                   Rate sensor
              1500 deg/s, CFC1000 filter
              Angular velocity (rad/s)
Run    +x  +Y   +Z   +R    -X       -Y  -Z    -R

1     0.0  1.4  0.0   5.8   0.0   -5.8   0.0  0.0
2     0.0  1.3  0.0  15.9  -0.1  -15.9  -0.1  0.0
3     9.7  1.2  4.5  10.5  -3.0   -0.4  -3.0  0.0
4    22.7  2.2  9.3  24.3  -4.4   -0.7  -6.8  0.0

Table 6. Positive and negative peak values for angular acceleration
for tests 1-4 as determined by angular rate sensor arrays with noise
added and then filtered to the listed cutoff criteria.



Run  Description

1    Frontal impact
2    Frontal impact (v39)
3    Lateral Impact
4    Lateral impact (v39)

            MADYMO output

            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R    -X     -Y    -Z  -R

1        0  282     0   2010     0  -2010     0   1
2        0  616     0   2540     0  -2540     0   0
3    12700  854  6260  14176  -272    -60  -453  20
4    11100  669  4940  12160  -685   -209  -712   9

                 3-2-2-2 array
            noise added, unfiltered
            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R    -X     -Y    -Z  -R

1        0  282     0   2053     0  -2053     0   1
2        1  617     0   2498    -1  -2498    -1   1
3    12687  857  5250  14165  -272    -61  -453  20
4    11079  669  4971  12119  -688   -210  -712   9

                  3-2-2-2 array
                  noise added, CFC1000
            Angular acceleration (rad/[s.sup.2])
Run     +x  +Y     +Z     +R    -X     -Y    -Z  -R

1        0  282     0   1960     0  -1960     0   1
2        1  617     0   2502     0  -2502     0   0
3    12657  856  6280  14140  -272    -60  -507  21
4    11078  668  4935  12124  -687   -209  -717   9

                  Rate sensor
            1500 deg/s, 300 Hz filter
            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R    -X     -Y    -Z  -R

1       28  290    28   1049   -40  -1049   -40   0
2       55  623    55   2323   -23  -2323   -24   0
3     6871  489  3250   7609  -292    -69  -419   0
4    10317  624  4589  11297  -697   -218  -764   0

                    Ratesensor
             18kdeg/s, 300 Hz filter
            Angular acceleration (rad/[s.sup.2])
Run     +X    +Y    +Z     +R     -X     -Y     -Z  -R

1      306   438   306   1075   -480  -1071   -480  0
2      493   838   493   2345   -286  -2344   -286  0
3     6954  1070  3337   7728   -554   -306   -601  0
4    10148   467  4433  11067  -1582  -1548  -1582  0

                     Ratesensor
            J211-allowed noise, 300 Hz filter
            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R  -X       -Y  -Z    -R

1       89  335    89   1080  -114  -1080  -114  0
2       90  630    89   2289  -243  -2289  -243  0
3     6892  510  3272   7639  -369   -132  -391  0
4    10312  634  4597  11290  -730   -339  -787  0

                  Ratesensor
            2500 deg/s, CFC1000 filter
            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R  -X       -Y    -Z  -R

1      359  547   359   1855  -390  -1848  -390  0
2      593  859   593   2693  -400  -2683  -401  0
3    12611  840  6182  14065  -690   -559  -629  0
4    11157  771  4989  12225  1062   -472  -853  0

                        Rate sensor
                     Least-sguare match
              Angular acceleration (rad/[s.sup.2])
Run    +X  +Y     +Z    +R    -X     -Y    -Z  -R

1      16  283    16   730   -21   -729   -21  0
2       3  558     3  1345   -32  -1345   -32  0
3    3654  286  1625  4005  -312    -71  -358  0
4    5084  279  2028  5471  -752   -229  -705  0

                         Ratesensor
                         Peakmatch
            Angular acceleration (rad/[s.sup.2])
Run     +X    +Y    +Z     +R     -X     -Y     -Z  -R

1      449   664   449   2010   -520  -1969   -520  0
2      456   896   455   2540   -541  -2538   -541  0
3    12677  1411  6268  14176  -1332  -1199  -1647  0
4    11076   743  5007  12160   -844   -562   -825  0

Table Al. Maximum resultant accelerations and HIC values for ATD tests
1-6. HIC36 values are the same as the reported HIC15 values, as all
durations were less than 15ms.

Test  Max Resultant (g)  HI[C.sub.15]  duration (ms)

1            79.6             37.9          1.2
2           174.5            247.5          0.9
3           181.3            206.0          0.8
4           279.0           1505.3          2.4
5           103.7             65.1          1.0
6           104.2             61.4          1.0

Table A2. Maximum resultant accelerations and HIC values for multibody
simulation tests 1-4. HIC36 values are the same as the reported HIC15
values, as all durations were less than 15ms.

Test  Max Resultant (g)  HI[C.sub.15]  duration (ms)

1           197.9           187.8           0.6
2           181.9           414.5           1.6
3           198.3           254.0           0.8
4           175.4           513.2           2.0


William R. Bussone

JP Research Inc.

Joseph Olberding and Michael Prange

Exponent Inc

doi:10.4271/2017-01-1465
Table 1. Cutoff frequencies for angular velocity data which best
matched least-square or peak-match selection criteria for
differentiation into angular accelerations. High and low correspond to
[+ or -]5% best match. Numerical differentiation (ND) methods 1-3 are
specified according to Eqs 8-10.

                 Least Square Match  Peak Match
                 Filter cutoff (Hz)  Filter cutoff (Hz)
Test  ND Method  Best  Low  High     Best  Low  High

          1      337   240  383      368   348  387
1         2      339   240  384      370   350  390
          3      336   239  381      367   347  387
          1      270   246  292      426   415  436
2         2      271   247  295      429   418  439
          3      269   246  293      425   414  435
          1      402   367  433      617   603  631
3         2      406   369  436      634   620  648
          3      401   365  431      617   604  630
          1      704   672  735      768   743  794
4         2      722   689  756      789   765  813
          3      701   670  731      764   742  786
          1      174   164  181      319   308  330
5         2      174   164  182      321   310  331
          3      173   164  181      319   308  329
          1      389   304  438      581   549  611
6         2      393   305  442      592   558  625
          3      389   303  437      579   546  610

Table 2a. Positive and negative peak values for tests 1-6 as
determined by 3-2-2-2 NAP array.

                      3-2-2-2 array
                      1000 Hz cutoff
               Angular acceleration (rad/[s.sup.2])
Test   +X     +Y    +Z     +R      -X      -Y      -Z    -R

1       401  3790    838  13859    -483  -13830    -432   0
2      8713  1407   1214  14966   -3495    -271  -12167   0
3     15905  1680    929  18183  -13350   -5754  -12282   0
4     34631  2323  13108  42051   -9907  -22329  -11894   0
5      2027  3461    828   4741   -4725   -1650   -2622   0
6     10264  3101    745  16405   -4218   -7386  -10633   0

Table 2b. Positive and negative peak values for tests 1-6 as
determined by angular rate sensor array filtered to the listed cutoff
criteria and numerically differentiated by various means. ND methods
1-3 are specified according to Eqs 8-10.

                                   Rate sensor
                                   Least square
                           Angular acceleration (rad/[s.sup.2])
Test  ND Method     +X     +Y     +z     +R      -X      -Y      -Z  -R

          1       1643   3905   2217  12790   -2449  -12701   -2063   0
1         2       1652   3891   2193  12817   -2419  -12706   -2049   0
          3       1651   3895   2202  12796   -2421  -12686   -2043   0
          1       3719   2609   3291   5251   -2987   -1504   -4231   0
2         2       3727   2569   3289   5214   -3001   -1470   -4241   0
          3       3718   2571   3255   5178   -2986   -1472   -4241   0
          1       6735   4161   5077   9369   -6020   -2643   -9248   0
3         2       6821   4191   5039   9396   -5961   -2658   -9281   0
          3       6829   4214   5038   9378   -5977   -2636   -9263   0
          1      34484  16362  22664  36766  -32365  -14539  -23284   0
4         2      34693  16419  22567  36625  -31981  -15268  -23719   0
          3      34460  16318  22607  36406  -32147  -15289  -23717   0
          1       1205   1488    900   2019   -1281    -344   -1377   0
5         2       1197   1486    896   2016   -1271    -342   -1372   0
          3       1189   1481    890   2006   -1264    -338   -1361   0
          1       6193   2983   3148  11858   -3621   -5237   -9252   0
6         2       6153   2987   3195  11882   -3644   -5229   -9247   0
          3       6167   2984   3222  11904   -3621   -5224   -9264   0

                       Rate sensor
                       Match peak
               Angular acceleration (rad/[s.sup.1])
Test     +x     +Y     +Z     +R      -X      -Y      -Z  -R

       1900   4015   2420  13873   -2755  -13814   -2411   0
1      1907   4011   2399  13870   -2718  -13791   -2418   0
       1911   4020   2414  13871   -2730  -13793   -2418   0
      10184   9604   9584  14999   -5473   -7742   -8788   0
2      9987   9567   9573  14996   -5440   -7690   -8793   0
      10024   9618   9610  14997   -5448   -7799   -8809   0
      14228   9176  10625  18179   -8726   -5029   17704   0
3     13823   9071  10289  18159   -8956   -5163  -17256   0
      13784   9088  10260  18158   -9025   -5056  -17243   0
      37867  19159  27500  42028  -36174  -17741  -26585   0
4     38499  19047  27384  42051  -35789  -18435  -26773   0
      38319  19060  27513  42064  -36047  -18494  -26830   0
       2475   2699   3029   4742   -3905   -1993   -3617   0
5      2469   2707   3037   4750   -3899   -1981   -3643   0
       2483   2714   3045   4750   -3902   -2001   -3649   0
       9037   4567   7545  16416   -5516   -7267  -13368   0
6      9102   4573   7653  16396   -5436   -7336  -13635   0
       9127   4563   7630  16416   -5436   -7360  -13624   0

                          Rate sensor
                        300 Hz cutoff
              Angula acceleration (rad/[s.sup.2])
Test    +X  +V    +Z       +R  -X         -Y  -Z     -R

      1360  3710  1998  11449  -2109  -11330  -1660  0
1     1340  3698  1960  11423  -2080  -11282  -1656  0
      1356  3709  1988  11487  -2100  -11350  -1672  0
      4742  3587  4484   6914  -3477   -2314  -4671  0
2     4693  3491  4422   6790  -3447   -2205  -4666  0
      4770  3578  4478   6889  -3468   -2276  -4706  0
      4410  2559  3900   7072  -4618   -1455  -6954  0
3     4343  2505  3878   7056  -4607   -1436  -6952  0
      4398  2544  3909   7100  -1634   -1454  -6997  0
      8417  3442  5964   8502  -7449   -2473  -6588  0
4     8386  3405  5953   8404  -7321   -2464  -6544  0
      8500  3435  5986   8518  -7433   -2478  -6594  0
      2342  2534  2744   4347  -3518   -1686  -3257  0
5     2327  2520  2724   4310  -3466   -1649  -3251  0
      2349  2541  2755   4346  -3503   -1691  -3288  0
      5001  2348  1919   9722  -2829   -4170  -7783  0
6     4998  2348  1916   9678  -2828   -4169  -7751  0
      5029  2361  1955   9736  -2847   -4191  -7778  0

                       Rate sensor
                      1000 Hz cutoff
              Angula  acceleration (rad/[s.sup.2])
Test   +x       +V  +Z     +R     -X      -Y      -Z      -R

      14313  16173  20421  34020  -18450  -29683  -15469  0
1     12643  15444  18974  32477  -16364  -29326  -14936  0
      13780  16643  20424  33945  -17928  -30377  -15986  0
      43614  39988  34889  54605  -33475  -52555  -30913  0
2     41310  38588  33838  51052  -30718  -48202  -29767  0
      44151  39921  36401  55210  -33571  -51804  -31258  0
      33463  25178  34975  50549  -20881  -17636  -45382  0
3     31684  21274  30803  48272  -18024  -17438  -41582  0
      34654  23273  34021  53253  -19539  -19666  -45468  0
      49216  31149  46251  63979  -52166  -32024  -38740  0
4     47042  31006  43626  60104  -51401  -29278  -36049  0
      48429  33811  47788  64290  -55860  -31964  -38288  0
      40638  21364  47241  69722  -41074  -21556  -56800  0
5     40037  18393  43884  6362b  -34771  -20543  -55439  0
      44965  20311  47734  70115  -39029  -22175  -61048  0
      20013   7303  19470  33977  -13349  -13899  -23681  0
6     19048   6973  19385  30710  -12383  -12550  -22993  0
      20371   7394  20704  32996  -13515  -13344  -24818  0

Table 3. Cutoff frequencies for angular velocity data which best
matched least-square or peak-match selection criteria for
differentiation into angular accelerations. High and low correspond to
[+ or -]5% best match.

                   Rate sensor
                 1000 Hz cutoff
                 Angular velocity (rad/s)
Test    +X  +Y     +Z    +R  -X       -Y     -Z  -R

1      3.7  8.7   2.3  14.9  -2.2  -14.6   -4.9  0.0
2      5.8  7.6   5.3  13.1  -9.1   -9.2   -9.5  0.0
3      4.0  4.1   7.1   9.7  -5.6   -3.7   -9.4  0.0
4     14.9  8.8  10.4  15.6  -9.4   -6.6   -5.2  0.0
5      6.7  9.0   9.1  12.8  -4.2   -1.8   -6.4  0.0
6      8.8  4.2   6.3  15.6  -5.1   -6.6  -14.9  0.0

                       Rate sensor
                       300 Hz cutoff
                  Angular velocity (rad/s)
Test  +x   +Y   +Z     +R  -X    -Y       -Z  -R

1     1.2  8.6  1.6  13.9  -1.1  13.8   -1.6  0.0
2     2.7  1.5  3.0   8.1  -5.9  -0.9   -7.0  0.0
3     3.5  1.7  3.4   8.3  -5.6  -0.6   -7.9  0.0
4     4.6  3.1  5.6   8.4  -6.5  -1.4   -3.0  0.0
5     2.9  5.8  0.6   5.9  -1.3  -1.8   -3.6  0.0
6     6.5  4.1  6.3  13.8  -5.1  -5.3  -12.2  0.0

                   3-2-2-2 array
                   1000 Hz cutoff
                   Angular velocity (rad/s)
Test     +X    +Y    +Z     +R     -X     -Y     -Z  -R

1       0.0   4.9  19.0   25.1   -3.3  -20.3    0.0  0.0
2       0.0   0.0  15.3   17.4  -11.3   -5.4    0.0  0.0
3       7.1   1.9   0.0   25.5  -18.5   -1.7  -20.2  0.0
4     110.3   0.8  38.4  138.8  -10.5  -82.2    0.0  0.0
5       0.0   6.4   6.6    9.3   -6.4   -1.3    0.0  0.0
6      21.6  21.2   0.0   62.9    0.0    0.0  -58.6  0.0

                            3-2-2-2 array
                      1000 Hz cutoff, detrended
                       Angular velocity (rad/s)
Test    +X    +y    +Z    +R     -X     -Y     -Z  -R

1      0.6   8.7   1.5  15.3   -1.0  -15.2   -1.4  0.0
2      3.2   1.1   2.3   9.9   -6.9   -1.0   -9.5  0.0
3     11.3   1.6   2.7  12.7   -9.7   -1.9   -9.4  0.2
4     37.5  25.7  15.5  52.7  -46.0  -24.6   -5.9  2.7
5      2.8   6.0   1.1   6.4   -1.9   -1.7   -1.5  0.1
6      8.2   4.4   6.5  14.5   -5.1   -6.1  -12.1  0.0

Table 4. Cutoff frequencies for MADYMO simulations for angular
velocity data which best matched least-square or peak-match selection
criteria for differentiation into angular accelerations.

                         Least Square Match      Peak Match
                         Filter cutoff (Hz)  Filter cutoff (Hz)
Test Description                Best                Best

Ml Frontal impact               129                 1009
M2 Frontal impact (v39)          48                  928
M3 Lateral Impact               178                 1889
M4 Lateral impact (v39)         111                  654

Table 5. Positive and negative peak values for angular velocity for
tests 1-4 as determined by angular rate sensor arrays with noise added
and then filtered via the listed cutoff criteria.

                                      MADYMO output

                                 Angular velocity (rad/s)
Run  Description             +X  +Y   +Z     +R    -X     -Y    -Z  -R

1    Frontal impact         0.0  1.4  0.0   5.8   0.0   -5.8   0.0  0.0
2    Frontal impact (v39)   0.0  1.3  0.0  15.9   0.0  -15.9   0.0  0.0
3    Lateral Impact         9.7  1.2  4.5  10.5  -3.0   -0.3  -3.0  0.0
4    Lateral impact (v39)  22.7  2.2  9.3  24.4  -4.4   -0.7  -6.7  0.0

                  3-2-2-2 array
             noise added, unfiltered
             Angular velocity (rad/s)
Run    +x  +Y   +Z     +R    -X     -Y    -Z  -R

1     0.0  1.4  0.0   5.8   0.0   -5.8   0.0  0.0
2     0.0  1.3  0.0  15.9   0.0  -15.9   0.0  0.0
3     9.7  1.2  4.5  10.5  -3.0   -0.3  -3.0  0.0
4    22.7  2.2  9.3  24.3  -4.3   -0.7  -6.8  0.0

                     3-2-2-2 array
                noise added, CFC1000
                Angular velocity (rad/s)
Run    +x  +Y   +Z     +R    -X     -Y    -Z

1     0.0  1.4  0.0   5.8   0.0   -5.8   0.0
2     0.0  1.3  0.0  15.9   0.0  -15.9   0.0
3     9.7  1.2  4.5  10.5  -3.0   -0.3  -3.0
4    22.7  2.2  9.3  24.3  -4.3   -0.7  -6.8

                  Rate sensor
           1500 deg/s, 300 Hz filter
           Angular velocity (rad/s)
Run    +X  +Y   +z     +R  -X       -Y  -Z    -R

1     0.1  1.4  0.1   5.9  -0.1   -5.9  -0.1  0.0
2     0.1  1.3  0.1  16.0  -0.1  -16.0  -0.1  0.0
3     9.7  1.3  4.5  10.6  -3.1   -0.4  -3.1  0.0
4    22.8  2.2  9.4  24.4  -4.4   -0.7  -6.8  0.0

                         Rate sensor
                     18k deg/s, 300 Hz filter
                     Angular velocity (rad/s)
Run    +x  +Y     +Z    +R  -X       -Y  -Z    -R

1     0.8  2.1   0.8   6.7  -0.8   -6.6  -0.8  0.0
2     0.8  1.9   0.8  16.7  -0.8  -16.7  -0.8  0.0
3    10.4  2.0   5.0  11.5  -3.8   -1.1  -3.8  0.0
4    23.4  2.9  10.0  25.4  -5.0   -1.4  -7.5  0.1

                   Rate sensor
           1211-allowed noise, 300 Hz filter
              Angular velocity (rad/s)
Run    +X  +Y   +z     +R  -X       -Y  -Z    -R

1     0.3  1.6  0.3   6.0  -0.3   -6.0  -0.3  0.0
2     0.3  1.5  0.3  16.2  -0.3  -16.1  -0.3  0.0
3     9.9  1.4  4.7  10.9  -3.3   -0.6  -3.2  0.0
4    23.0  2.4  9.6  24.6  -4.6   -0.9  -7.0  0.0

                   Rate sensor
              1500 deg/s, CFC1000 filter
              Angular velocity (rad/s)
Run    +x  +Y   +Z   +R    -X       -Y  -Z    -R

1     0.0  1.4  0.0   5.8   0.0   -5.8   0.0  0.0
2     0.0  1.3  0.0  15.9  -0.1  -15.9  -0.1  0.0
3     9.7  1.2  4.5  10.5  -3.0   -0.4  -3.0  0.0
4    22.7  2.2  9.3  24.3  -4.4   -0.7  -6.8  0.0

Table 6. Positive and negative peak values for angular acceleration
for tests 1-4 as determined by angular rate sensor arrays with noise
added and then filtered to the listed cutoff criteria.



Run  Description

1    Frontal impact
2    Frontal impact (v39)
3    Lateral Impact
4    Lateral impact (v39)

            MADYMO output

            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R    -X     -Y    -Z  -R

1        0  282     0   2010     0  -2010     0   1
2        0  616     0   2540     0  -2540     0   0
3    12700  854  6260  14176  -272    -60  -453  20
4    11100  669  4940  12160  -685   -209  -712   9

                 3-2-2-2 array
            noise added, unfiltered
            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R    -X     -Y    -Z  -R

1        0  282     0   2053     0  -2053     0   1
2        1  617     0   2498    -1  -2498    -1   1
3    12687  857  5250  14165  -272    -61  -453  20
4    11079  669  4971  12119  -688   -210  -712   9

                  3-2-2-2 array
                  noise added, CFC1000
            Angular acceleration (rad/[s.sup.2])
Run     +x  +Y     +Z     +R    -X     -Y    -Z  -R

1        0  282     0   1960     0  -1960     0   1
2        1  617     0   2502     0  -2502     0   0
3    12657  856  6280  14140  -272    -60  -507  21
4    11078  668  4935  12124  -687   -209  -717   9

                  Rate sensor
            1500 deg/s, 300 Hz filter
            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R    -X     -Y    -Z  -R

1       28  290    28   1049   -40  -1049   -40   0
2       55  623    55   2323   -23  -2323   -24   0
3     6871  489  3250   7609  -292    -69  -419   0
4    10317  624  4589  11297  -697   -218  -764   0

                    Ratesensor
             18kdeg/s, 300 Hz filter
            Angular acceleration (rad/[s.sup.2])
Run     +X    +Y    +Z     +R     -X     -Y     -Z  -R

1      306   438   306   1075   -480  -1071   -480  0
2      493   838   493   2345   -286  -2344   -286  0
3     6954  1070  3337   7728   -554   -306   -601  0
4    10148   467  4433  11067  -1582  -1548  -1582  0

                     Ratesensor
            J211-allowed noise, 300 Hz filter
            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R  -X       -Y  -Z    -R

1       89  335    89   1080  -114  -1080  -114  0
2       90  630    89   2289  -243  -2289  -243  0
3     6892  510  3272   7639  -369   -132  -391  0
4    10312  634  4597  11290  -730   -339  -787  0

                  Ratesensor
            2500 deg/s, CFC1000 filter
            Angular acceleration (rad/[s.sup.2])
Run     +X  +Y     +Z     +R  -X       -Y    -Z  -R

1      359  547   359   1855  -390  -1848  -390  0
2      593  859   593   2693  -400  -2683  -401  0
3    12611  840  6182  14065  -690   -559  -629  0
4    11157  771  4989  12225  1062   -472  -853  0

                        Rate sensor
                     Least-sguare match
              Angular acceleration (rad/[s.sup.2])
Run    +X  +Y     +Z    +R    -X     -Y    -Z  -R

1      16  283    16   730   -21   -729   -21  0
2       3  558     3  1345   -32  -1345   -32  0
3    3654  286  1625  4005  -312    -71  -358  0
4    5084  279  2028  5471  -752   -229  -705  0

                         Ratesensor
                         Peakmatch
            Angular acceleration (rad/[s.sup.2])
Run     +X    +Y    +Z     +R     -X     -Y     -Z  -R

1      449   664   449   2010   -520  -1969   -520  0
2      456   896   455   2540   -541  -2538   -541  0
3    12677  1411  6268  14176  -1332  -1199  -1647  0
4    11076   743  5007  12160   -844   -562   -825  0
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Author:Bussone, William R.; Olberding, Joseph; Prange, Michael
Publication:SAE International Journal of Transportation Safety
Article Type:Report
Date:Jul 1, 2017
Words:13936
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