Adaptive head impact protection via a rate-activated helmet suspension
Devon J. Spinelli a,b, Thomas A. Plaisted a, Eric D. Wetzel a,⁎a U.S. Army Research Laboratory, Materials and Manufacturing Science Division, Aberdeen Proving Ground, MD 21005, United Statesb Drexel University, Department of Materials Science and Engineering, Philadelphia 19104, PA, United States
H I G H L I G H T S
• A helmet suspension containing shearthickening fluid is designed and testedunder conditions representative ofhead impacts.
• The helmet suspension exhibits rate-sensitive behavior, increasing in resis-tance to extension as impact velocityincreases.
• Impact accelerations for the rate-sensitive suspension are ~50% lowerthan observed for a conventionalsuspension.
• Model calculations reveal ideal suspen-sion characteristics: a rate-sensitiveforce that is steady over largedisplacements.
G R A P H I C A L A B S T R A C T
a b s t r a c ta r t i c l e i n f o
Article history:Received 9 January 2018Received in revised form 6 April 2018Accepted 30 April 2018Available online 04 May 2018
The design of an adaptive helmet suspension system that provides optimized head protection under variable im-pact conditions is reported. The adaptive response is achieved through the use of rate activated tethers (RATs), aflexible strap-like material that uses shear thickening fluids to generate speed-sensitive extensional resistance.The RATs are integrated into a helmet by replacing the webbing system of a traditional construction hardhatwith a network of RATs, and performing impact attenuation testing over a range of velocities. Impacts to thecrown region of the helmet demonstrate a 50% reduction in peak acceleration experienced by the headformfor impact velocities between 1.5 and 3.5 m/s compared to the conventional webbing system, and comparableresponse to the conventional system at 4.5m/s. Complementary RAT extensional testing and low velocity helmetcompression tests confirm that the rate-sensitive response of the RATs contributes significantly to improved sys-tem performance. Additionally, calculations for suspension model systems show that the steady yield force ex-hibited by RATs over long strokes is a critical feature for minimizing head acceleration, and that the RATsuspension systems are achieving responses remarkably close to the ideal suspension response.
Headprotection is a vital requirement formilitary, sports, and indus-trial safety. Most helmets are primarily designed to avert fatality by
preventing severe injuries such as intracranial bleeding and skull frac-ture [1,2]. Helmet test standards, such as NOCSAE performance stan-dards [3], FMVSS 2218 [4], ASTM F1446-13 [5], and ANSI/ISEA Z89.1-2014 [6], were created to determine a helmet's ability to attenuate im-pact forces, where the acceptance criteria were derived from the obser-vation of cranial fracture in cadaveric studies [2]. Generally, thesestandards require mounting the helmet to an instrumented headform,
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subjecting the head and helmet to an impact, measuring translationalhead response, and then comparing head response to injury limits.Most helmet test standards report translational acceleration metrics,both peak value and time-integrated, which correlate well with likeli-hood of skull fracture [1,7]. Due to the emphasis on averting fatal inju-ries, these tests evaluate helmets under relatively high impactenergies where skull fracture is likely to occur for an unprotectedhead and seldom provide additional requirements for reducing headloads at lower impact energies. As a result, many helmets are designedwith relatively stiff webbed or foam suspension systems that are opti-mized for high energy impact, but provide little compliance duringlower energy events.
More recently, the medical community has recognized that concus-sion and repetitive brain trauma are also a serious health concern[8–10], leading to interest in designing helmets that also reduce thelikelihood and severity of brain injuries [11]. Translational accelerationhas been shown to correlate with concussion [12–15], although newerhypotheses propose that rotational loads more directly generate braintissue strains that are believed to induce axonal damage associatedwith long term diffuse brain injury [16–19]. In the present study wefocus on translational dynamics, consistent with most existing headprotection performance requirements. Studies have also suggestedthatmultiple impacts, even if each impact is at lowenergy, can cause cu-mulative effects that lead to serious injury [20–24]. Therefore, anemerging goal for head protection is tominimize head loads at all veloc-ities, rather than designing helmets to meet only maximum thresholdsat high energies.
Tominimize translational head accelerations at all impact velocities,an energy absorberwith uniquely tuned properties is required. First, theenergy absorber must yield at a near constant force over the full strokeof the energy absorber [25–28]. A constant resistance force induces con-stant head deceleration, which minimizes peak and time-averaged ac-celeration metrics when the translational body is brought to rest atthe point of maximum stroke. The stroke of the energy absorber is typ-ically limited by a fundamental geometric constraint such as the shell-to-skull gap in a helmet, and there is no penalty associated with usingthat full stroke during the impact event. Second, the energy absorbershould generate a resistance force that is proportional to impact energy,thereby minimizing acceleration by using the full stroke of the energyabsorber under all impact conditions. This requirement is met by de-signing an energy absorber that is compliant under lower velocity im-pacts, but becomes increasingly resistive to displacement as impactvelocity increases. This combination of properties – a velocity-sensitive yield force, which, upon yielding, displaces at a constantforce – is rarely found in energy absorbers, including those used in cur-rent helmet technologies.
The threemain components of a helmet are the shell, the suspensionsystem, and the retention system. The suspension system maintains astandoff distance between the shell and the skull, and provides the pri-marymeans of absorbing energy during impact. The twomost commonsuspension designs are a webbed suspension or compression pads. Themain advantage of a webbed design is improved thermal comfort, madepossible by the large air gap between the suspension and the helmetshell. Compression pads have proven to bemore efficient for energy ab-sorption and can provide a better helmet fit [29]. For these reasons, pad-ded suspensions are currently used inmost military and sports helmets.Compression pads include open cell and closed cell foams [30–32],pneumatic pads [33,34], and elastomeric trusses [35–38]. Closed cellpolystyrene foams are commonly used in bicycle and motorcycle hel-mets, but are only designed to withstand a single impact event andare then discarded. For most sports and military applications, suspen-sion energy absorbers (both webbed and pad systems) must retaintheir protective attributes over multiple impacts. Other suspension ap-proaches include slip layers to reduce rotational loads, and hybrid sys-tems that combine both webbing and compression pads into a singlesystem.
In the present study, we evaluate a new helmet webbing suspensiondesign, in which the web elements consist of “rate activated tethers”(RATs) [39,40]. RATs are flexible straps that exhibit a low force, elasticresponse at low velocities, but exhibit increasing resistance to extensionas extension rates increase. RATs consist of an outer elastic tube, withtwo enclosed ribbons, immersed in a shear thickening fluid (STF)(Fig. 1). The STF imbues speed sensitivity to the device, and consists ofcolloidal particles stabilized at high volume fraction in a carrier fluid.At low shear rates, the STF exhibits a flowable low viscosity state, butat high shear rates the STF becomes solid-like [41–43]. These uniqueproperties have been exploited for a range of energy absorbing applica-tions, including protective textiles [44–48], shock absorbers [49,50], andfoam pads [51–53]. For the RATs, the detailed mechanism of interactionbetween the STF and ribbons is uncertain, but it is likely that thetransitioned STF transfers load between ribbons through a combinationof viscous forces and particle-particle force chains [54–56]. IncreasingRAT extension rates lead to high shear rates between the ribbons,resulting in the aforementioned STF transition which drives an in-creased resistance to RAT extension. For typical RATs, a 10–100× in-crease in extension forces are possible with an increase in extensionrate of 10–100×. Factors such as RAT diameter, length, ribbon material,and STF composition can be selected to tunemechanical response to suitspecific application needs.
The objective of this study is to compare the impact response of ahelmet with a RAT-based suspension system to a conventional web-based suspension. First, details on the construction of the RATs, andtheir assembly into a helmet suspension, are provided. Then, mechani-cal and impact testing of RATs and RAT-based helmet suspensions areconducted and compared with a conventional helmet suspension sys-tem. A construction hard hat is used as a simple and low cost test plat-form, and testing follows the ANSI/ISEA Z89.1-2014 test standard.Impact results are then compared with analytical impact models toidentify key features of system response, and quantify the behavior ofthe helmet suspensions relative to ideal performance.
2. Experimental methods
2.1. Materials
The RAT assemblies utilized two commercially available STFs, STF-PO-52 and STF-PO-50 (STF Technologies, New Castle, DE). These fluidsconsist of precipitated calcium carbonate (PCC) particles with a meansize of 600 nm, suspended at 52 vol% and 50 vol%, in a paraffinic oil car-rier fluid. Fig. 2 shows viscosity (resistance to flow) as a function ofshear rate and shear stress, measured by an AR2000 (TA Instruments,NewCastle, DE) stress-controlled rheometerwith 40mmdiameter par-allel plates and a 1 mm gap. Both fluids exhibit steady shear thinning, areduction in viscosity as shear stress increases, at low shear rates. Forboth fluids, the viscosity reaches a minimum value at a “critical shearrate”, γc, as marked on Fig. 2a. The 0.52 volume fraction STF shows asharp rise in viscosity, or shear thickening, at higher shear rates. Addi-tionally, a similar viscosity rise would be expected at higher shearrates for the 0.50 volume fraction STF, but the rheometer is unable tooperate at shear rates higher than 100 s−1. The critical shear rate foreach fluid occurs at a common critical shear stress value, τc, of 100 Paas indicated in Fig. 2b. Two ascending and two descending shear ratesweeps were performed for each STF to confirm repeatability of theSTF formulations. Critical shear stress values over these four sweeps ex-hibited a coefficient of variation of 10% for the 0.50 volume fraction STF,and 11% for the 0.52 volume fraction STF.
RATs were constructed using elastomeric tubing, a polymer spacer,steel ribbons, heat shrink tubing, and epoxy (Fig. 3a). Viton tubes(10 mm inner diameter and 12 mm outer diameter, cut to 127-mmlengths; McMaster Carr, Robbinsville, NJ, product number 5102K15)were pre-conditioned in an oven at 100 °C for 24 h to remove anymois-ture, and placed in a bag with desiccant for storage prior to RAT
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assembly. Steel ribbons (93× 7.3mmactive area)were cut out of springsteel sheets (0.64-mm-thick, type 1095, “blue state”; McMaster Carrproduct number 9072K14) using a waterjet. Once cut, they wereprepped with a Scotch-Brite surface conditioning disc (3M, St. Paul,MN), and then grit blasted using 60 grit aluminum oxide blast media.Immediately after grit blasting, the steel ribbonswere placed inmineraloil for storage to ensure strong oil wetting to the ribbon surface, and toprevent ribbon oxidation. A spacer was designed to the same dimen-sions of the ribbons, with three 3-mm-high projections to maintain aconstant gap distance between ribbons (Fig. 3b). The spacer was 3Dprinted from poly(lactic acid) (PLA) thermoplastic, placed in an ovenat 212 °F for 24 h, then placed in a bag with desiccant for storage. Thespacer was adhered to one ribbon using epoxy. Heat shrink tubing(19 mm inner diameter, McMaster Carr product number 8195K35)and five-minute cure, two-part Gorilla epoxy (Sharonville, OH) wasused as received.
A construction hardhat (Peak View-PV50, Portwest, Carrowbeg,Ireland) shell was used as a low cost test platform for helmet suspen-sions. The shell is transparent polycarbonate, which allowed for bettervisualization during testing.
2.2. Tether fabrication
Two types of RATs were constructed and tested. “T52” RATs use the0.52 volume fraction STF, while “T50” RATs use the 0.50 volume fraction
STF. To fabricate a RAT, one Viton tube was placed over the first ribbon(with adhered spacer). The tubing was stretched open with reverse-action pliers to pull the tubing past the barbed section of the ribbon. Re-moving the pliers then allowed the tubing to contract andmechanicallylock around the ribbon barb. The assembly was then positioned verti-cally, with the ribbon head at the top, and a measured volume of STFwas injected through the bottom (open) end of the assembly, allowingair to vent from the top of the assembly as the device was filled. Afterfilling, to seal the first ribbon in place, epoxy was injected between theribbon barb and the tubing. Heat-shrink tubing was placed over thebarb and heat was applied using a heat gun to activate the heat-shrinktubing and accelerate the epoxy cure, locking and sealing the first rib-bon in place. The assembly was then flipped and degassed on a vortexmixer to remove trapped air. The second ribbon was then carefullyinserted past the barb, and the adhesive and heat shrink processwas re-peated to bond and seal the second ribbon in place, creating the finalRAT device (Fig. 3c).
2.3. Suspension system fabrication
Four types of suspension systems were tested, and are denoted as“conventional”, “T52”, “T50”, and “empty”. The conventional suspensionis the as-received hardhat suspension, consisting of nylonwebbing con-figured in a six-spoke design (Fig. 4a, b). This design will serve as thebenchmark when evaluating RAT suspension performance.
Fig. 1. Construction and operation of a rate-activated tether (RAT).
Fig. 2. Rheological behavior of PCC-oil STFs at particle volume fractions of 0.52 and 0.50. (a) Viscosity versus shear rate, and (b) viscosity versus shear stress.
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Preliminary impact testing using a 6-spoke array of RATs resulted indisplacements significantly less than the acceptable displacementlimits, indicating that the suspension was resisting with forces thatwere greater than optimal. The RAT suspension design was thenchanged to a 4-spoke array, which resulted in lower peak accelerationvalues within acceptable displacement limits for the test conditions.Based on these results, for all testing the T52 and T50 suspensionsconsisted of four T52 or T50 RATs assembled into a 4-spoke, cross-
shaped assembly (Fig. 4c). A steel coupling piece was cut and attachedto the free end of each tether, allowing it to hook into the existingshell mounting points (Fig. 4d), like the conventional system. No mod-ifications were made to the shell.
The empty suspension was made using the same steps as the RATsuspensions, but without filling the RATs with STF. The performanceof an empty suspension provides a control condition that clarifies thecontribution of the STF to the performance of the RAT suspensions.
Viton tubing
heat-shrink tubing
steel ribbons 20 mm
spacer
(a) (c)
(b)
Fig. 3. (a) RAT components, (b) ribbon and spacer assembly, and (c) assembled RAT.
Fig. 4. Conventional suspension system (a) isolated and (b) integratedwith helmet shell. 4-spoke RAT suspension system (c) isolated and (d) integratedwith helmet shell. Front and sideviews of the conventional and RAT systems, respectively, mounted on a headform can be seen in Figs. 5b and 6b.
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Each RAT suspension design was constructed such that there was a30–40 mm gap between the crown of the headform and the interiorof the helmet shell, consistent with the gap for the conventional helmetdesign.
2.4. Extensional testing
Tensile testing was performed on individual T52, T50, and emptyRATs using an Instron 8871 servo-hydraulic load frame (Norwood,MA) with a 5 kN load cell and WaveMatrix control software. This loadframe reduces ringing in the force data by means of “inertial dampen-ing”, using an integrated accelerometer on the load cell to measureand subtract inertial effects from the reported sample loads.Mechanicalscrew-action grips were used to clamp the ends of the specimens. A se-ries of tensile experiments were conducted in random order, consistingof five repetitions at six rates: 0.833, 8.33, 16.7, 50.0, 83.3, and167 mm/s. Data acquisition rates were set for each extension rate sothere were force values recorded for at least every 0.1mmof extension,and experiments were conducted to extensions of 40 mm. Data is re-ported as force-displacement, rather than stress-strain, because theconcept of an “average stress” or “average strain” acting over thesemulti-material devices is not rigorously meaningful; additionally, theforce-displacement data is more directly applicable for designing aRAT-based suspension that results in a desired headform acceleration-displacement behavior during helmet impact. The data was analyzedusing Matlab software (Mathworks, Natick, MA) where average forceresponse histories at each extension rate were created by binningeach set of five repetitions onto discrete 0.1 mm displacement steps,up to 30 mm. To account for inconsistent specimen pre-tension orslack, zero displacement was defined as the last point where force wasless than 3.3 N. The consistency of the RAT response was characterizedby calculating a coefficient of variation (CV), over the five experimentalrepetitions at each rate, for the binned force values at each displacementfor each rate. The global average of these CV values during low rate ten-sile testingwas 7.1% for the T52 RAT and 7.5% for the T50 RAT, indicatingthat the force-displacement response of the RATs is very repeatable ateach extension rate. Tether energies at each extension rate were calcu-lated by integrating the force-displacement data from 0 to 30 mm.
An additional tensile test sequence was performed on a sample ofnylon webbingwith a gauge length of 145mm, extracted from the con-ventional suspension andmounted to the load frame using capstan fab-ric grips. The webbing was tested at the same five repetitions and sixrates as the RAT testing protocol, but only to a peak displacement of
10 mm due to the high stiffness of the webbing. Because the responseis not speed sensitive, only one force history (the third repetition) perextension rate will be reported.
High rate tensile testing was performed on T52 and T50 RATs usinganMTS 810 servo-hydraulic load frame with MTS 458.20 MicroConsole(Eden Prairie, MN) and a 2 kN load cell. The tethers were mounted oncustom pin grips inserted through closely fitting holes of each RATend effector. An output signal was processed with an oscilloscopeusing Win600e software (Hi-Techniques, Madison, WI). Each tetherwas subject to a randomized matrix of 24 extensional tests, consistingof four repetitions of six different extensional rates: 160, 250, 500,750, 1000, and 1250 mm/s. Data acquisition rates at each speed wereset to record force histories for at least every 0.1 mm extension. UsingMatlab software, average force data was binned into displacementsteps in a similarmanner as the low rate extensional test data. Zero dis-placementwas defined as the last pointwhere forcewas less than 1.1N.The global average of the force CV values during high rate tensile testingwas 10.2% for the T52 RAT and 8.6% for the T50 RAT. Force-displacement, force-time, and energy-rate curves were generated andmerged with those of the low rate extensional test.
2.5. Helmet low velocity testing
Suspension designs were tested at low speeds in an Instron 8871load frame using a 5 kN load cell (Fig. 5). A Department of Transporta-tion (DOT) headform size C (CADEX, QC, Canada) was mounted to thebase of the load frame, using mechanical screw-action grips and ashort length of aluminum extrusion fastened to the headform. A helmetcontaining one of the suspension assemblies was placed on theheadform, and the crosshead moved downward at constant velocity tocompress the helmet system to 20 mm of displacement. Experimentswere run at six different rates in random order: 0.833, 8.33, 16.7, 50,83.3, and 167mm/s. Force versus displacement histories were recordedat each rate and reported up to 15mm. Experimentswere performed onthe conventional, T52, T50, and empty RAT suspensions.
2.6. Helmet impact testing
Impact testing experiments were conducted based on the “ImpactEnergy Attenuation” requirements and test descriptions provided intheAmericanNational Standards (ANSI) and International Safety Equip-ment Association (ISEA) standard Z89.1-2014 [6]. Helmets weremounted on a DOT headform size C on a guided uniaxial monorail
Fig. 5. Helmet system low velocity test platform. (a) Schematic and (b) photograph, also showing front view of conventional suspension system mounted on headform.
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(CADEX, QC, Canada) (Fig. 6), constrained to allow motion in the verti-cal direction. The total drop assembly, withmass 4.955 kg, was raised toa prescribed drop height, then released to impact a hemi-spherical anvilwith a radius of 48 mm and a chord length of 76 mm. The helmet andheadform were oriented to achieve a crown hit on the helmet, and theanvil was positioned directly in line with an accelerometer (model353B18 with ±500 g peak from PCB Piezotronics, Depew, NY) inte-grated into the drop assembly. Impact velocity was measured using anoptical sensor offset 2 mm above the horizontal plane of impact. Ahigh speed video camera (Photron FASTCAM SA1.1, San Diego, CA,USA) operating at 2000 frames per second (1 frame per 500 μs) was po-sitioned to view the back of the helmet upon impact.
To achieve a passing test, the standard requires that the peak accel-eration of the headformmust not exceed 150 g for an impact velocity of3.5 m/s. For the present study, experiments at additional impact veloc-ities were conducted to determine amore comprehensive evaluation ofsuspension response. RAT suspensions were tested at 1.5, 2.5, and3.5 m/s with three replicates at each velocity, in a random order, andone replicate at 4.5 m/s. These target velocities correspond to heightsof 11.9, 35, 65 cm, and 107 cm respectively. After each impact, helmetswere removed and inspected for damage. Approximately 10–15 minelapsed between successive impacts on the same suspension system,allowing for a cursory data review, test sample inspection, and resettingthe test system. An empty tether suspension was tested with three rep-etitions of 1.5 m/s, but only one test at each higher speed to reduce thelikelihood of damaging the shell. The conventional helmet was alsotested with three repetitions at 1.5 and 2.5 m/s, but only one test atthe higher speeds, as damage to the shell and suspension (most typi-cally at the connection point between shell and suspension) was ob-served for drops at 3.5 and 4.5 m/s. For any experiment that resultedin helmet damage, the helmet was not tested further and was replacedwith an untested helmet. Therefore the conventional helmet behaviorsat 3.5 and 4.5 m/s could include some energy absorption from irrevers-ible damage mechanisms, unlike the RAT systems which were fully re-settable at all velocities.
Raw acceleration data was sampled at 33 kHz, and filtered using aCFC 1000 low-pass filter with a corner frequency of 1650 Hz [57]. To re-port data, the acceleration histories for the three replicates at each dropvelocity were averaged at each time increment, and then this averagedresponse was integrated to create representative displacement histo-ries. Comparison of integrated accelerometer data with displacementhistories measured from analysis of high speed video confirmed the ac-curacy of the integrated displacement data to within video resolution,approximately 1 mm. Video analysis revealed that the headform hadtypically displaced 2–3 mm towards the crown of the helmet beforethe headformaccelerometer registered over 1 g. Therefore, our reported
headform displacement values are likely to be 2–3mm smaller than thetotal vertical suspension travel relative to its initial position for allexperiments.
Helmet impact data was also characterized via head injury criterion(HIC) values, calculated as
HIC ¼ Max t2−t1ð Þ 1t2−t1ð Þ
Zt2t1
a tð Þdt
264
3752:58><
>:9>=>; ð1Þ
where a is acceleration, t is time, and integration occurs over the timeinterval from t1 to t2 where t1 is selected to provide the maximum HICvalue. The HIC captures the combined effects of intensity and durationof acceleration, and provides a complementary metric to peak accelera-tion values. A t2 − t1 value of 15 ms (HIC-15) was used for all calcula-tions, consistent with prior studies showing a correlation between thismetric and concussion risk [12,58].
Fig. 6. Helmet impact test platform. (a) Schematic and (b) photograph, also showing side view of RAT suspension system mounted on headform.
Fig. 7. Conventional suspension strap tensile behavior at different extension rates.
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3. Results
3.1. Extensional response of suspension components
Fig. 7 shows the force response as a function of displacement for thenylon webbing from the conventional suspension. The force is linearwith displacement, and shows negligible rate dependence. A springconstant was calculated by taking an average of the spring constantsat each extension rate between 2 mm to 5 mm, giving a value of 133± 6 N/mm.
The extensional behavior of RAT T52 is shown in Fig. 8. The responseis highly rate-sensitive, with a peak force of 319 N at 4 mm of
displacement for an extension rate of 1250mm/s,while at the same dis-placement the force is only 11 N at 0.83 mm/s. At extension rates of17mm/s and below, the loads are small and increase graduallywith dis-placement. At higher extension rates, the force shows an initial peakvalue followed by a long, relatively steady plateau force. The value ofthis plateau force increases gradually with extension rate up to750 mm/s. At extensional velocities of 750, 1000, and 1250 mm/s, theinitial peak forces continue to increase, but the force values at higherdisplacement are similar. The peak force value at 1250 mm/s occurs ata time of 7 ms, demonstrating the extremely fast response time forRATs, and suggesting their relevance for impact applications. Thecause of the fluctuations in the plateau force with displacement is not
Fig. 9. Tether T50 response as a function of extension rate. (a) Force versus displacement and (b) force versus time.
Fig. 8. Tether T52 response as a function of extension rate. (a) Force versus displacement and (b) force versus time.
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known, but could be associated with stick-slip behavior at the STF-ribbon interface.
The extensional behavior of RAT T50 is shown in Fig. 9. Compared tothe T52 RAT, the T50 RATs shows lower forces at similar extension rates,while not exhibiting a strong initial force peak. The highest force valueat 1250 mm/s for the T50 RAT is 190 N at a displacement of 18 mmand time of 17 ms. While the T50 force values are lower than the T52force values, at velocities of 500 mm/s and above a prominent, steadyforce plateau over long displacements is apparent. At very low speedsof 50 mm/s and below, the T50 RAT exhibits a low force, linear elasticresponse, similar to that of the T52 RAT at 0.83 mm/s.
In contrast, the tensile response of the empty RAT was linear elasticat all extension rates (Fig. 10). At a displacement of 30 mm, the emptytether demonstrated an average force of 30N at each rate. This responseis nearly identical to the low rate responses for T52 and T50 RATs,confirming that at low rates, RAT response is equivalent to the responseof the elastic tubing.
Fig. 11 shows the total extensional energies at each extension ratefor each RAT. Extensional energy is compared, rather than for exampleplateau force, since the extensional energy better captures the averagesystem response over long strokes. Comparing the STF filled RATs, T52consistently absorbed greater energy than T50 at the same speed. At1250 mm/s, the T52 and T50 RATs absorbed 8.0 and 4.4 J of energy, re-spectively. However, the energy plots suggest that the T52 responsecould be plateauing, while the T50 energy is continuing to increasewith extension rate. It is possible that at higher velocities (such as3.5 m/s) the T50 and T52 exhibit similar energy absorption. In contrast,the energy absorbed by the empty tether at 167mm/s is 0.72 J. The per-cent increase of absorbed energy between extension rates of at0.83 mm/s to 1250 mm/s is 1200% for T52, and 800% for T50, while forthe empty tether it is only 23%.
3.2. Helmet system compression response
Fig. 12 shows the force-displacement curves of the different helmetsystems tested in low velocity compression. The conventional suspen-sion is consistent with the extensional behavior of the nylon webbing,demonstrating high forces with linearly elastic response, and insignifi-cant rate dependence (Fig. 12a). Forces reach 2000 N at 15 mm of
displacement. In contrast, the T52 assembly shows a strong speed sen-sitivity (Fig. 12b). A transition in the response is evident between50 mm/s and 83.3 mm/s for this assembly, where force values jumpfrom 200 N to 360 N, respectively, at 15 mm displacement. The T50 as-sembly also shows clear speed sensitivity, but with lower forces and amore gradual dependence on velocity (Fig. 12c). The maximum forcesreported for the T52 and T50 suspensions at 167 mm/s and 15 mm ofdisplacement are 413 and 186 N, respectively. The empty assemblyshows a small amount of compression rate dependence. At 15 mm dis-placement, this system reaches a peak force of 179 N at 167 mm/s.
Fig. 11. Comparison of energy versus extension rate for T52, T50, and empty tethersystems.
Fig. 10. Empty tether response as a function of extension rate. (a) Force versus displacement and (b) force versus time.
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3.3. Helmet system impact response
Figs. 13 and 14, and Table 1, summarize the impact test results at thefour target velocities of 1.5, 2.5, 3.5, and 4.5 m/s. In Table 1, standard de-viation values are reported for cases where three drop repetitions wereperformed.
We first consider the response of the conventional, T52, and T50 sus-pensions at 3.5 m/s (Fig. 13). All suspensions meet the ANSI/ISEAhardhat requirement of that the headform experience less than 150 gpeak acceleration. The conventional helmet suspension displays apeak acceleration of 51 g and a peak displacement of 21 mm(Fig. 13a). Only one drop was performed at 3.5 m/s, leading to evidentplastic deformation of the components that connect the nylon webbingto the helmet shell (this shellwas not tested further). The T52 system, incontrast, shows a more prominent force plateau and hysteresis. The
average peak acceleration of the T52 assembly was 38 g, or 25% lowerthan the conventional suspension, at a peak displacement of 28 mm.The T50 provided an even more compliant response, with a very longforce plateau and peak displacement of 37 mm. The peak accelerationfor the T50 system was only 25 g, or 51% lower than the conventionalsuspension. All systems show a rapid response, achieving significantforces within 5 ms of impact (Fig. 13b).
Fig. 14a shows the conventional suspension response for all impactvelocities. A small amount of hysteresis is present in the acceleration-displacement curves up to 3.5 m/s. At 4.5 m/s, more hysteresis and alonger force plateau is noted. The sudden, short peak in force at30 mm is most likely due to contact between the headform and the in-side of the helmet shell. At 4.5 m/s failure of components was apparent.For the conventional helmet, the plastic buckles that connect the strapsto the shell failed while, for the RAT systems, the slots molded into the
Fig. 12. Force versus displacement at different helmet displacement rates during low velocity testing, for (a) conventional suspension system, (b) T52 system, (c) T50 system, and(d) empty system.
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helmet shell that hosted the steel RAT-to-shell couplers broke. Com-bined with the observation of component deformation for the impactat 3.5 m/s, it is likely that the conventional helmet is designed to absorbenergy via deformation and failure of components, and is meant to bediscarded after a serious impact.
A different response is apparent for the T52 suspension system(Fig. 14b). A force plateau and significant hysteresis is apparent ateach impact velocity, with the plateau force value increasingwith veloc-ity. Compared to the conventional system, the accelerations are lower,and displacements are higher. Similarly, the T50 system (Fig. 14c)shows long plateaus with lower force and higher displacement thanthe conventional or T52 system. At 4.5 m/s, the T50 system extends tothe point that the headform contacts the helmet shell, while the T52system does not. Finally, the empty RAT system shows very little forcebelow 30 mm of displacement, and high peak displacements. Theheadform contacts the shell at velocities above 2.5 m/s.
Table 1 also tabulates head injury criterion (HIC) values for each ve-locity and design. The results show that HIC values for the RAT designsare significantly lower than for the conventional suspension at all veloc-ities. At 3.5 m/s, for example, the HIC values for the T52 and T50 designsare 69% and 82% lower than the conventional suspension HIC,respectively.
4. Discussion
4.1. Relationship between STF rheology, RAT behavior, and RAT suspensionresponse
The rheology of the STFs (Fig. 2) shows that the 0.52 STF has a lowercritical shear rate compared to the 0.50 STF, higher viscosity at all shearrates, and a steeper rise in viscosity above the critical shear rate. Thesefeatures are consistent with prior studies on volume fraction effects onSTF rheology [59], and appear to directly determine RAT behavior. As-suming that the ribbon-to-ribbon gap is determined by the spacerheight of 3 mm, then the rheological critical shear rates of 13 s−1 and33 s−1 correspond to critical extension rates of 39 mm/s and 99 mm/sfor the T52 and T50 RATs, respectively. For the T52 system, extensionforces begin to increase significantly between 17 and 50mm/s of exten-sion rate, and for the T50 system significant forces appear between 83
and 170 mm/s, in agreement with the rheological predictions (Figs. 8and 9). Furthermore, the rise in forces and energy for the T52 RATwith respect to extension rate ismore rapid than the T50RAT,which ex-hibits amore gradual rise in extensional resistance (Fig. 11). This behav-ior correlates with the steeper rise in viscosity typical for higher particleloaded STFs, and suggested by the rheology data in Fig. 2 (although,however, further rheology data for the 0.50 system above 100 s−1
would be necessary to confirm this relationship). We also note thatthe empty tether system shows low forces and negligible rate depen-dence (Fig. 10), confirming that the STFs are responsible for the unusualbehaviors of the RATs.
Of less certainty is the relationship between STF rheology and RATplateau force. The T52 RAT generates approximately 250 N at1250 mm/s, while the T50 RAT resists with around 150 N (Figs. 8 and9). It is tempting to correlate the higher forces in the T52 system to itshigher viscosities measured in the rheology data. However, we do notbelieve this correlation to be rigorously meaningful. At an extensionalrate of 1.25 m/s, the STF is at a shear rate of around 400 s−1 (assuminga ribbon gap of 3 mm), well beyond the critical shear rates and mea-sured rheology data. Under these conditions, the STF is likely to be in asolid-like state. Prior studies have shown that dilatancy during shearthickening causes particles to push outward from the bulk STF and pro-trude from free fluid surfaces, creating a rough surface that appears“dry” due to scattering [43,55,60]. Therefore, during rapid RAT exten-sion, RAT forces are generated by particle-particle force chains bridgingbetween the ribbons, and mediated by particle-ribbon friction andfluid-ribbon wetting. The quantitative origin of these forces is elusive,and indeed the physics and mechanics of the interactions betweentransitioned, confined STFs and solid surfaces is not well understood.
The low velocity helmet suspension response data also correlateswell with the extensional RAT data (Fig. 12). For the T52 system, aclear jump in force is observed between helmet compression rates of50 and 83 mm/s. If we assume that the RATs are at a 45° angle relativeto the compression direction, then these compression rates correspondto tether extension rates of 30–60mm/s, similar to the critical rates ob-served in extension experiments. For the T50 system, similar argumentswould suggest that the compression force should increase at compres-sion rates of around 140mm/s, which is reasonably consistent with ob-servations. We also note that the steeper force and viscosity responses
Fig. 13. Comparison of different helmet designs subject to impact at 3.5 m/s. (a) Acceleration versus displacement, and (b) acceleration versus time.
162 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169
measured for the 0.52 STF and T52 RATs also agree well with the moredistinct jump in force observed for the T52 system during compression.The empty RAT system, consistent with prior data, show no rate depen-dence, although the compression tests results show surprisingly highresistance forces of 200 N at 15mmof displacement. This behavior sug-gests that friction between the ribbons is significant for the empty sys-tem when subject to transverse compression from the headform.
Based on these supporting experimental results, the behaviors ob-served during helmet impact are not surprising. The T52 system, com-pared to the T50 system, demonstrates higher acceleration values andlower displacements, consistent with the higher forces observed in ex-tensional and compression experiments. The empty tether systemshows poor performance in impact, demonstrating that ribbon-to-ribbon friction is not sufficient to resist impact in these experiments.
We also note that the impact velocities of 1.5–4.5 m/s correspond to ex-tensional rates of approximately 1–3m/s, above the limits of our exten-sional RAT data. The extensional energy data (Fig. 11) suggests thatresistance to extension rises steadily within this range, and this increas-ing resistance is likely responsible for the increasing head accelerationplateaus with increasing drop velocity.
4.2. Comparison of RAT and conventional webbed suspensions
The webbing used in the conventional suspension system exhibitsrate-insensitive, linear elastic behavior, with loads that extrapolate toapproximately 3500 N at 30 mm of displacement (Fig. 7). In contrast,RATs exhibit highly rate-sensitive behavior, requiring only 30 N to ex-tend to 30mm at low rates (below 10mm/s), and up to 300 N at higher
Fig. 14. Acceleration versus displacement for helmets subject to impact testing at different impact velocities. (a) Conventional helmet, (b) T52 system (c) T50 system, and (d) emptysystem.
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rates (above 1 m/s). Importantly, RATs also exhibit force plateaus overlong distances, rather than a linearly proportional force displacementtrend. These effects lead directly to 10× lower forces during low velocityhelmet compression, and 2× lower acceleration values during helmetimpact, compared to the conventional suspension.
Given the even greater differences between webbing and RATs dur-ing tensile testing at low speeds (i.e. 100× lower forces for RATs at10mm/s), wewould expect evenmore dramatic differences in peak ac-celeration values at impact velocities below 1.5 m/s. Unfortunately, wewere unable to collect data at impact velocities lower than 1.5 m/s forour test platform due to limitations in the control software. Althoughthe head accelerations would be relatively minor (less than 20 g forthe conventional helmet) at these low speeds, these impacts are farmore common than the 3.5 m/s impacts required in the test standard.Data collected using instrumented headforms during football indicatesthat typical players receive roughly 1000 significant head impacts perseason, and nearly half of those impacts result in head accelerations ofless than 20 g [15]. As concern grows about the role of repetitive low en-ergy head impacts in long term brain health, technologies that can re-duce head accelerations at low velocities could prove criticallyimportant.
4.3. Behavior of suspension systems relative to ideal behaviors
Consider amassMwith an initial velocity Vo, decelerating due to theaction of a constant force Fc. Standard dynamics calculations predict that
the force required to bring the body to rest at a distance H is (seeAppendix A).
Fc ¼ M g þ V2o
2H
!ð2Þ
where g is gravitational acceleration. This body is decelerated at a con-stant value ac, given by
a xð Þ ¼ ac ¼ −V2o
2Hð3Þ
We will refer to this behavior as a “constant force” system. Relatingthis model to our helmet problem, we assume that the fixed distanceH is equal to the standoff distance between the helmet shell and the sus-pension, and there exists an Fc value at each impact velocity thatwill de-celerate the headform to zero velocity utilizing this full standoffdistance. This constant forcemodel represents the best possible suspen-sion system response, by minimizing peak acceleration at every impactvelocity.
Now consider a mass M with initial velocity Vo, decelerating due tothe action of a spring force that increases linearly with displacement ac-cording to F = K · x, where K is a spring constant. The spring constantnecessary to bring the mass to rest at a distance H is (see Appendix A).
K ¼ 2MgH
1þ V2o
2gH
!ð4Þ
The acceleration of this body as a function of distance is
a xð Þ ¼ g−2g 1þ V2o
2gH
!xH
ð5Þ
Fig. 15. Acceleration versus displacement for ideal “constant force”, “optimal spring”, and“invariant spring” systems for a mass of 5 kg at an initial velocity of 1.5, 2.5, 3.5, and4.5 m/s. For the constant force and optimal spring models, the mass is brought to rest ata distance of 30 mm. For the invariant spring model, a spring constant of K =133 N/mm is used and each labeled data point indicates the final displacement andacceleration when the mass is brought to rest.
Table 1Summary of impact results, and comparison with model systems. Standard deviationvalues are calculated from sets of three drops; table entries without standard deviationvalues indicate that a single experimentwas performed, or that results are exact analyticalsolutions. A spring constant of K= 133 N/mm is used for the invariant spring model.
164 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169
and the peak acceleration, which occurs at x = H, is
ac ¼ − g þ V2o
H
!ð6Þ
In this treatment, which we refer to as the “optimal spring” system,the spring constant varies as a function of initial velocity (but remainsconstant during an impact event) to bring the body to rest at a selecteddistance ofH. This model represents the best possible performance for asuspension system consisting of a linear elastic strap or linear elasticpad suspension system, in which the spring constant is optimallytuned for each impact velocity.
The same mass-spring solution can be re-expressed for the case of aspring constant K that does not change with initial velocity, resulting in
a different deceleration distance H for each initial velocity. In this ap-proach, which we refer to as the “invariant spring” system, the dropheight is given by
Fig. 16. Comparison of model acceleration versus displacement responses with experimental data for conventional, T52, and T50 designs at (a) 1.5 m/s, (b) 2.5 m/s, (c) 3.5 m/s, and(d) 4.5 m/s. Experimental data is plotted up to the peak displacement value for each case.
165D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169
The invariant spring model best represents conventional elastic andlinear elastic pad suspension systems, which exhibit a linear force-displacement behavior that is relatively invariant with deformationrate.
Fig. 15 shows acceleration versus displacement for the constantforce, optimal spring, and invariant spring systems. Peak accelerationand displacement values, as well as HIC values (see Appendix A for der-ivation), are tabulated in Table 1. A value of M = 5 kg is used here andfor all subsequent calculations, and for the constant force and optimalspring systems the mass is brought to rest at a distance of H =30 mm. For the invariant spring case, we use a spring constant of K =133 N/mm, consistent with the slope of the force-displacement datafor the conventional helmet subject to quasistatic loading (Fig. 7). Theinvariant spring model shows that stopping distance increases withdrop velocity. For all velocities up to 4.5 m/s, the final displacement isless than 30mm, leading to peak acceleration values significantly higherthan the optimal spring model. Comparing the optimal spring and con-stant force models, the constant force system brings the drop mass torest at approximately half the peak acceleration value of the optimalspring system. These model comparisons reveal the two key featuresof an ideal suspension system,with the goal ofminimizing peak acceler-ation at every impact speed: (i) a resistance force that is constantthroughout the deceleration distance, and (ii) a resistance force that in-creases proportionallywith impact velocity. The invariant spring systemperforms the worst, because it does not meet either of these ideal char-acteristics; the constant spring system meets the second criterion (re-sponse proportional to speed), but does not provide a resistance forcethat is constant with displacement.
Fig. 16 superimposes experimental acceleration versus displacementbehavior for the conventional, T52, and T50 suspension systems relativeto the model systems. The conventional suspension system exhibits abehavior that is most similar to the invariant spring model, with aroughly linear acceleration-displacement response and a consistentslope for impacts at 1.5–3.5 m/s. At 4.5 m/s, interestingly the conven-tional suspension system exhibits a more ideal force plateau response.This force plateau could be due to one-time mechanical damage of thesuspension components, as the second impact at this velocity causedcatastrophic failure at the strap-to-shell mounting locations. The T52and T50 RAT suspension systems, in comparison, show behavior moresimilar to the ideal, constant force response. Both of these systemsshow force plateaus, with plateau values that increase with increasingimpact velocity. The T50 most closely matches the ideal peak accelera-tion values for the constant force system, although the significant accel-eration rise period (5–10mm) leads to peak displacements that exceedthe 30 mm limit at higher velocities.
The fact that the RAT responses so closely follow ideal response isprimarily the result of empirical tuning. A number of other RAT suspen-sion configurations (e.g. smaller diameter tubing, a six-spoke design,long straps parallel to the coronal plane of the head) and RAT compo-nent details (RAT length, ribbon material, ribbon spacing) wereattempted before arriving at the presently reported designs. Additionaldesign modifications could be performed to continue to tune perfor-mance empirically and approach ideal response even more closely. Acomputational design and optimization approach would be possibleby collecting RAT extensional data through 5 m/s, and coupling theseconstitutive responses with a more detailed model of the impactevent. A dynamic model would need to explicitly treat the RATs asrate-dependent elements, capture the details of headform engagementwith the suspension system, andmodel compliance and deformation ofthe helmet shell. This model would provide a more detailed under-standing of the suspension response, as the analytical treatment ofRATs as constant force elements during the entire impact event is anoversimplification that does not consider the continuous decrease inRAT extensional velocity as the headform decelerates.
Table 1 also compares theoretical HIC values (see Appendix A forderivation)withmeasured values. The RAT suspensions are remarkably
similar to the ideal constant force response, in some cases achievingHICvalues below the ideal value due to displacements beyond the 30 mmmodel limit. Previous studies associate HIC values of 250 [12] to 700[58] with a significant probability of concussion, so that only theempty RAT suspension at 4.5 m/s impact velocity introduces a signifi-cant risk of concussion (HIC of 543). However, the significant reductionin HIC for RAT suspensions relative to conventional webbing, and theirsimilarity to ideal behavior, suggests that these designs could signifi-cantly improve outcomes from low velocity head impact.
4.4. Comparison of RAT suspension relative to foam pads
Although the current tests use a webbed suspension as a control,RAT suspensions may also offer superior energy attenuating character-istics compared to energy attenuating foams. Helmets designed formulti-impact will use paddingmaterials, such as expanded polypropyl-ene or vinyl nitrile foam, that retain their compression characteristics
Fig. 17. (a) RAT suspension design for protection in multiple impact directions.(b) Suspension between inner and outer helmet shells, to provide energy absorptionduring rotational loading. (c) Suspension design with RAT outside of the helmet shell.
166 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169
over repetitive impacts. Thesematerials exhibit a compression responsecharacterized by an initial linear region followed by yielding that ex-tends over a “plateau region” as the cellular structure collapses [30].This plateau is rarely a true constant force response, as closed cellfoams usually exhibit some degree of strain hardening over the plateauregion as the entrapped gas is progressively pressurized [31]. Instead,unless the pads have been carefully designed and optimized [35–37],their behavior tends to be intermediate between a “constant force”and “invariant spring” response. Therefore, a well-tuned RAT systemhas a high potential to provide amore ideal response, and consequentlylower head accelerations, than current foam pad systems.
In addition, although some pads have viscoelastic responses thatprovide a moderate increase resistance with impact velocity [31],these resistance changes are difficult to tune and do not appear tohave been highly optimized for head protection. Generally foams thatare highly rate sensitive are also highly temperature sensitive whichcan limit the conditions over which the helmet protects the head.These factors suggest that our rate-dependent suspension system hasstrong potential to exceed the impact performance of foam pad suspen-sion systems over a wide range of impact energies. We also expect thatcreating a hybrid protection system with webbed suspension, RAT sus-pension, and foam pads in parallel or series could provide advantagesfor tuning impact response and comfort. Practically, we note that RATsare relatively straightforward to manufacture and do not require exoticmaterials, but likely would be somewhat more expensive to producecompared to existing resettable foam pads. Therefore we would expectcost-performance tradeoffs to be one consideration when designing acommercial helmet using a RAT-based suspension.
5. Conclusions
This study has identified two key features for optimal helmet energyabsorbers: constant resistance force over large displacements, and in-creasing resistance with increasing impact velocity. The experimentaldata shows that RATs provide both of these characteristics, leading tosignificantly lower head accelerations compared to a conventionalwebbed suspension system. Rheology, extensional testing, and low ve-locity helmet compression testing demonstrate behavior correlations
that allow us to relate RAT suspension behavior back to details of theenclosed STF and RAT design features. Therefore, great opportunity ex-ists to further improve RAT suspension performance relative to thepresently reported designs.
The present testing uses plastic industrial hard hats, but we expectthe results to carry over to other helmet systems including militaryand sports helmets. While the present experiments focus on the ANSI“impact energy attenuation” test requirement, we would also expectgood performance for the “force transmission” requirement, in whicha falling mass strikes a stationary, helmeted headform. The currentstrap configuration appears best suited for crownhelmet strikes, but ad-ditional strap configurations could be designed to resist loading frommultiple directions (Fig. 17a). We also hypothesize that RATs could beused to control rotational accelerations, for example by creating innerand outer helmet shells that are coupledwith an array of RATs, orientedparallel to the shell walls (Fig. 17b).
The use of extensional energy absorbers such as RATs also opens upcompletely new suspension concepts, such as coupling conventionalwebbing inside the helmet to RATs positioned outside of the helmetshell (Fig. 17c). This ability to transmit head loads to a remote energyabsorber has broad design implication and, importantly, can potentiallylead to dramatically increased head displacement allowables withincurrent helmet shell envelopes. Current foam pad suspensions beginto significantly densify at compressions of 50–75%, so 25–50% of thehead-to-shell standoff is not available for deceleration distance. Withthin webbing and a remote RAT, the full head-to-shell standoff gapwould be available for managing head acceleration, leading to evenlower injury risk within existing helmet profiles.
Acknowledgements
The authors would like to acknowledge John Lim and MattLangenstein for their contributions to preliminary helmet suspensionexperiments; Mike Neblett, Larry Long, and Ryan Dunn for fabricationassistance; Mike Leadore for high rate testing support; and RichDombrowski of STF Technologies for the rheology data. Mr. Spinelliwas supported in part by an appointment to the Undergraduate Re-search Participation Program at the U.S. Army Research Laboratory
Fig. 17 (continued).
167D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169
administered by the Oak Ridge Institute for Science and Educationthrough an interagency agreement between the U.S. Department of En-ergy and USARL.
Appendix A
A.1. Solution for constant force deceleration of a falling mass
For the “constant force” system shown in Fig. A1a, the acceleration,velocity, and displacement history are given by
a tð Þ ¼ g−FcM
ð10Þ
v tð Þ ¼ g−FcM
� �t þ Vo ð11Þ
x tð Þ ¼ 12
g−FcM
� �t2 þ Vot ð12Þ
From Eq. (11), we find that the velocity reduces to a value of zero atthe critical time, tc, of
tc ¼ VoFcM
−g� �−1
ð13Þ
For the displacement to beH at a time of tc, then combining Eqs. (10)and (12) reveals that Fc must be equal to the value given in Eq. (2).Substituting this value back into Eq. (10) gives the constant accelerationvalue of the mass given in Eq. (3).
(a) (b)
H
gVo M
xF=KxH
gVo M
xFc
Fig. A1. Schematic of (a) constant forcemodel and (b) optimal spring and invariant springmodels
A.2. Solution for optimal spring deceleration of a falling mass
For the “optimal spring” system shown in Fig. A1b, the accelerationis dependent on the position x of the mass according to
a tð Þ ¼ g−KxM
ð14Þ
Solving this differential equation with the initial conditions that a=g, v = Vo and x = 0 yields.
a tð Þ ¼ g cos ωtð Þ−ωVo sin ωtð Þ ð15Þ
v tð Þ ¼ gω
sin ωtð Þ þ Vo cos ωtð Þ ð16Þ
x tð Þ ¼ gω2 1− cos ωtð Þð Þ þ Vo
ωsin ωtð Þ ð17Þ
where ω2 = K/M. If the mass comes to rest at time tc then, according toEq. (16),
0 ¼ gω
sin ωtcð Þ þ Vo cos ωtcð Þ ð18Þ
or.
tan ωtcð Þ ¼ −Voωg
ð19Þ
Eq. (19) can be re-expressed as.
sin ωtcð Þ ¼ −Voωg
1þ V2oω
2
g2
!−1=2
ð20Þ
cos ωtcð Þ ¼ 1þ V2oω
2
g2
!−1=2
ð21Þ
At time tc the mass is at position +H.
H ¼ gω2 1− cos ωtcð Þð Þ þ Vo
ωsin ωtcð Þ ð22Þ
Substituting Eqs. (20) and (21) into this equation yields.
H ¼ gω2 1− 1þ V2
oω2
g2
!−1=20@
1A−
Vo
ωVoωg
1þ V2oω
2
g2
!−1=2
ð23Þ
This equation can be solved for ω, giving.
ω2 ¼ V2o
H2 þ2gH
ð24Þ
or
K ¼ 2MgH
1þ V2o
2gH
!ð25Þ
This equation provides the spring constant value necessary for theoptimal spring system to bring a body of mass M and initial velocity Voto rest at a distance H. Substituting this value back into Eq. (15) givesthe acceleration as a function of displacement as given in Eq. (5).
A.3. Solution for invariant spring deceleration of a falling mass
For the case of an invariant spring, the solution to the equations ofmotion are identical to the optimal spring case, but we now invertEq. (25) to calculate thefinal drop distanceH as a function of spring con-stant K as given in Eq. (7). The solution has two roots, with the largerroot being equal to the downward deflection H (the second, negativeroot with slightly smaller magnitude, is the value of the upward re-bound deflection). Acceleration as a function of distance is simplygiven by Eq. (8) using the fixed K value.
A.4. Head injury criterion (HIC) values
For the constant force scenario, acceleration is constant and the HICfrom Eq. (1) reduces to.
HIC ¼ Δtð Þa5=2c ð26Þ
where ac is given by Eq. (3).For the constant and invariant spring models, the acceleration is si-
nusoidal with respect to time, and reaches peak acceleration at a timeof t = tc, where tc is given by Eq. (19). We can therefore calculate a
168 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169
HIC by substitution a(t) from Eq. (15) into Eq. (1), and integrating from(tc − Δt/2) to (tc + Δt/2), giving.
HIC ¼ Δtð Þ 1Δt
2gω
cos ωtcð Þ sin ωΔt=2ð Þ−2Vo sin ωtcð Þ sin ωΔt=2ð Þ� �� �5=2
ð27Þ
which reduces to.
HIC ¼ Δtð Þ 2g sin ωΔt=2ð ÞωΔt
1þ V2oω
2
g2
!1=224
355=2
ð28Þ
Substituting values of ω from Eq. (24), for the constant spring or in-variant spring cases, gives HIC values for each spring model scenario.
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