adaptive impact absorption

Mathematical Problems in Engineering

Adaptive Impact Absorption

Guest Editors: Jan Holnicki-Szulc, Mohamed Ichchou, Zhongdong Duan, and ukasz Jankowski

Mathematical Problems in Engineering


Guest Editors: Jan Holnicki-Szulc, Mohamed Ichchou,Zhongdong Duan, and ukasz Jankowski

Copyright 2016 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in Mathematical Problems in Engineering. All articles are open access articles distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

Editorial Board

Mohamed Abd El Aziz, EgyptFarid Abed-Meraim, FranceSilvia Abraho, SpainPaolo Addesso, ItalyClaudia Adduce, ItalyRamesh Agarwal, USAJuan C. Agero, AustraliaRicardo Aguilar-Lpez, MexicoTarek Ahmed-Ali, FranceHamid Akbarzadeh, CanadaMuhammad N. Akram, NorwayMohammad-Reza Alam, USASalvatore Alfonzetti, ItalyFrancisco Alhama, SpainJuan A. Almendral, SpainLionel Amodeo, FranceSebastian Anita, RomaniaRenata Archetti, ItalyFelice Arena, ItalySabri Arik, TurkeyFumihiro Ashida, JapanHassan Askari, CanadaMohsen Asle Zaeem, USAFrancesco Aymerich, ItalySeungik Baek, USAKhaled Bahlali, FranceLaurent Bako, FranceStefan Balint, RomaniaAlfonso Banos, SpainRoberto Baratti, ItalyMartino Bardi, ItalyAzeddine Beghdadi, FranceAbdel-Hakim Bendada, CanadaIvano Benedetti, ItalyElena Benvenuti, ItalyJamal Berakdar, GermanyEnrique Berjano, SpainJean-Charles Beugnot, FranceSimone Bianco, ItalyDavid Bigaud, FranceJonathan N. Blakely, USAPaul Bogdan, USADaniela Boso, ItalyAbdel-Ouahab Boudraa, FranceFrancesco Braghin, Italy

Michael J. Brennan, UKMaurizio Brocchini, ItalyJulien Bruchon, FranceJavier Buldu', SpainTito Busani, USAPierfrancesco Cacciola, UKSalvatore Caddemi, ItalyJose E. Capilla, SpainAna Carpio, SpainMiguel E. Cerrolaza, SpainM. Chadli, FranceGregory Chagnon, FranceChing-Ter Chang, TaiwanMichael J. Chappell, UKKacem Chehdi, FranceChunlin Chen, ChinaXinkai Chen, JapanFrancisco Chicano, SpainHung-Yuan Chung, TaiwanJoaquim Ciurana, SpainJohn D. Clayton, USACarlo Cosentino, ItalyPaolo Crippa, ItalyErik Cuevas, MexicoPeter Dabnichki, AustraliaLuca DAcierno, ItalyWeizhong Dai, USAPurushothaman Damodaran, USAFarhang Daneshmand, CanadaFabio De Angelis, ItalyStefano de Miranda, ItalyFilippo de Monte, ItalyXavier Delorme, FranceLuca Deseri, USAYannis Dimakopoulos, GreeceZhengtao Ding, UKRalph B. Dinwiddie, USAMohamed Djemai, FranceAlexandre B. Dolgui, FranceGeorge S. Dulikravich, USABogdan Dumitrescu, FinlandHorst Ecker, AustriaAhmed El Hajjaji, FranceFouad Erchiqui, CanadaAnders Eriksson, Sweden

Giovanni Falsone, ItalyHua Fan, ChinaYann Favennec, FranceGiuseppe Fedele, ItalyRoberto Fedele, ItalyJacques Ferland, CanadaJose R. Fernandez, SpainSimme D. Flapper, NetherlandsThierry Floquet, FranceEric Florentin, FranceFrancesco Franco, ItalyTomonari Furukawa, USAMohamed Gadala, CanadaMatteo Gaeta, ItalyZoran Gajic, USAUgo Galvanetto, ItalyAkemi Glvez, SpainRita Gamberini, ItalyMaria Gandarias, SpainArman Ganji, CanadaXin-Lin Gao, USAZhong-Ke Gao, ChinaGiovanni Garcea, ItalyFernando Garca, SpainLaura Gardini, ItalyAlessandro Gasparetto, ItalyVincenzo Gattulli, ItalyOleg V. Gendelman, IsraelMergen H. Ghayesh, AustraliaAnna M. Gil-Lafuente, SpainHector Gmez, SpainRama S. R. Gorla, USAOded Gottlieb, IsraelAntoine Grall, FranceJason Gu, CanadaQuang Phuc Ha, AustraliaOfer Hadar, IsraelMasoud Hajarian, IranFrdric Hamelin, FranceZhen-Lai Han, ChinaThomas Hanne, SwitzerlandTakashi Hasuike, JapanXiao-Qiao He, ChinaM.I. Herreros, SpainVincent Hilaire, France

Eckhard Hitzer, JapanJaromir Horacek, Czech RepublicMuneo Hori, JapanAndrs Horvth, ItalyGordon Huang, CanadaSajid Hussain, CanadaAsier Ibeas, SpainGiacomo Innocenti, ItalyEmilio Insfran, SpainNazrul Islam, USAPayman Jalali, FinlandReza Jazar, AustraliaKhalide Jbilou, FranceLinni Jian, ChinaBin Jiang, ChinaZhongping Jiang, USANingde Jin, ChinaGrand R. Joldes, AustraliaTadeusz Kaczorek, PolandTamas Kalmar-Nagy, HungaryTomasz Kapitaniak, PolandHaranath Kar, IndiaKonstantinos Karamanos, BelgiumChaudry Khalique, South AfricaDo Wan Kim, Republic of KoreaNam-Il Kim, Republic of KoreaOleg Kirillov, GermanyManfred Krafczyk, GermanyFrederic Kratz, FranceJurgen Kurths, GermanyKyandoghere Kyamakya, AustriaDavide La Torre, ItalyRisto Lahdelma, FinlandHak-Keung Lam, UKAntonino Laudani, ItalyAime Lay-Ekuakille, ItalyMarek Lefik, PolandYaguo Lei, ChinaThibault Lemaire, FranceStefano Lenci, ItalyRoman Lewandowski, PolandQing Q. Liang, AustraliaPanos Liatsis, UAEPeide Liu, ChinaPeter Liu, TaiwanWanquan Liu, AustraliaYan-Jun Liu, ChinaJean J. Loiseau, France

Paolo Lonetti, ItalyLuis M. Lpez-Ochoa, SpainVassilios C. Loukopoulos, GreeceValentin Lychagin, NorwayFazal M. Mahomed, South AfricaYassir T. Makkawi, UKNoureddine Manamanni, FranceDidier Maquin, FrancePaolo Maria Mariano, ItalyBenoit Marx, FranceGe'rard A. Maugin, FranceDriss Mehdi, FranceRoderick Melnik, CanadaPasquale Memmolo, ItalyXiangyu Meng, CanadaJose Merodio, SpainLuciano Mescia, ItalyLaurent Mevel, FranceYuri V. Mikhlin, UkraineAki Mikkola, FinlandHiroyuki Mino, JapanPablo Mira, SpainVito Mocella, ItalyRoberto Montanini, ItalyGisele Mophou, FranceRafael Morales, SpainAziz Moukrim, FranceEmiliano Mucchi, ItalyDomenico Mundo, ItalyJose J. Muoz, SpainGiuseppe Muscolino, ItalyMarco Mussetta, ItalyHakim Naceur, FranceHassane Naji, FranceDong Ngoduy, UKTatsushi Nishi, JapanBen T. Nohara, JapanMohammed Nouari, FranceMustapha Nourelfath, CanadaSotiris K. Ntouyas, GreeceRoger Ohayon, FranceMitsuhiro Okayasu, JapanJavier Ortega-Garcia, SpainAlejandro Ortega-Moux, SpainNaohisa Otsuka, JapanErika Ottaviano, ItalyAlkis S. Paipetis, GreeceAlessandro Palmeri, UK

Anna Pandolfi, ItalyElena Panteley, FranceManuel Pastor, SpainPubudu N. Pathirana, AustraliaFrancesco Pellicano, ItalyHaipeng Peng, ChinaMingshu Peng, ChinaZhike Peng, ChinaMarzio Pennisi, ItalyMatjaz Perc, SloveniaFrancesco Pesavento, ItalyMaria do Rosrio Pinho, PortugalAntonina Pirrotta, ItalyVicent Pla, SpainJavier Plaza, SpainJean-Christophe Ponsart, FranceMauro Pontani, ItalyStanislav Potapenko, CanadaSergio Preidikman, USAChristopher Pretty, New ZealandCarsten Proppe, GermanyLuca Pugi, ItalyYuming Qin, ChinaDane Quinn, USAJose Ragot, FranceKumbakonam Ramamani Rajagopal, USAGianluca Ranzi, AustraliaSivaguru Ravindran, USAAlessandro Reali, ItalyOscar Reinoso, SpainNidhal Rezg, FranceRicardo Riaza, SpainGerasimos Rigatos, GreeceJos Rodellar, SpainRosana Rodriguez-Lopez, SpainIgnacio Rojas, SpainCarla Roque, PortugalAline Roumy, FranceDebasish Roy, IndiaRubn Ruiz Garca, SpainAntonio Ruiz-Cortes, SpainIvan D. Rukhlenko, AustraliaMazen Saad, FranceKishin Sadarangani, SpainMehrdad Saif, CanadaMiguel A. Salido, SpainRoque J. Saltarn, SpainFrancisco J. Salvador, Spain

Alessandro Salvini, ItalyMaura Sandri, ItalyMiguel A. F. Sanjuan, SpainJuan F. San-Juan, SpainRoberta Santoro, ItalyIlmar Ferreira Santos, DenmarkJos A. Sanz-Herrera, SpainNickolas S. Sapidis, GreeceEvangelos J. Sapountzakis, GreeceAndrey V. Savkin, AustraliaValery Sbitnev, RussiaThomas Schuster, GermanyMohammed Seaid, UKLotfi Senhadji, FranceJoan Serra-Sagrista, SpainLeonid Shaikhet, UkraineHassan M. Shanechi, USASanjay K. Sharma, IndiaBo Shen, GermanyBabak Shotorban, USAZhan Shu, UKDan Simon, GreeceLuciano Simoni, ItalyChristos H. Skiadas, GreeceMichael Small, AustraliaFrancesco Soldovieri, ItalyRaffaele Solimene, ItalyRuben Specogna, ItalySri Sridharan, USA

Ivanka Stamova, USAYakov Strelniker, IsraelSergey A. Suslov, AustraliaThomas Svensson, SwedenAndrzej Swierniak, PolandYang Tang, GermanySergio Teggi, ItalyAlexander Timokha, NorwayRafael Toledo, SpainGisella Tomasini, ItalyFrancesco Tornabene, ItalyAntonio Tornambe, ItalyFernando Torres, SpainFabio Tramontana, ItalySbastien Tremblay, CanadaIrina N. Trendafilova, UKGeorge Tsiatas, GreeceAntonios Tsourdos, UKVladimir Turetsky, IsraelMustafa Tutar, SpainEfstratios Tzirtzilakis, GreeceFilippo Ubertini, ItalyFrancesco Ubertini, ItalyHassan Ugail, UKGiuseppe Vairo, ItalyKuppalapalle Vajravelu, USARobertt A. Valente, PortugalPandian Vasant, MalaysiaMiguel E. Vzquez-Mndez, Spain

Josep Vehi, SpainK. Veluvolu, Republic of KoreaFons J. Verbeek, NetherlandsFranck J. Vernerey, USAGeorgios Veronis, USAAnna Vila, SpainRafael J. Villanueva, SpainUchechukwu E. Vincent, UKMirko Viroli, ItalyMichael Vynnycky, SwedenJunwu Wang, ChinaShuming Wang, SingaporeYan-WuWang, ChinaYongqi Wang, GermanyDesheng D. Wu, CanadaYuqiang Wu, ChinaGuangming Xie, ChinaXuejun Xie, ChinaGen Qi Xu, ChinaHang Xu, ChinaXinggang Yan, UKLuis J. Yebra, SpainPeng-Yeng Yin, TaiwanIbrahim Zeid, USAHuaguang Zhang, ChinaQingling Zhang, ChinaJian Guo Zhou, UKQuanxin Zhu, ChinaMustapha Zidi, France

Contents

Adaptive Impact AbsorptionJan Holnicki-Szulc, Mohamed Ichchou, Zhongdong Duan, and ukasz JankowskiVolume 2016, Article ID 4871549, 2 pages

Pneumatic Adaptive Absorber: Mathematical Modelling with Experimental VerificationGrzegorz Mikuowski and Rafa WiszowatyVolume 2016, Article ID 7074206, 13 pages

Crashworthiness of InflatableThin-Walled Structures for Impact AbsorptionCezary Graczykowski and Jan Holnicki-SzulcVolume 2015, Article ID 830471, 22 pages

Designing of Elastoplastic Adaptive Truss Structures with the Use of Particle SwarmOptimizationJacek Szklarski and Marcin WikoVolume 2015, Article ID 652824, 14 pages

Prestress Accumulation-Release Technique for Damping of Impact-Born Vibrations: Application toSelf-Deployable StructuresArkadiusz Mrz, Jan Holnicki-Szulc, and Jan BiczykVolume 2015, Article ID 720236, 9 pages

Impact Safety Control Strategy for the Battery System of an Example Electric BusZhen-po Wang, Jia Liu, Hai-tao Li, and Lei ZhangVolume 2015, Article ID 123626, 12 pages

EditorialAdaptive Impact Absorption

Jan Holnicki-Szulc,1 Mohamed Ichchou,2 Zhongdong Duan,3 and Aukasz Jankowski1

1 Institute of Fundamental Technological Research (IPPT PAN), Ulica Pawinskiego 5B, 02-106 Warsaw, Poland2Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France3Shenzhen Graduate School of Harbin Institute of Technology, Xili, Shenzhen University Town, Shenzhen 518055, China

Correspondence should be addressed to Jan Holnicki-Szulc; [email protected]

Received 10 December 2015; Accepted 14 December 2015

Copyright 2016 Jan Holnicki-Szulc et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The surging quest for safety is a clearly visible trend inmodern societies. One of the related areas of research is thedesign of systems protecting against heavy dynamic loadssuch as low and medium velocity traffic-related impacts andenvironmental loadings. Commonly applied passive systemsare typically designed to withstand a specified, well-definedheavy load scenario, which limits their performance overany wider range of loads, including the less heavy loads thatare encountered in the lifetime of a typical structure muchmore often than themaximum limiting loads. As an example,such a situation is particularly evident in the case of aircraftlanding gears, which are required by the regulations to bedesigned for the maximum specified touchdown load, whichalmost never occurs in practice. As a result, a typical landinggear during its lifetime operates constantly in a nonoptimizedmode and transfers to the fuselage much higher loads than itwould be technically possible with a proper adaptation [1].Another illustrative example is provided by the traditionallyrigid support structures of railway carriages: as demon-strated in [2], their crashworthiness can be significantlyimproved by employing controllable adaptive members. Inthese and many other applications, embodying structureswith a certain amount of intelligence, understood here asthe interconnected abilities of sensing (real-time recognitionand identification of the loading conditions) and optimumself-adaptation, can considerably increase dissipation of loadenergy, reduce operational loads transferred to the structure,and thus contribute to limiting the risk of structural failure orthe fatigue factor.

Basically, the related idea of optimum counteraction tothe excitation is one of the fundamental concepts behindmany successful applications of the control theory, such as

piezo-based damping of space structures [3]. However, typi-cal formulations lead to problems of active structural control,which require the actuators to be able to generate and transferto the structure significant external loads. Although effective,such an approach requires high energy supply, and in caseof power or hardware failures it brings the risk of danger-ous instabilities, which needs to be separately addressed.These shortcomings of active control approaches can besignificantly reduced by implementing semi-active controlstrategies and adaptive impact absorption (AIA) systems.Thegeneral concept of an AIA system, considered for the firsttime probably in [4], refers to an online adaptation of anenergy-absorbing structure to an extreme overloading by

(1) real-time recognition of an imminent impact load andidentification of its crucial characteristics,

(2) application of semi-actively controllable dissipaters ofa various nature (magnetorheological fluids, piezo-actuated devices, pneumatic systems, etc.) in order tooptimally dissipate the energy of the impact.

Such an approach allows the structural capacity for absorp-tion of unexpected extreme loads to be significantly enlarged.Good examples of potential practical applications for AIAsystems are adaptive road barriers, adaptive landing gears,and adaptive airbags for emergency landing scenarios.

The field of AIA generates a number of original researchproblems related to the optimum design of AIA systems,real-time impact load identification, globally optimum semi-active control, local modelling of the involved dissipaters,and so forth. This special issue is focused on mathematicalmodeling and optimization of AIA systems considered in

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 4871549, 2 pageshttp://dx.doi.org/10.1155/2016/4871549

http://dx.doi.org/10.1155/2016/4871549

2 Mathematical Problems in Engineering

their entireties and at the level of their adaptive components.Three of the submitted papers investigate the problemsrelated to the global design and control of AIA systems.J. Szklarski and M. Wiklo demonstrate in their contributionDesigning of Elastoplastic Adaptive Truss Structures withtheUse of Particle SwarmOptimization the applicability andeffectiveness of a heuristic algorithm of particle swarm opti-mization (PSO) to the problemof global design of an adaptiveimpact-absorbing structure, including placement of adaptivefuses and redistribution of mass and yield stress limits. Incomparison to the classical gradient-based optimization, theparticularly appealing feature of PSO is its ability to providealternative and equally good solutions, as well as its flexibilitywith respect to the definition of the objective function. Inthe paper titled Prestress Accumulation-Release Techniquefor Damping of Impact-Born Vibrations: Application to Self-Deployable Structures, A. Mroz et al. study an interestingadaptive technique (prestress accumulation-release, PAR)intended for damping of impact-born vibrations of framestructures. The crucial point providing for its effectivenessis its ability to convert the vibration energy from lightlydamped, lower order vibrationmodes into high-ordermodes,which are highly damped by means of standard mechanismsof material damping. Z. Wang et al. consider in the contribu-tion Impact Safety Control Strategy for the Battery System ofElectric Bus an important practical application and developa control strategy for a battery system in an electric busto ensure high-voltage safety during side impacts in trafficconditions.

The other group of papers in this special issue focuses onlocal modelling of actuators dedicated to AIA applications.In the paper Pneumatic Adaptive Absorber: MathematicalModelling with Experimental Verification, G. Mikuowskiand R. Wiszowaty propose, study, and experimentally vali-date a mathematical model of a pneumatic adaptive absorber.An important advantage of the proposed solution is itsdouble-chamber, closed design, which allows the device tobe used repetitively without reinflation. Finally, in the contri-bution Crashworthiness of InflatableThin-Walled Structuresfor Impact Absorption, C. Graczykowski and J. Holnicki-Szulc investigate the crashworthiness of inflatable thin-walledstructures in terms of their local response, as well as interms of their influence on the global structural response.Theadaptive control is implemented in two ways: the preimpactinflation to a desired pressure and then a gradual release ofthe pressure during impact reception.

Compared to passive impact-handling systems, thesignificantly better performance of AIA systems can beattributed to the paradigm of smart self-adaptivity, which isubiquitous in nature, but still sparsely applied in structuraland transport engineering.We hope that this special issue, bystimulating a concerted effort in facing a number of relatedresearch challenges, will contribute to their ultimate success:the dream of structural safety coming true.

Acknowledgments

We would like to thank all the authors for supportingthis special issue with their excellent contributions. We

express also our appreciation to all the reviewers for theirinsightful and constructive comments. Financial supportof the Polish National Science Centre (Project AIA, DEC-2012/05/B/ST8/02971) is gratefully acknowledged.

Jan Holnicki-SzulcMohamed IchchouZhongdong Duanukasz Jankowski

References

[1] G. Mikuowski and . Jankowski, Adaptive landing gear: opti-mum control strategy and potential for improvement, Shockand Vibration, vol. 16, no. 2, pp. 175194, 2009.

[2] J. Holnicki-Szulc and L. Knap, Adaptive crashworthiness con-cept, International Journal of Impact Engineering, vol. 30, no. 6,pp. 639663, 2004.

[3] A. Preumont, J.-P. Dufour, and C. Malekian, Active dampingby a local force feedback with piezoelectric actuators, Journalof Guidance, Control, and Dynamics, vol. 15, no. 2, pp. 390395,1992.

[4] J. Holnicki-Szulc, A.Mackiewicz, and P. Koakowski, Design ofadaptive structures for improved load capacity, AIAA Journal,vol. 36, no. 3, pp. 471476, 1998.

Research ArticlePneumatic Adaptive Absorber: Mathematical Modelling withExperimental Verification

Grzegorz MikuBowski and RafaB Wiszowaty

Institute of Fundamental Technological Research, Ulica Pawinskiego 5B, 02-106 Warszawa, Poland

Correspondence should be addressed to Grzegorz Mikuowski; [email protected]

Received 15 April 2015; Accepted 30 November 2015

Academic Editor: Zhongdong Duan

Copyright 2016 G. Mikuowski and R. Wiszowaty. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Many of mechanical energy absorbers utilized in engineering structures are hydraulic dampers, since they are simple and highlyefficient and have favourable volume to load capacity ratio. However, there exist fields of applications where a threat of toxiccontamination with the hydraulic fluid contents must be avoided, for example, food or pharmacy industries. A solution here canbe a Pneumatic Adaptive Absorber (PAA), which is characterized by a high dissipation efficiency and an inactive medium. Inorder to properly analyse the characteristics of a PAA, an adequate mathematical model is required.This paper proposes a conceptfor mathematical modelling of a PAA with experimental verification. The PAA is considered as a piston-cylinder device with acontrollable valve incorporated inside the piston. The objective of this paper is to describe a thermodynamic model of a doublechamber cylinder with gas migration between the inner volumes of the device. The specific situation considered here is that theprocess cannot be defined as polytropic, characterized by constant in time thermodynamic coefficients. Instead, the coefficients ofthe proposedmodel are updated during the analysis.The results of the experimental research reveal that the proposedmathematicalmodel is able to accurately reflect the physical behaviour of the fabricated demonstrator of the shock absorber.

1. Introduction

Mechanical energy dissipation is an important task desiredin many industry applications [1, 2]. Currently, efforts aregiven to increase productivity of automated plants and thespeed of transportation on production lines. In parallelto increasing the transportation speed, effective means ofstopping the objects on the lines are required, which isespecially evident in the production processes, where thebraking distance is limited due to packaging reasons [3]. Themost popular technique is based on hydraulic dampers due totheir effectiveness, durability, and favourable volume to forceratio [49]. However, in some applications the utilizationof fluid-based devices is undesirable due to the possibilityof toxic contamination of the goods being produced, forexample, in the pharmaceutical or food industry [10, 11]. Insuch cases damping solutions based on pneumatics can beapplied with chemically inactive gases [10, 1214].

There are known techniques for pneumatic cylindricalshock absorbers used in aeronautical or food industries [10,15]. However, due to the low viscosity of gases and theircompressibility, the energy dissipation efficiency of thesedevices does not exceed 40%, while the hydraulic dampersare characterized by the efficiency of 80% [5, 15]. Thereexist a number of patents that propose ways to increase theeffectiveness of the cylindrical pneumatic shock absorbers[1618]. Most of the solutions are based on a double-stagealgorithm of operation. After the initial compression of thegas in the cylinder, a mechanically operated valve releases themedium out of the cylinder to the surroundings. By this waythe energy accumulated in the compressed gas is dissipatedand the spring-back effect is diminished. These solutionsincrease the effectiveness of the pneumatic absorbers, butthey are limited to a single, strictly defined impact energy.When the impact energy is too low, the absorber does notrelease the gas at the proper moment and the absorption


http://dx.doi.org/10.1155/2016/7074206


does not take place. Another disadvantage is related to thefact that the compressed gas is released to the surroundings,which introduces the necessity of refilling the device aftereach working cycle.

An improved solution considered here is based onintroduction of a controlled flow between the chambers inthe cylinder via the piston. In this way it is possible todissipate energy of various magnitudes with the efficiencycomparable to hydraulic devices (80%). Moreover, the gas isnot released to the surroundings, which allows the deviceto be operated in a repeatable way. Recent developmentsin functional materials technology allow us to consider anovel approach to adaptive pneumatic shock absorber withutilization of a piezoelectric material for actuation of thedevice. A piezoelectric multilayered actuator is applied in aminiature valve positioned in the pneumatic cylinder piston[19]. In this paper we focus onmathematical modelling of thecylindrical pneumatic shock absorber with a controlled flowbetween the internal volumes.

Mathematical modelling of pneumatic actuators is ademanding task due to the necessity of taking into accountthe thermodynamic properties of the gas and the nonlinear-ities present in this kind of mechanical system. The nonlin-earities exhibit themselves mostly due to the compressibilityof the gas, internal friction, and energy transfer by heat.

Pneumatic systems are typically utilized in three domainsof applications: suspensions for vibration isolation, actuationin automatics, and mechanical absorbers. Methods of themodelling are strictly related to the field of application.

Many pneumatic systems for isolation and vibrationmitigation are developed for suspension of precisemeasuringinstrumentation [20], as well as for large structures: seismicprotection of buildings or large installations [21, 22]. Theprinciple for these systems is to suspend the protectedobject on double chamber interconnected pneumatic springs.In these cases the devices are capable of eliminating orlimiting vibrations of small amplitudes in comparison to thescale of the entire structure [23]. Since the devices can beassumed to operate in vibration of small amplitudes, severalsimplifications to the modelling approach can be assumed;for example, many authors investigate pneumatic systemsoscillating with small amplitudes around the equilibriumposition, which allows them to assume linearity of themechanical response [24]. The second important physicalphenomenon modelled in these pneumatic structures is thegas flow between the internal volumes of the structures,which has a direct influence on the dissipation properties. Inmany cases it is acceptable to assume a simplifiedmodel of thecapillary flow based on the Poiseuille model, which is derivedfor viscous fluid [24].This model assumes very lowmass flowrates of the fluid, laminar flow, and low average velocity of thefluid.

In contrast to the mentioned analyses, the mathematicalmodel of the PAA investigated in this paper must considerthe state of the gas and the internal flow between the volumesenforced in the conditions of PAA under large displacementsand high velocities. Such a process is nonstationary andincludes large deflections of the piston and time-variantsubsonic flow through the valve.

z

D F C

Figure 1: Schematic cross section of the considered adaptivepneumatic shock absorber.

When the pneumatic actuators are to be utilized asactuators in control of applications, the mathematical modelstend to be simplified in order to find a linearised version ofthe plant representation, which allows the further analysisand development of a controller to be based on the classicalcontrol theory that operates most efficiently with linear, timeinvariant plants [2528]. In these cases the simplification ofthe models is an advantage. In contrast to utilization of thepneumatic devices as actuators in automation systems, herewe consider them as dampers of energy, and we need toprecisely analyse the dissipation process from the point ofview of its effectiveness. Therefore, a precise thermodynamicmodel of the structure is developed.

The thermodynamic systems are commonly describedwith polytropic relation and an assumption of a constantvalue of the polytropic coefficient. This coefficient is stronglyrelated to the heat exchange in the system. Therefore, aconstant value of the polytropic coefficient can be assumedonly if the temperature of the object is stabilized, which wasnot the case for the considered pneumatic shock absorber.

For these reasons, in this paper we propose a numer-ical method for mathematical modelling of a cylindricalpneumatic dissipater with a controlled flow between internalchambers where the heat transfer, energy balance, and orificeflow are taken into account and thermodynamic state of gasis updated every calculation step.

Thepaper is divided into six sections, which are organisedas follows. Section 2 introduces the structure of the absorberand the principle of its operation. Then in Section 3 analysisof the system is presented and amathematicalmodel based onthermodynamic analysis is proposed. Experimental methodsand hardware are introduced in Section 4, and in Section 5the results of an experimental verification are given before theconclusions are stated in Section 6.

2. Structure and Principle of Operation ofthe Adaptive Pneumatic Absorber

Theconceptual pneumatic adaptive shock absorber is consid-ered as a piston-cylinder device equippedwith a fast operatedvalve positioned in the piston. A schematic structure of theconsidered device is presented in Figure 1.

In principle, the system is analysed as being able totransfer the mechanical energy of a moving body connectedto the piston rod into the internal energy of the gas and thento dissipate it by heat.

Mathematical Problems in Engineering 3

2.1. The Dissipation Process. The dissipation process of theexternal mechanical energy by means of the absorber is con-ducted within three phases. The first phase is a conversion ofthemechanical energy into thermodynamic energy of the gasin the process of a simultaneous expansion and contractionof the media in the two internal volumes of the absorber, and (Figure 1). In the subsequent phase, in order tocounteract the releasing of the accumulated energy via thespring-back effect, a flow through the piston is allowed in acontrolledmanner, which results in a spontaneous expansionof the gas within the cylinder volumes.The effect is a decreaseof the pressure difference on the piston and a limitationof the reaction force generated by the absorber. The finaldissipation phase is the cooling of the gas in the cylinder byheat transfer to the surroundings.Themacroscopic effect thatis intended to be achieved is an elastoplastic-like responsewith a controllable level of plastic yielding.

2.2. Idea of the Control Algorithm. The adjustable gas expan-sion process allows the value of the reacting force to becontrolled. The magnitude of the absorbed energy is adaptedaccording to the applied algorithm.The flow process betweenthe volumes is conductedwithin a period estimated as severalmilliseconds for the analysed range of impact velocities (upto 5m/s). Technically it is possible to realize such a taskby employing a fast operated piezoelectric valve. With thistechnique, it is possible to control the absorption process byadjusting the level of the mechanical energy dissipated by thesystem and to control the deceleration and forces acting onthe protected objects.

The control algorithm for the considered dissipativeadaptive system based on the absorbers can be designed torespond adequately to a wide spectrum of excitations. Forthe analysed conceptual, one-dimensional adaptive absorp-tion system the process is based on three-stage operation.During the first stage, the energy of the moving object isestimated with a system of electronic noncontact sensors ina few milliseconds before the impact event. The task canbe performed with a real-time system for velocity deter-mination [29] or more advanced systems devoted to massidentification [30]. After the magnitude of the energy to bedissipated is determined, the mechanical energy of the objectis converted into an increase of the enthalpy of the gas inthe absorber. In the third stage, an electronically controlledprocess of the accumulated energy dissipation is conductedvia a thermodynamically irreversible process of spontaneousgas expansion between the internal chambers of the absorber.This process was monitored and controlled by means ofelectronic pressure and temperature sensors positioned inthe absorber cylinder. The piezoelectrically driven valve isused to adjust the process of the gas expansion and thereforeto maintain the magnitude of the converted energy on apredefined level in accordance with the piston position. Thealgorithm is implemented in a dedicated control modulethat is provided with signals related to the pressures andtemperatures levels in the chambers.

The presented configuration of the absorber enables us togenerate the reaction force on a desired level in dependence

Fe + Fa

Fe + Fa

pA

pA

TDD

pD

TCCpC

z + dz

TD + dTDD + dD

pD + dpD

TC + dTCC + dCpC + dpC

D

CD

C

z

z

Figure 2: Scheme of the absorber.

on the magnitude of the energy to be dissipated. Therefore,the device can be considered as adaptive.

3. Mathematical Model of the PneumaticShock Absorber

In order to properly reflect the mechanical response ofthe system, it is analysed in terms of the mechanical andthermodynamic processes. The analysis is divided into threesections devoted to dynamics of the piston treated as a rigidbody, thermodynamics of the gas in the absorber chambers,and gas flow through the valve. Two control volumes aredistinguished inside the absorber, and , as depicted inFigure 2.

3.1. Forces Acting on the Piston Treated as a Rigid Body. Thetotal force equilibrium of the piston can be defined as

()

(, , ) + (, , ) +

() = 0,

(1)

where the symbols are denoted as , external excitation;

,

force resulting from pressure in chamber ;

, force

resulting from pressure in chamber ;

, force resulting

from ambient pressure; , friction force; , displacement

of the piston; , , gas temperatures in corresponding

volumes; , mass of the gas in volume ;

, mass of the

gas in volume.The forces acting on the piston can be formulated as

= (, ) 1, (2)

= (, ) (1 2) , (3)


= 2, (4)

= sgn () , (5)

where the following symbols are used: , , gas densities in

chambers and;1, effective piston area;

2, area of piston

rod radial cross section; , ambient pressure;

, friction

coefficient.

3.2. Thermodynamics of the Gas in the Absorber Chambers.The thermodynamic processes in the cylinder are describedwith energy balance equations for the chambers: and ,the gas state equation and the relations governing the com-pressible fluid flow through the applied valve. The followingassumptions are introduced during the analysis.

3.2.1. Ideal Gas Law. The gas filling the volumes and is dry nitrogen, which is operated in the temperature above200K. Therefore, the assumption of the ideal gas is approvedto be valid as the state of the gas is not approaching the criticalpoint:

V = . (6)

3.2.2. Uniform-State, Uniform-Flow Process. Volume of theabsorber chambers and the speed of sound determine thetime necessary for the gas to reach the uniform state. Since theconsidered chambers have dimensions of the order of 0.1mand the speed of sound in normal conditions is approximately340m/s, the time it takes for the gas to reach a uniformpressure is negligibly small. Furthermore, the gas is assumedto mix instantaneously in the chambers, so the fluid isdescribed by a uniformly distributed temperature in eachchamber.

The further assumptions regarding the gas dynamics areformulated as follows:

(i) the state of the mass entering the valve is constantwithin the time steps and uniform over the volumeof the valve where the flow occurs;

(ii) the state of the mass within the control volumes canchange with time, but at any instant of time the stateis uniform over the entire control volume;

(iii) the changes in the kinetic energy of the gas arenegligible;

(iv) the inertia and gravity forces of the gas are negligible.

3.2.3. Thermodynamic Processes Assumption. During theoperation of the absorber, the gas is simultaneously com-pressed and decompressed in the chambers separated by thepiston.The processes that take place cannot be defined clearlyas isothermal or adiabatic, since the expected rate of theprocess varied in dependence on the work conditions. Forlow velocity operations the processes can be treated as closeto isothermal, since in such a case there is enough time torelease the energy by heat from the gas and its temperaturecan be assumed to remain unchanged. Otherwise, during afast process the time is too short for the energy transfer of a

significant magnitude to occur and therefore the process canbe treated as adiabatic. In between the mentioned cases theprocess can be assumed to be polytropic:

= const., (7)where is pressure, is volume, and is polytropiccoefficient. Adequately, = 1 for an isothermal process and = 1.4 for an adiabatic process.

In the analysed case, the model being developed isexpected to reflect the behaviour of the absorber under awiderange of operational conditions and to be valid irrespectiveof the piston kinematics. For that reason, it is not adequate todescribe the gas processes with the polytropic model with aconstant parameter .

Therefore, within each discrete time step of the computa-tion, the state of the gas in the control volumes is calculatedin the following manner: first, the gas is assumed to changeits state adiabatically ( = = 1.4) and then the gasstate parameters are recalculated based on an analysis of theinternal energy balance with the heat exchange and the massexchange between the chambers taken into account. Thatapproach allows us to update the final state of the gas.

During each time step the following analysis is conducted:(i) determination of the gas state change due to the

volume change on the basis of the adiabatic model,(ii) determination of the internal energy of the gas,(iii) determination of the heat exchange between the

control volume and the surroundings (with the actualarea of the cylinder walls interfacing the gas com-puted),

(iv) determination of the energy balance in the controlvolume with the mass and heat exchange taken intoconsideration,

(v) recalculation of the gas state parameters on the basisof the energy balance equation.

This method allows us to account for the changes in thethermodynamic processes in time and therefore to reflectthe polytropic-like process in dependence on the operatingconditions.

The assumption of the ideal gas allows us to calculate thethermodynamic state parameters as

() =

1(

1

1 1)

,

() =

1(

1

1 1)

1

,

(8)

where 1

is the initial volume of chamber , 1

is theinitial pressure in chamber ,

1is the initial temperature

in chamber , and is the adiabatic coefficient.Also,

() =

1(

1

1+ (1 2) )

,

() =

1(

1

1+ (1 2) )

1

,

(9)


where1

is the initial volume of chamber,1

is the initialpressure in chamber , and

1is the initial temperature in

chamber.

3.2.4. Mass Continuity and Energy Balance. For case of ageneral volume with the uniformity assumptions, the masscontinuity equations for volumes and take, respectively,the following forms:

+ = 0,

+ = 0,

(10)

where is the mass of the gas in the control volume ,

is the mass leaving the control volume , and is the

mass entering the control volume . The symbols with thesubscript denote the same quantities in volume.

The energy balance in the control volumes can be statedas [31]

+ = + +

() ,

+ = + +

() ,

(11)

where is the heat transferred to the control volume,

is

the specific enthalpy of the gas occupying control volume ,is the work done by the gas in the control volume , and

is the specific internal energy of the gas in volume . The

symbols with index describe the same quantities related tovolume.

The values of

and

depend on the differencebetween the temperatures of the gas in the respective controlvolume and the surroundings, the area of cylinder walls thatthe gas is in contact with, and the material constants,

= () (

()

()) ,

= () (

()

()) ,

(12)

where () is the area of cylinder being in contact with gas

in volume , is the heat transfer coefficient, and () is the

ambient temperature.The symbols with index describe thesame quantities related to volume.

According to the continuity principle, the mass of gasleaving the control volume is equal to the mass of gasentering the control volume and vice versa:

=

, (13)

where

={

{

{

> 0 when

>

< 0 when

<

(14)

and

={

{

{

> 0 when

>

< 0 when

< .

(15)

Let us denote the transferred mass of gas as

= =

. (16)

The specific enthalpiesandof the gas in the volumes

and , respectively, are different in general. Therefore,when the assumption of the isenthalpic flow through thevalve holds true, the specific enthalpy of the gas transferredbetween the volumes depends on the flow direction, and it isequal to

= , when the gas leaves volume ,

= , when the gas leaves volume ,

(17)

where is the specific heat of the gas at the constant pressure.

In the considered range of temperatures between 200Kand 400K the specific heat of the gas is assumed to beconstant.

The work done by the gas is defined for the particularcontrol volumes as

= ()

,

= ()

.

(18)

3.3. Mass Flow Rate through the Valve. The valve is assumedto operate in a bistable, on-off mode. In the opened positionthe mass flow rate

(16) depends on the Mach number Ma

and on the gas state parameters at the inlet of the valve 0, 0.

In themodel, the values are taken as equal to the actual valuesof the parameters in the control volumes as

0={

{

{

when >

when < ,

0={

{

{

when >

when < .

(19)

The flow is assumed to be an adiabatic process (thereis no heat exchange between the gas and the walls of thevalve but the mechanical losses are considered [19]). Theflow losses are described with the discharge coefficient

treated as a characteristic design parameter of the valve [32].In accordance with the throttled flow theory [31, 32], the flowis assumed to be choked when the Mach number Ma is closeto 1. The mass flow rate of the gas exchanged between thechambers can be thus expressed in the form [19]

=

{{{{{

{{{{{

{

Ma0/

0

[1 + ( 1)Ma2/2](+1)/2(1), if Ma < 1

0

0

(2

+ 1)

(+1)/2(1)

, if Ma = 1,

(20)

where is the cross section of the flowduct, is the isentropiccoefficient, and is the gas constant.


Figure 3: View of the absorber and the piezoelectric valve.

CylinderValve PistonJoint Joint

Thermocouple Thermocouple Piston rodGas flow through the valve

Pressuretransducer

Pressuretransducer

Electricalconnection

to thepiezo valve

Figure 4: Main components of the absorber.

FForce

transducerInvestigated absorber

Cup-and-balljoint

Actuator

Cup-and-balljoint

Figure 5: Scheme of the testing stand.

3.4. Governing Equations. By substituting (18) and (10) to theequations of energy balance (11), and taking into considera-tion (14), (15), and (17), the internal energy of the gas in thecontrol volumes and can be calculated as follows:

for >

()+ () 1

+ = 0,

() () (

1 2)

= 0,

(21)

for <

()+ () 1

= 0,

() () (

1 2)

+ = 0.

(22)

iiiii

iii

iv

v

vi

viivii

Figure 6: Laboratory testing stand.

3.5. Control Algorithm. The control procedure for the pneu-matic adaptive shock absorber is aimed at maintaining apredefined level of difference between forces (2) and (3)acting on the piston. The pressure of the gas has a directimpact on the reaction force generated by the absorber. Thevalve opening control function has the following form:

() ={

{

{

open, if () > ref +

close, if () < ref ,(23)


IRF9640

IRF644

Piezo-actuator

Valve

74HC0474HC04

74HC04

IRF644

MCULPC1769

PWMout

Pa

Pressuretransducer

PaPressure

transducer

ADCin

ADCin

190V190V

0.5W

5W

5V1

22k

10k

10k

27

2710W

10W

7F

Figure 7: Scheme of the laboratory control system.

where symbols are denoted as open/close, signal of open-ing/closing the valve;

(), reaction force; ref , reference

level; , tolerance range.

4. Experimental Program

Thedata required to verify the proposedmethod ofmodellingis acquired by means of a small scale device. The consideredconcept of the adaptive pneumatic absorber is demonstratedin laboratory scale with a demonstrator presented in Figure 3.The device is designed in dimensions: 32mm diameter,300mm length, and 100mm stroke, which allowed it todissipate the energy of 40 J per stroke.

The demonstrator is a piston-cylinder device equippedwith a valve positioned in the piston. In order to ensure theadequately short time response of the system, the valve isactivated with a piezoelectric stack. The closed loop controlsystem of the demonstrator is fed with signals from twopressure and two temperature sensors positioned within theabsorber housing as depicted in Figure 4.

The testing campaign of the absorber is conducted bymeans of a hydraulic excitation system. The system consistsof an electronically controlled servohydraulic actuator (MTS242.01 actuator, Eden Prairie, MN, USA), positioned hori-zontally in line with the tested adaptive absorber (Figure 5).The actuator-absorber connections are realized via cup-and-ball joints in order to prevent the transmission of bendingmoments and shear forces into the structure of the testedspecimen. The actuation system enables us to examine theabsorber under periodic axial loading with the displacementreference signal.

The complete experimental stand is depicted in Figure 6and consists of

(i) adaptive absorber,(ii) pressure transducers,(iii) electronic control unit,

(iv) piezoelectric valve supplier,(v) hydraulic excitation system,(vi) force cell,(vii) hydraulic grips.

The testing program is aimed at verification of theproposed mathematical model under a variety of excitationconditions. The independent parameters are the rate ofdisplacement, the initial pressure inside of the cylinder, andthe expected (controlled) magnitude of the reaction force.The testing program is defined as follows:

(i) kinematic excitation with triangle signal, amplitude40mm, frequency computed based on the expectedvelocity of the piston,

(ii) velocity of the piston: 0.25m/s,(iii) initial gas pressure in the cylinder: 0.3MPa, 0.5MPa,(iv) ambient pressure: 100 kPa,(v) ambient temperature: 292K,(vi) magnitude of the reaction force: 100N, 200N, 300N,

400N, and 500N.

4.1. Electronic Control Module and Control Algorithm Realiza-tion. The proposed control algorithm is realized in labora-tory conditions with an intentionally designed and fabricatedcontrol system that operates in closed loop. The system con-sists of a microcontroller, a power supply for the piezoelectricvalve, and a set of transducers (Figure 7).

The control of the reaction force in the absorber isachieved through the use of the feedback loop that iscomposed of

(i) control module, the circuit with the microcontrollerunit (MCU),

(ii) voltage supplier,


Interruptrequest

Loading the resultof the last ADC conversion

Force calculation:F = p1 S1 p2 S2 patm. S3

S1

S2

S3

No

No

No

No No

Yes

YesYes

Yes Yes

F > 0

Openingthe valve

Openingthe valve

Closingthe valve

Closingthe valve

Clearingthe interrupt

F < FUPPER LIMIT F > FUPPER LIMIT

F > FLOWER LIMIT F < FLOWER LIMIT

Figure 8: Algorithm executed in the interrupt service routine.

(iii) voltage relay, the circuit adjusting the driving signalaccording to the specific characteristics of the piezo-electric stack in the valve.

MCU peripherals include analog to digital converters (ADC)that in the assembled system are connected to the outputs ofthe pressure transducers. On the basis of the pressure mea-surements, the forces acting on the piston are computed inthe MCU (denoted by LPC 1769 in Figure 7). The calculatedvalues act as the input signals for a Pulse Width Modulation(PWM) control algorithm, which operates the valve in

a bistable mode. The PWM output is amplified by means ofthe voltage relay and feed to the piezoelectric actuator.

The algorithm implemented in the MCU is presentedin Figure 8. Every time step of operation, the followingroutine is executed: after the interrupt request comes fromthe PWM clock, the most recent ADC conversion result isstored in the variables representing pressure values insidethe cylinder (see Figure 8). In the subsequent steps of theinterrupt handler, the force coming from the gas acting on thepiston is calculated and checked if it is positive or negative.Then it is compared to the two previously defined limits


ExperimentSimulation

ExperimentSim., n = 1.1Sim., adapted

Vol. C: exp.Vol. D: exp.C: sim., n = 1.1

D: sim., n = 1.1C: sim., adaptedD: sim., adapted

105

0

5

10

Pres

sure

(Pa)

0.1 0.2 0.3 0.40Time (s)

0.1 0.2 0.3 0.40Time (s)

400

200

0

200

400

Forc

e (N

)

0.1 0.2 0.3 0.40Time (s)

0

0.02

0.04

0.06

0.08

0.1D

ispla

cem

ent (

m)

Figure 9: Comparison of simulation results obtained with two thermodynamic models: with a constant value of the polytropic coefficient(blue plot) and with a recalculated value of the polytropic coefficient (red plot). Excitation path (displacement) and mechanical response(reaction force and pressure) of the adaptive absorber at init = 0.5MPa, CTRL = 300N, and = 20Hz.

UPPER LIMIT and LOWER LIMIT, which define the referencerange for the desired force level.

5. Results

Thedata acquired during the experimental tests are comparedto the results coming from the numerical simulation. Thecomparison is conducted for several operational conditionsof the absorber in accordance with a defined testing program.The principle aim of this task is to verify the proposedmathematicalmodel versus the experimentally obtained data.The parameters of the numerical model utilized in theexample analysis are summarised in Table 1.

The conducted verification campaignhas three objectives:first, to reveal the advantage of the proposed method ofmodelling in comparison to a model with a constant valueof the polytropic coefficient; second, to exhibit a properresponse of themodel under various conditions of excitation;third, to prove the correct reaction of the model to a changeof control input signals.

A comparison between the response of the proposedmodel and a one with a constant value of the polytropiccoefficient is depicted in Figure 9. The presented resultsare obtained for a sinusoidal displacement excitation with

Table 1: Model parameters.

Piston area 1 8.042 04 [m2]

Piston rod area 2 7.854 05 [m2]

Cylinder wall thickness 0.002 [m]Cylinder diameter 0.032 [m]Initial volume ini 9.402 05 [m

3]Initial volume ini 1.165 05 [m

3]Adiabatic constant 1.4 []Gas constant 296.8 [J/(kgK)]Specific heat by constantvolume V 743 [J/(kgK)]

Specific heat by constantpressure 1039 [J/(kgK)]

Heat transfer coefficient 20 [W/(m2K)]Valve discharge coefficient 0.6 []Ambient pressure 1003 [Pa]Ambient temperature 292 [K]

the velocity range of 00.5m/s. The conditions of the test areinitial pressure in the cylinder ini = 0.5MPa, control signalinput CTRL = 300N, displacement amplitude = 0.04m,


Press. CexperimentPress. Dexperiment

Press. CsimulationPress. Dsimulation

105

0.1 0.2 0.3 0.4 0.5 0.6 0.70Time (s)

0

5

10

Pres

sure

(Pa)


0.1 0.2 0.3 0.4 0.5 0.6 0.70Time (s)

400

200

0

200

400

Forc

e (N

)


0.1 0.2 0.3 0.4 0.5 0.6 0.70Time (s)

0

0.02

0.04

0.06

0.08

0.1D

ispla

cem

ent (

m)

(a)

Press. CexperimentPress. Dexperiment

Press. CsimulationPress. Dsimulation

105

0

5

10

Pres

sure

(Pa)

0.05 0.1 0.15 0.2 0.25 0.3 0.350Time (s)


400

200

0

200

400

Forc

e (N

)

0.05 0.1 0.15 0.2 0.25 0.3 0.350Time (s)


0

0.02

0.04

0.06

0.08

0.1

Disp

lace

men

t (m

)

0.05 0.1 0.15 0.2 0.25 0.3 0.350Time (s)

(b)

Figure 10: Verification of simulation results versus experimental data under sinusoidal excitation signal. Excitation path (displacement)and mechanical response (reaction force and pressure) of the adaptive absorber at init = 0.5MPa and CTRL = 300N. (a) = 10Hz. (b) = 20Hz.

and excitation frequency = 20Hz. The modelling resultsprove that the proposed model with adapted polytropiccoefficient is able to reflect the absorber behaviour with asignificantly increased accuracy.

The second part of the verification is devoted to provingthe method of modelling the PAA under varying excitationconditions. The absorber is excited with sinusoidal signals offrequencies 10Hz and 20Hz and amplitude 0.04m, which,respectively, lead to the excitation velocities 00.25m/s and00.5m/s. As it is depicted in Figure 10, the proposedmethodof modelling allowed the mechanical response of the PAA tobe reflected with minor discrepancy.

The third part of the experimental verification is aimed atproving that the proposedmodel is able to accommodate alsoa variety of control signal inputs and initial internal pressurelevels. Figure 11 depicts the absorbers response in the domainof time under displacement excitation with a triangle signal.The test parameters are as follows: initial pressure: ini =0.3MPa and 0.5MPa, excitation amplitude: = 0.04m, andcontrol input signal: CTRL = 100N, 200N, 300N, 400N,and 500N. Figure 12 depicts the results of verification inthe domain of displacement. The presented plots reveal thatthe proposed model is relevant and stable in reflecting themechanical behaviour of the PAA in the variety of conditions.



0

0.02

0.04

0.06

0.08D

ispla

cem

ent (

m)

0.1 0.2 0.3 0.4 0.5 0.60Time (s)


0.1 0.2 0.3 0.4 0.5 0.60Time (s)

500

0

500

Forc

e (N

)

CexperimentCsimulation

DexperimentDsimulation

105

0

5

10

Pres

sure

(Pa)

0.1 0.2 0.3 0.4 0.5 0.60Time (s)

(a)


0

0.02

0.04

0.06

0.08

Disp

lace

men

t (m

)

0.1 0.2 0.3 0.4 0.5 0.60Time (s)


500

0

500

Forc

e (N

)0.1 0.2 0.3 0.4 0.5 0.60

Time (s)

105

0

5

10

Pres

sure

(Pa)

0.1 0.2 0.3 0.4 0.5 0.60Time (s)

CexperimentCsimulation

DexperimentDsimulation

(b)

Figure 11: Verification of simulation results versus experimental data under triangle excitation signal. Excitation path (displacement) andmechanical response (reaction force and pressure) of the adaptive absorber at V = 0.25m/s, CTRL = 100, 200, 300, 400, and 500N. (a)init = 0.3MPa. (b) init = 0.5MPa.

6. Concluding Remarks

In this paper a method for mathematical modelling of anadaptive pneumatic absorber is presented. The absorber isa new conceptual device that might be utilized in variousfields of application and therefore, from the design point ofview, it is crucial to develop a reliable numerical method forreflecting its physical behaviour. The presented method ofmodelling with recalculation and updating of the polytropiccoefficient every time step proves to be effective in reflectingthe mechanical behaviour of the PAA. An advantage of thepresented method is its simplicity and the relatively smalldemand for computation resources in comparison to theCFD methods of modelling. Therefore, it can be successfullyutilized as a complementary simulation tool to the CFD

approach. The model can be used for simulation of extendeddynamic systems containing Pneumatic Adaptive Absorbers.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The support of Structural Funds in the Operational Pro-gramme Innovative Economy (IE OP) financed from theEuropean Regional Development Fund Projects Modern



500

400

300

200

100

0

100

200

300

400

500Fo

rce (

N)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080Displacement (m)

(a)


500

400

300

200

100

0

100

200

300

400

500

Forc

e (N

)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080Displacement (m)

(b)

Figure 12: Verification of simulation results versus experimental data under triangle excitation signal. Reaction force of the absorber indomain of the piston displacement at V = 0.25m/s, CTRL = 100, 200, 300, 400, and 500N. (a) init = 0.3MPa. (b) init = 0.5MPa.

Material Technologies in Aerospace Industry (POIG.0101.02-00-015/08) and Polish National Science Centre Project AIA(DEC 2012/05/B/ST8/02971) is gratefully acknowledged.

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Research ArticleCrashworthiness of Inflatable Thin-Walled Structures forImpact Absorption

Cezary Graczykowski and Jan Holnicki-Szulc

Institute of Fundamental Technological Research, Pawinskiego 5b, 02-106 Warsaw, Poland

Correspondence should be addressed to Cezary Graczykowski; [email protected]

Received 15 June 2015; Revised 20 September 2015; Accepted 27 September 2015

Academic Editor: Roberto Montanini

Copyright 2015 C. Graczykowski and J. Holnicki-Szulc. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The paper describes application of innovative, inflatable thin-walled structures for absorption of the impact loading and thoroughlyinvestigates their crash characteristics. The proposed concept assumes inflation of thin-walled structures with compressed gas ofappropriately adjusted pressure in order to improve their basic mechanical properties, enhance energy dissipation capabilities, andincrease corresponding durability to impact loading. In the first part of the paper the influence of compressed gas on mechanicalcharacteristics of aluminium beverage can is analysed experimentally and by the corresponding numerical simulations. Thefollowing section proposes and numerically verifies three diverse engineering applications of inflatable thin-walled structuresfor impact absorption. Finally, the last part introduces the concept of adaptive inflatable barrier and briefly presents threesimple strategies of pressure control. Both the performed basic experiment and the conducted numerical simulations show theadvantageous influence of compressed gas and prove the feasibility of using inflatable thin-walled structures for impact absorption.

1. Introduction

Thin-walled structures are commonly used in transport andmechanical industry because of their large stiffness, dura-bility, and small weight. Additionally, thin-walled structuresmade of aluminium or steel effectively absorb the energy ofaxial loading due to the process of plastic folding. The abovefeatures cause that thin-walled elements are efficiently appliedin crushing zones of trains, cars, and other energy absorbingstructures.

Mechanics of thin-walled structures subjected to largedeformations can be analysed by both analytical and numer-ical methods. Despite many complex phenomena whichoccur during crushing of thin-walled structures (includingmaterial and geometrical nonlinearities such as plasticity,buckling, and contact) many analytical and semiempiricalsolutions were derived thanks to application of simplifyingassumptions concerning material models and kinematics ofdeformation; comparewell-known solution for axisymmetriccrushing of circular tube by Alexander [1]. Comprehensiveapproach to mechanics of thin-walled profiles, methods

of determination of their folding patterns, and estimationof impact absorption capabilities were presented in booksby Jones [2] and conference proceedings by Jones andWierzbicki [35]. The research in the field of thin-walledstructures had also influenced development and stimulatedprogress in design of complex crashworthy structures such aspassenger vehicles, trains, and ships [6].

The numerical approach to impact problems was pos-sible owing to contemporary understanding of structuraldynamics and development of finite element method as anefficient tool for solving complex problems of nonlinearmechanics [7, 8]. The recent finite element codes dedicatedto crashworthiness problems [9] utilise explicit methods forintegrating equations of motion [10, 11].The choice of explicitmethods is motivated by the ease of their numerical imple-mentation and the possibility of obtaining efficient and robustsolution of strongly nonlinear crashworthiness problems. Onthe other hand, the main drawback of explicit methods is therequirement of using relatively short time steps in order toensure numerical stability and the existence of critical timestep beyondwhich numerical instabilities occur.Thus, in case


http://dx.doi.org/10.1155/2015/830471


of rigorous stability conditions leading to extremely shorttime step size the explicit methods are sometimes replaced bythe implicit ones. The implicit methods have, however, theirown limitations including more difficult implementation forstrongly nonlinear problems, complex computations at asingle time step, and common difficulties with convergenceof the numerical solution.

In addition to the above classical numerical methodsthe alternative so-called semianalytical approaches to crash-worthiness problems were proposed and developed. Themethods of this group provide substantial simplification ofthe crashworthiness problem and corresponding reductionof its computational cost. The examples are the methodsbased on discretisation of the structure into the so-calledmacroelements with predefined folding patterns developedby Abramowicz [13].

Despite many years of research and development themechanical properties and energy absorption capabilities ofthin-walled absorbers of various shapes and cross sectionsas well as their practical applications are still studied, forexample, byKimandWierzbicki [14, 15] orHan andYamazaki[16]. A full review of various types of conventional thin-walled impact energy absorbers will not be performed here,since it can be found in subject literature, for example,in comprehensive paper by Alghamdi [17], Ph.D. thesisby Lee [18], or Zhang [19]. Instead, the attention will befocused on innovative methods of enhancement and controlof crashworthy capabilities of thin-walled structures.

Crashworthiness of thin-walled structures subjected toaxial loading was improved in variousmanners. In particular,filling thin-walled structure with granular material (like sandor grain) and taking the advantage of friction forces generatedbetween granules during impact were proposed by Lee [18].In turn, Zhang [19] examined the usage of buckling initiatorsactivated just before expected impact in order to reduce initialpeak of crushing force. Moreover, thin-walled absorberscomposed of two sections joined by pyrotechnic connectors,which can be detached during impact in order to reduceglobal stiffness, were proposed and tested experimentally byOstrowski et al. [20]. Another important concept is fillingaxially loaded circular tubes with compressed gas in orderto take advantage of the effect of gas compression duringimpact and to affect the shape of deformation of thin-walledabsorber; compare Zhang [19]. The above concept was alsostudied by Gren [21] who had developed precise analyticalmodel of the process of longitudinal crushing assisted bygas pressure. Finally, the application of the above concept tocontrol stiffness of the vehicle longitudinal frontal memberswas studied by Pipkorn and Haland [22].

In contrast to previously mentioned solutions, the con-cept presented in this paper is focused on thin-walled struc-tures subjected to lateral impact. In case of lateral loading,thin-walled structure easily undergoes buckling and localplastic yielding and, as a result, only a small part of theimpact energy is dissipated. However, as it will be shownin the following sections, filling thin-walled structure withcompressed gas of properly adjusted pressure may signifi-cantly improve its mechanical properties and increase globaldurability to impact load. The paper starts with a simple

experiment where aluminium beverage can is subjected tothe action of transverse force and the influence of internalpressure on buckling phenomenon is observed. In addition,mechanical response of the inflated can is studied withthe use of various numerical models. Further, three diverseapplications of inflatable thin-walled structures for impactabsorption are proposed. The corresponding simulationsof impact process are conducted and general conclusionsconcerning the effects of structure inflation are drawn.Finally, the last section proposes the concept of adaptiveinflatable barrier with real-time pressure control. The cor-responding simplified two-dimensional model is applied todemonstrate the potential of compressed gas in minimiz-ing generated internal forces, reducing impacting objectdeceleration, and obtaining desired final deformation of thestructure.

Several ideas presented within this paper are included inthe patent claim [23], while initial and immature versions ofthe proposed concepts were presented in conference paper[24] dedicated to buckling of the aluminium can and designof the inflatable barrier.

2. Basic Experiment: Buckling ofInflated Thin-Walled Can

Basic experiment confirming the beneficial influence offilling thin-walled structure with compressed gas was per-formed with the use of aluminium beverage can (the exper-iment was performed by Mr. Rafa Chmielewski, who isgratefully acknowledged for his work). In the conducted teststhe right end of the empty can was clamped around thecircumference, while the left end was reinforced by specialring and subjected to action of vertical force F acting upwards(Figure 1). As a result of applied boundary conditions andexternal force, the can was acting as a cantilever. The appliedloading caused bending of the can, tension of its lower wall,and compression of the upper one.

In the initial experiment the canwas not sealed so internalpressure was permanently equal to ambient pressure. Duringthe performed test the value of vertical force was graduallyincreased. Sudden collapse of the structure was observedwhen vertical force reached 155N and it was caused bybuckling of cylinder sidewalls. The buckling region covereda large part of the can located between its middle part andthe support (Figure 1(a)). Moreover, buckling shape wasapproximately symmetrical on both sides of the cylinder.The collapse of the structure occurred at relatively smalldisplacement of the left edge, which indicates small workdone by external force before buckling.

In the second stage of the experiment the can was sealedand inflated with compressed gas to pressures of 0,2MPa,0,4MPa, 0,6MPa, and 0,8MPa. During the experiment thepressure of gas was not externally changed or controlled.Theapplied internal pressure was expected to cause increase ofthe critical force since it generates additional surface loading,which acts against buckling phenomenon. Two particulareffects of internal pressure, which can be distinguished, areas follows:


(a) (b)

(c)

Figure 1: Deformation of the beverage can with overpressure: (a) 0MPa, (b) 0,6MPa, and (c) 0,8MPa.

(i) reduction of compressive stresses arising at prone-to-buckling upper wall,

(ii) counteracting inward deformation which occurs dur-ing buckling of the empty can.

According to the above expectations, the value of critical forcecausing buckling of the structure was gradually rising (upto 610N for internal pressure of 0,6MPa). Along with anincrease of critical force, the area of buckling was decreasingand moving into the direction of the support, Figure 1(b).Moreover, the buckling phenomenon had occurred at appar-ently larger displacement of the left end of the canwhich indi-cates significant increase of work done by external force andcorresponding increase of energy absorbed by the inflatedcan.

In the last considered case of pressure of 0,8MPa, themaximal value of applied force was approximately equal to700N; however a sudden burst of the can had occurred brieflyafter buckling because of material rapture in the vicinity ofthe support (Figure 1(c)). Unfortunately the limitations ofthe conducted experiment, in particular insufficient qualityand frame rate of applied camera, did not allow identifyingthe location where material rapture and corresponding burstwere initiated. Thus, the only available data concerningthe form of can destruction was circumferential shape ofcan rupture in the vicinity of the support (Figure 1(c))and irregular rapture of disconnected part in longitudinaldirection (not presented).

The explanation of the phenomenon of sudden burst ofthe can at the end of the process may be twofold. The mostdirect hypothesis is that burst was initiated in the regionlocated at the bottom of the can in the vicinity of the supportand it was caused by exceeding maximal allowable value

of tensile stress, which induced peripherally propagatingrupture of the material. According to alternative concept theburst was initiated at a certain point of the buckling area, forexample, top or side of the can where buckling phenomenonis manifested by distinct indentations (cf. Figure 1(b)), whichcan provoke the occurrence of material rupture.

The experiment reveals two important features of theinflatable thin-walled structures. Primarily, the durability ofthin-walled structures to lateral loading and the amount ofdissipated energy can be substantially increased by the use ofcompressed gas. On the other hand, application of excessiveinternal pressure is related to a danger of sudden destructionor blast of the structure. This implicates the necessity ofprecise adjustment of the initial value of pressure or eventhe requirement of its real-time control during the process ofdeformation.

3. Numerical Analysis ofInflated Thin-Walled Can

Theproblem of bending of inflated canwas simulated numer-ically with the use of finite element method. The interactionof the internal gas and the structurewasmodelled by applyinguniformly distributed surface loading perpendicular to thesurface of the can. Since in considered simulations the inter-nal volume of the can (and thus the volume of gas) changesinsignificantly the corresponding changes of gas pressure areexpected to be negligible. Consequently, the action of internalpressure was modelled by distributed loading of a constantvalue. The following numerical analyses were conducted:

(1) static analysis of bending of empty and inflated can,


(2) linear buckling analysis of empty and inflated canduring bending,

(3) nonlinear analysis of bending, buckling, and burstingof the can,

(4) linear buckling analysis of the can during axial com-pression,

(5) modal analysis of empty and inflated can.

Thin-walled cylindrical shell structure considered in numer-ical simulations had the dimensions of the aluminium can:length = 170mm, radius of the base = 33mm, and thicknessof the wall = 0,1mm.Thickness of both bases of the cylinderwas equal to 0,5mm tomodel large stiffness of the reinforcingring from the experiment. Since the imperfections of the canfrom cylindrical shape and indentations of its lateral walls,which substantially decrease global bending resistance of thestructure, are probabilistic in nature and hard for estimationthey were not directly introduced into the model. Instead,their influence on the response of the can was modelledin a simplified way by decreasing the standard value ofaluminium Youngmodulus by 20% (from 70GPa to 56GPa).Such approach provides agreement of the numerical analysisand the buckling experiment for the cases of both empty andinflated aluminium can.

The main solver applied for the numerical simulationswas commercial finite element codeABAQUS (both Standardand Explicit) and the main finite element applied in simu-lations was doubly curved thin shell element with reducedintegration, hourglass control, and finite membrane strains(S4R). Different sizes of finite elements were used dependingon the type of conducted analysis.

3.1. Static Analysis. Static analysis of bending of the inflatedcan was aimed at investigating the influence of internalpressure on distribution of internal stresses in cylinderwalls. Internal pressure was modelled as distributed loadingapplied perpendicularly to all internal walls of the cylinder.In turn, the applied vertical load was distributed along thecircumference of the left base of the cylinder. At this stageof analysis, characteristics of the material were assumed aslinear elastic ones; however, the equilibrium equations wereconsidered in actual configuration.

The problem of bending of the inflated can consists of thestep of inflation (the increase of pressure loading) and the stepof bending (the increase of external force). In finite elementnotation the problem solved reads as follows:

Step 1: K (Qp, q) q = Qp (, q) , (1a)

Step 2: K (Qmaxp ,QF, q) q

= Qp (max

, q) +QF (, q) ,(1b)

where K is the stiffness matrix dependent on actual appliedloading Q and actual deformation of the structure q. Thequantity Qp(, q) is the load vector caused by internal pres-sure p (in particularQmaxp = Qp(

max, q)) and QF(, q) is the

load vector caused by external force F of a magnitude F. Both

load vectors depend on actual deformation of the structuresince pressure loading is perpendicular to the walls of thecan and external force is assumed to follow the structuredeformation. Let us note that the case when both forces areapplied simultaneously

K (Qp,QF, q) q = Qp (, q) +QF (, q) (2)

is not equivalent to problem defined by (1a), (1b), whichis reflected in different arguments of the stiffness matrixand consequently different path of structure equilibrium.Significant simplification of the above problem is obtainedby setting equilibrium equations in initial (nondeformed)configuration. In such a case the problem solved reads

Kq = Qp () +QF () (3)

and can be decomposed into two simpler problems:

Kqp = Qp () ,

KqF = QF () ,(4a)

q = qp + qF, (4b)

where qp and qF indicate displacements caused by internalpressure and vertical force. The total displacement q can becalculated as a sum of the two above displacements accordingto the superposition principle. Equations (4a), (4b) reveal,in a simplified manner, the influence of internal pressure ondistribution of internal forces in considered structure andcorresponding shape of deformation.

The corresponding numerical simulations were con-ducted with the use of ABAQUS Standard. The exemplarynumerical results are related to the case of bending forceequal to 165N (the value of buckling force obtained fromnumerical linear buckling analysis, cf. Section 3.2) and twovalues of internal overpressure: 0MPa and 0,4MPa. In caseof zero internal overpressure, the distribution of longitudinalstress is very regular with tension region at the lower sideand compression regions at the upper side of the cylinder(Figure 2(a)). Although deformation of the cylinder is rela-tively small, the influence of geometry change on distributionof internal forces is reflected in slightly unsymmetricaldistribution of longitudinal stress on the bottom and on thetop of the cylinder.

In turn, the loading caused by application of internal pres-sure results in nearly uniform longitudinal and circumferen-tial tensile stresses in the sidewalls of the cylinder (except theperipheral regions of clamping and free end). Consequently,the interaction of internal pressure and bending force leadsto reduction of longitudinal compressive stresses in theupper part of the cylinder with simultaneous increase oflongitudinal tensile stresses in the lower part (Figure 2(b)).

Although the finite element analysis reveals vague effectof bending of cylinder walls (slight variation of stress acrossshell thickness), the distribution of stresses can be estimatedby means of membrane shell theory [25], which assumesthat equilibrium of shell is provided without the presenceof bending forces. Analytical formulae defining stress in


+4.295e + 08+8.708e + 07+7.207e + 07+5.706e + 07+4.205e + 07+2.705e + 07+1.204e + 072.970e + 061.798e + 073.299e + 074.800e + 076.300e + 077.801e + 079.302e + 076.663e + 08

(a)

+4.295e + 08+8.708e + 07+7.207e + 07+5.706e + 07+4.205e + 07+2.705e + 07+1.204e + 072.970e + 061.798e + 073.299e + 074.800e + 076.300e + 077.801e + 079.302e + 076.663e + 08

(b)

Figure 2: Distribution of the longitudinal stress [Pa] at outer side of the shell caused by bending force of 165N: (a) empty can, (b) internaloverpressure of 0,4.

the longitudinal direction at critical points on the top andbottom of the cylinder in the vicinity of the support read asfollows.

(1) First Load Case. The case with force acting at the end ofthe cantilever = 165N and no internal overpressure is

top =

=

(/4) (4 ( )4

)

= 82,36 MPa,

bottom = 82, 36MPa.

(5)

(2) Second Load Case. The case with force = 165N andinternal overpressure = 0,4MPa is

top =

+

2= 16,4 MPa,

bottom =

+

2= 148,5 MPa.

(6)

In the above formulae is the bending moment caused byapplied force, while is the moment of inertia of the circularcross section, which can be expressed in terms of radius ofthe can r and thickness of the wall . Both the above resultsare in good agreement with the results of FEM simulationspresented in Figure 2. The obtained results indicate thatobserved in the experiment buckling of the empty can occursat relatively low value of stress (below plastic limit of thealuminium) and the process can be treated as elastic buckling.On the other hand, when the cylinder is subjected to actionof internal pressure of 0,8MPa and bended by the force of700N (extreme conditions in the experiment) the maximallongitudinal stress calculated according to (6) equals 481MPaand exceeds yield strength of aluminium. Therefore, thephenomenon obtained in the experiment in case of highinternal pressure should be rather treated as elastoplasticbuckling.

3.2. Linear Buckling Analysis. The next stage of considera-tions was linear buckling analysis of empty and inflated can

during bending by the vertical force. Analysis of buckling ofinflated structure requires two subsequent steps:

(i) initial prestressing where distributed loading mod-elling gas pressure is applied,

(ii) eigenvalue buckling analysis where critical value ofthe bending load is searched.

The equations governing two parts of the problem read asfollows:

Step 1: Kqp = Qp

or K (Qp, qp) qp = Qp,(7a)

Step 2: [K (Qmaxp ) KQF] k = 0

or [K (Qmaxp , qp) KQF (qp)] k = 0.(7b)

The first step of simulation is a standard static analysis,which can be executed either as a geometrically linear oras a nonlinear one. The result of this analysis is prestressedstate corresponding to initial or deformed configurationof the structure. The second step of simulation is linearbuckling analysis performed either for prestressed nonde-formed configuration defined by stiffness matrixK(Qmaxp ) or,alternatively, for deformed configuration defined by stiffnessmatrix K(Qmaxp , qp). In buckling analysis the global stiffnessmatrix is composed of base stiffness matrix K and loadstiffness matrix KQFcorresponding to external vertical loadof a unit value. The step is aimed at finding value of externalbending load for which (7b) has nontrivial solutions, thatis, for which global stiffness matrix of the system becomessingular. The eigenvalues are determined as solution of theequation:

det [K (Qmaxp ) KQF] = 0 (8a)

or det [K (Qmaxp , qp) KQF (qp)] = 0 (8b)

which further allows finding value of critical bending load-ing QF, total value of critical loading Qp + QF, andeigenvectors k.


+4.274e 03+3.918e 03+3.562e 03+3.206e 03+2.850e 03+2.493e 03+2.137e 03+1.781e 03+1.425e 03+1.069e 03+7.124e 04+3.562e 04+0.000e + 00

U(m

agni

tude

)

(a)

+4.274e 03

+3.562e 03+3.206e 03+2.850e 03+2.493e 03+2.137e 03+1.781e 03+1.425e 03+1.069e 03+7.124e 04+3.562e 04+0.000e + 00

U(m

agni

tude

)

+3.918e 03

(b)

Figure 3: Change of buckling shape as a result of increase of internal overpressure in the cylinder: (a) = 0MPa, (b) = 0,35MPa.

The corresponding numerical simulations were con-ducted with the use of two finite element solvers (ABAQUSStandard and ANSYS Mechanical) where four- and eight-node shell elements were applied. In case of noninflatedcylinder the value of the critical bending force correspondingto the first buckling mode computed by the two above solverswas equal to 171N and 165N, which is only slightly higherthan the value of critical force obtained from the experiment.Determined buckling shape (Figure 3(a)) includes longitu-dinal deformations arising on both sides of the cylinder.The fact that the subsequent values of the critical forcesare close to each other indicates that the loss of stabilityof real structure may involve combination of several initialbuckling shapes

adaptive impact absorption

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