DIFFERENTIAL MODEL AND IMPACT RESPONSE OF A FLEXIBLE BEAM ATTACHED TO A RIGID SUPPORTING STRUCTURE. A Thesis. Presented to

Transcription

1 DIFFERENTIAL MODEL AND IMPACT RESPONSE OF A FLEXIBLE BEAM ATTACHED TO A RIGID SUPPORTING STRUCTURE A Thesis Presened o The Graduae Faculy of he Universiy of Akron In Parial Fulfillmen of he Requiremens for he Degree Maser of Science Harish Chandra May, 8

2 DIFFERENTIAL MODEL AND IMPACT RESPONSE OF A FLEXIBLE BEAM ATTACHED TO A RIGID SUPPORTING STRUCTURE Harish Chandra Thesis Approved: Acceped: Advisor Dr. D. Dane Quinn Dean of he College Dr. George K. Harios Faculy Reader Dr. T.S.Srivasan Dean of he Graduae School Dr. George R. Newkome Deparmen Chair Dr. Celal Baur Dae ii

3 ABSTRACT Ofen elecronic componens such as lapops and cellular phones are dropped accidenally during usage and eensive damage is developed due o he impulsive force generaed a he conac poin. While eernal damage is easy o deec, inernal damage o he elecronic circuiry go undeeced, ye may cause failure of he sysem. An accurae descripion of he impulsive force is necessary o undersand he dynamics of he sysem. This research sudy involves developmen of a differenial model of a fleible beam aached o a rigid supporing srucure and sudying is response due o impacs. An Euler Bernoulli beam heory is used o model he beam, and Rouh s graphical mehod for wo dimensional impacs is used o calculae he impulse a he conac poin. The dynamics of impac a he conac poin is used o develop he boundary condiions and Galerkin s approach is used o find an approimae soluion. An eample is presened in which he response due o drop a differen angles of approach is sudied. The posiion of he beam on he frame, he coefficien of fricion µ ) and he coefficien of resiuion e ) are varied o see heir influence on he beam response. Finally he influence of he boundary condiions on he sresses and srains developed in he beam is discussed. iii

4 ACKNOWLEDGEMENTS Firs and foremos I would like o hank my advisor Dr.Dane Quinn for his invaluable guidance, paience and suppor hroughou he course of my sudy. I remain indebed o him for awarding me a Research Assisanship during my firs semeser and helping me ge a Teaching assisanship hereafer ill he end of my sudy. I would like o hank he Deparmen of Mechanical Engineering a he Universiy of Akron [Dr. Celal Baur] for awarding me a Teaching Assisanship during my graduae sudy for he Maser of Science degree. I would like o hank Dr. T.S.Srivasan and Dr.Graham Kelly for serving on my hesis commiee. I appreciae all he suppor I go from he saff members specially Sacy and Sephanie. I also eend warmes hanks o my parens and friends who have been wih me and encouraged me during my sudy. iv

5 TABLE OF CONTENTS Page LIST OF FIGURES... viii CHAPTER I. INTRODUCTION. Overview... Moivaion....3 Collisions 3.3. Rigid body collisions Collision law properies Rigid body assumpions Mahemaical modeling Seps in Mahemaical Modeling..9 II. LITERATURE REVIEW III. MATHEMATICAL MODELING.7 3. Inroducion 7 3. Problem formulaion Boundary condiions Iniial Condiions Transformaion of boundary condiions... v

6 3.4 The dynamics of impac. 3.5 Impac velociies Rouh s mehod o find he impulse Procedure o calculae impulse The impac process diagram Galerkin Reducion Galerkin mehod as applied o he beam model Sresses and Srains Review..36 IV. THE DYNAMICS OF IMPACT AND IMPACT VELOCITIES Overview An impac problem Rouh s graphical mehod Using he impulse o find change in velociies Mode shapes equaions Resuls.45 V. RESULTS AND CONCLUSION Overview Angle of approach v/s impac Influence of µ ) on he impulse a conac poin Influence of he coefficien of resiuion e) on he impulse Influence of he angle of impac on he beam velociy..54 vi

7 5.6 Influence of he angle of impac on he deflecion, sress and srain Influence of he locaion of he beam on he boundary condiions Change in sress and srain wih change in posiion of beam Conclusion.6 5. Underlying simplificaions Recommendaions for fuure work 6 REFERENCES..63 vii

8 LIST OF FIGURES Figure Page. Impac configuraion. Single degree of freedom sysem.5 3. Rigid frame wih a fleible beam Illusraion of Impac Posiion of Mass cener Conac velociies Impac process diagram illusraing Slip-Sick Impac process diagram illusraing Sick An Impac Problem Impac process diagram Maimum Deflecion Maimum Sress Maimum Srain Angle of Impac v/s Magniude of impac.48 o 5. Impulse pah a θ = o 5.3 Impulse pah a θ = o 5.4 Impulse pah a θ = µ v/s P Impulse pah forµ =. 5 viii

9 5.7 Impulse pah forµ = Impulse pah forµ = Impulse pah forµ = 5 5. e v/s P Angle of impac v/s Change in magniude of velociy a end A Angle of impac v/s Change in magniude of velociy a end B Angle of impac v/s Deflecion Angle of impac v/s Sress Angle of Impac v/s Srain Frame wih beam aached a differen locaions Locaion of beam v/s change in magniude of velociy a A Locaion of beam v/s change in magniude of velociy a B Sress v/s Locaion of beam Srain v/s Locaion of beam...58 i

10 CHAPTER INTRODUCTION. Overview Impac can be described as he ineracion beween wo or more bodies and plays a vial role in many mechanical engineering applicaions. An accurae descripion of his ineracion is necessary o undersand he dynamics of mechanical sysems. The objecive of his hesis is o develop a differenial model of he response of a fleible beam aached o rigid supporing srucure upon impac. Figure 3. illusraes he problem being sudied. Q ê ê B A P y) G, R ĵ î S Figure. Impac configuraion This work sudies he effec of he impulse produced a he conac poin on he response of he beam. The process of impac is described by Rouh s model, a simplified descripion of he pos - collision velociy of he body can be prediced, given he pre-

11 collision velociy and geomery. The impac hen specifies boundary condiions on he fleible beam, so ha he effec of he impac on he sress, srain and deflecion of he beam are presened. The firs chaper of his hesis inroduces he reader o rigid body collisions and impacs. A few basic erms as applied o rigid body dynamics are eplained. The second chaper discusses he previous work ha has been done on collision models and heir resuls. The main work done in his hesis is eplained in chapers hree, four and five. In Chaper 3 he basic equaions ha form he differenial model of he beam are developed. A simplified model for he approimaion of impac is presened. The relaionship beween impac and he pos-collision velociies is esablished. In Chaper 4 he equaions developed in he previous chaper are used and an impac problem is eplained in deail. The sresses and srains developed in he beam due o changes in velociy upon impac are sudied. Chaper 5 concludes he hesis wih he sudy of he influence of he angle of impac of he block on he pos-collision velociies and hence he sresses and srains developed and heir locaion. Finally limiaions and recommendaions for possible fuure work are saed.. Moivaion Ofen elecronic componens are dropped accidenally during usage and can cause eensive damage. The damage could be eerior in naure such as dens or cracks on he surface or may lead o failure of he elecronic circuis ha play a vial role in is funcioning. While eernal damage may do lile o affec he performance of he objec, inernal damage may cause he objec o cease proper orienaion. Therefore he proper

12 funcioning of porable elecronic producs such as mobile phones, lapops, and digial cameras depends largely on he inegriy of he inernal componens and he elecronic circuiry. Due o he rapid progress made in he echnological field in he las few years elecronic devices have become smaller and ligher and he inernal componens have become so comple ha hey are more suscepible o damage due o accidenal drop. When an objec is accidenally dropped an impulsive force is creaed when i makes conac wih he ground. This is a poenial cause of failure. Under hese circumsances heir reliabiliy due o shock and impac becomes a criical issue affecing performance. They are also eposed o shock during various sages of heir producion and operaion. When he produc is accidenally dropped on he ground, impac forces are ransmied from he produc case o he prined circui board PCB) and oher componens wihin he case. This can cause fracure problems. Also someimes he produc may fail due o moion of he inernal componens. Wih escalaing producion and manufacuring coss i would help if he damage caused o hese componens can be idenified or prediced before i is delivered o cusomers. This would help in he producion of componens ha are more reliable under shock and more resisan o damage due o drops. One of he mehods used o analyze he dynamics of he problem is mahemaical modeling. I is a powerful and useful approach o idenify he poenial weakness of producs in he iniial sages of fabricaion..3 Collisions Collisions can be defined as he acion of wo or more bodies coming ogeher or sriking one anoher for a small amoun of ime. A collision resuls in an impulse. In he 3

13 even of a collision he mass and he change in velociy is easily measured bu he impac force is no. Once he ime of collision is known he average impac force can be calculaed. In oher words one of he firs seps in order o find he impac force is collision deecion. Collisions are classified as single poin and mulipoin depending on he way hey come ino conac. If he bodies come ino conac a a single poin i s called a single poin collision. Oherwise i s called a mulipoin collision. During a collision beween wo bodies here is large force acing for a very brief period of ime. Inegraion of ha force over ha brief ime gives he impulse. Collisions can be eiher elasic or inelasic. A perfecly elasic collision is defined as one in which here is no loss of kineic energy in he collision. Elasic collisions occur only if here is no conversion of kineic energy o oher forms. Collisions beween aoms are perfec eamples of elasic collisions. In an elasic collision he oal kineic energy is he same before and afer collision, hence: m u mu mv = m v Momenum balance is mainained in all collisions. Therefore oal momenum remains consan afer collision: m = u mu mv mv An inelasic collision is one in which here is some form of energy conversion during he collision. During an inelasic collision par of he kineic energy is convered o inernal energy. Finally, in a plasic collision he objecs sick ogeher afer collision. Impac is characerized by a sudden high force or shock applied over a shor inerval. Across a collision only he impulse is imporan. Impac has been sudied in 4

14 deail in a lo of previous works since i forms he subjec of a wide variey of engineering applicaions. The opic of ineres for mos design and srucural engineers is he reacion forces ha develop during collision and he response of srucures o hese forces. There are various mehods o deermine he response of a sysem o a shock load. The wo mos popular ones are he frequency domain approach or he shock response specrum and he ime domain approach or he ime hisory of he sysem. In he frequency domain approach he seady-sae maimum response of he sysem o a given shock pulse is deermined. This is accomplished by considering he response of a single degree of freedom model composed of a spring mass and damper. In he ime domain approach he equaions of moion governing he sysem are firs wrien, hey are hen inegraed wih respec o ime o deermine he response. Mechanical componens can ofen be modeled wih parial differenial equaions; hese equaions are discreized o yield a sysem of ordinary differenial equaions in ime which can hen be inegraed. Figure.: Single degree of freedom sysem A shock is defined as a ransien physical eciaion ha causes a sudden jump in velociy. I can be caused by a drop, collision wih anoher objec, sudden disurbance like an earhquake or eplosion. The magniude of shock is measured using an 5

15 acceleromeer. A shock response specrum is used for evaluaing he response o a mechanical shock. I is a graph depicing he response of a single degree of freedom sysem such as a spring-mass-damper sysem o an arbirary ransien acceleraion as inpu. The horizonal coordinae represens he naural frequency of any given single degree of freedom sysem and he verical coordinae represens he acceleraion response. Shocks of high magniude have he poenial of damaging an enire srucure. Also he damage caused depends upon wheher he maerial is brile or ducile. A brile maerial under shock fracures whereas a ducile maerial bends. An eample of he damage caused o a brile maerial under shock is when a crysal glass is dropped o he floor i shaers. Whereas when a copper picher, a ducile iem is dropped on he floor i bends. Some maerials are no damaged by a single shock bu eperience faigue failure under numerous low level shocks..3. Rigid Body Collisions A rigid body is a solid objec ha has a definie shape and canno be deformed in any way. In oher words is shape does no change during collision. In order o find he pos collision velociies of wo colliding rigid bodies cerain laws are used. These laws are referred o as collision laws. Given he velociies of he ceners of mass and he angular velociies of he bodies a he insan before collision, a collision law is a rule which predics he corresponding velociies afer he collision. A collision law requires physical deails of he colliding bodies such as maerial properies, geomeric characerisics of he bodies, fricion properies in he region of conac, ec. 6

16 .3. Collision Law properies. A collision law should be consisen wih he fundamenal laws of mechanics and dynamics like conservaion of mass and momenum, linear and angular momenum balance.. I should consider bodies wih arbirary shape, mass disribuions, maerial and surface properies, orienaions and velociies. 3. I should be able o give resuls ha are in agreemen wih observed behavior for simple models. 4. I should be consisen wih oher simpler laws like he laws of fricion and oher phenomena which are modeled using lesser known laws. 5. I should be dependen on reasonably small number of inpu parameers and should involve simple calculaions. 6. I should be able o capure a wide variey of observed behavior for given inpu parameers. 7. The inpu parameers should have simple physical inerpreaions..3.3 Rigid Body assumpions The colliding bodies are reaed as rigid before and afer collision. This means ha he colliding bodies move almos like rigid bodies before and afer collision wih deformaion being negleced in he calculaion of linear and angular momenum. Any non rigid behavior aking place during collision causes small deformaions for majoriy of he body wih larger deformaions occurring a he conac area. Collision occurs for a very shor duraion wih displacemens and roaions being negligible, acceleraions being large wih definie changes in velociies. Impulses oher han hose occurring a he 7

17 conac area are negleced wih oher body forces also being negligible. The mass, momens of ineria and dimensions of a leas one colliding body are finie and known. Poin conac beween he bodies is assumed. Even hough in realiy conac occurs over a region, his region is assumed o have smaller dimensions when compared o he lengh of he smaller colliding body. There is no kineic energy creaed in a collision. The ne kineic energy of rigid body moion in he bodies afer collision is less or equal o he kineic energy before collision. The bodies do no pass hrough each oher and here is no inerpeneraion as his violaes he poin conac assumpion. These are reaed as reasonable assumpions according o he laws of laws of mechanics..4 Mahemaical Modeling Engineers ofen use heir knowledge of science, mahemaics, and appropriae eperience o find suiable soluions o a problem. Creaing an appropriae mahemaical model of a problem allows hem o analyze i, and o es poenial soluions. Usually muliple reasonable soluions eis, so engineers mus evaluae he differen design choices on heir meris and choose he soluion ha bes mees heir requiremens. A mahemaical model usually describes a sysem by a se of variables and a se of equaions ha esablish relaionships beween he variables. The values of he variables can be real or ineger numbers, boolean values or srings. The variables represen some properies of he sysem, for eample, sysem oupus in he form of signals, iming daa, couners, even occurrences ec. Mahemaical models are of grea imporance in physics. I is common o use idealized models in physics o simplify hings. I is a effecive ool ha engineers use o anlyse, conrol and opimize physical sysems Throughou hisory, 8

18 more and more accurae mahemaical models have been developed. The laws of physics are represened wih simple equaions such as Newon's laws; Mawell's equaions ec.these laws form a basis for making mahemaical models of real siuaions. Many real siuaions are very comple and hus modeled approimaely on a compuer; a model ha is compuaionally feasible o compue is made from he basic laws or from approimae models made from he basic laws. Mahemaical modeling problems are ofen classified ino black bo or whie bo models, according o how much background informaion is available of he sysem. A black-bo model is a sysem of which here is no background informaion available. A whie-bo model is a sysem where all necessary informaion is available. Pracically all sysems are somewhere beween he black-bo and whie-bo models, so his concep only works as an inuiive guide for approach. Usually i is preferable o use as much background informaion as possible o make he model more accurae. Therefore he whie-bo models are usually considered easier, because if you have used he informaion correcly, hen he model will behave correcly..4. Seps in Mahemaical Modeling. Problem idenificaion: The sysem o be modeled is isolaed from is surroundings and he effecs of he surroundings are noed. Known consans and variables are idenified.. Assumpions: Assumpions are made o simplify he modeling. Considering all effecs of he sysem resuls in compleiy and hence a mahemaical soluion becomes difficul. Assumpions should only be made if hey yield resuls ha are simpler han hose go wihou he assumpions. Someimes cerain implici 9

19 assumpions are made which are aken for graned and are seldom menioned. Eamples include assuming physical properies as coninuous funcions, considering all maerials o be linear, homogeneous and isoropic, ignoring relaivisic, chemical, nuclear and oher effecs. 3. Basic laws of naure: A basic law is a physical law ha applies o all physical sysems regardless of he maerial from which he sysem is consruced. They are laws ha can be observed bu no derived from more fundamenal laws. Eamples include conservaion of energy, conservaion of momenum; he second and a hird law of hermodynamics.among he above only conservaion of momenum plays a significan role in problems involving vibraing sysems. 4. Consiuive equaions: They provide informaion abou he maerials of which a sysem is made. They are used o develop force-displacemen relaionships for mechanical componens used in vibraion problems. 5. Geomeric consrains: They are essenial in compleing he mahemaical model of a sysem. They can be in he form of kinemaic relaionships beween velociy, acceleraion and dispalcemen.geomeric consrains are used o formulae he boundary condiions and iniial condiions once he differenial equaions are developed. 6. Mahemaical soluion: Once he mahemaical problem is obained, he modeling is no complee ill he appropriae mahemaics is used o obain a soluion. Eac analyical soluions, if hey eis are preferable o numerical or approimae

20 soluions. Usually eac soluions eis for only for linear problems bu for very few nonlinear problems. 7. Physical inerpreaion of resuls: This is he ne sep afer he modeling of he desired sysem is complee and he mahemaical soluion is obained. In cerain insances i may involve drawing general conclusions from he mahemaical soluion or developmen of design curves or jus require simple arihmeic o arrive a a conclusion.

21 CHAPTER II LITERATURE REVIEW. Recen developmens in he elecronics indusry over he las few years have a brough abou a huge increase in he usage of porable elecronic producs like cellular phones, personal digial assisans and digial cameras. As hese producs ge smaller for ease of handling, hey become more suscepible o drop impacs. Impac loads are he main cause of mechanical failures in producs [-8]. There can be significan damage caused o he par as a resul of he impac forces developed during drop. Mos of he common elecronic devices have a similar srucure consising of an upper srucure and lower srucure conneced by a hinge device. The upper srucure includes he LCD module, covers, meal frames and he lower srucure includes a keyboard, housings, baery, moherboard and chipses [3]. There can be serious damage caused o elecronic producs when dropped. The impac force produced can cause he inernal componens o fail or cracking of he ouer cover. Some of he damage can also include housing fracure, join breaking, connecor disconnecion or complee componen failure []. Of hese he one ha is of grea concern is he funcional failure of he par. The cause of such failure is normally due o he loss of elecrical conacs arising from he breakage of inerconnecions beween componens wihin he produc []. The resuling sresses and srains can cause he housings o deform, cause assemblies o come apar and cause he

22 liquid crysal display LCD) o crack []. As a resul, sudying he response of hese elecronic componens subjeced o drop impacs is vial in predicing he life of he componen. Over he years various researchers have used differen mehods o sudy he effec of drop on he performance of he sysem. The radiional mehod of analysis is carrying ou physical ess using a prooype. Bu in using such a mehod i is difficul o capure he impac even as i happens over such a shor span of ime and involves a lo of rial and error [5]. Also i has proven o be epensive, ime consuming and requires a lo of effor []. Anoher mehod ha is very popular and widely used is simulaion of he impac even using a sofware package such as FEA and comparing i eperimenal or analyical resuls [, ]. Tan e al [] used Finie elemen FE) simulaion o model a ball grid array BGA) package ha consiss of a inegraed circuiic) conneced o a fleible prined circui board PCB) subjeced o drop impac. He used hree differen kinds of mesh o model he BGA package and sudied he deviaion in resuls wih he change in meshing. The parameers ha were compared were he deflecion of he PCB and he sresses ha develop wihin he solder balls. He concluded ha he ype of mesh used o model a componen has an influence on he sresses and srains developed. Lim e al [] sudied he feasibiliy of using FE simulaion for he drop impac response of a pager by performing a numerical simulaion and using ABAQUS and verified numerical resuls wih eperimens. He concluded hrough his eperimenal and simulaion resuls ha he impac orienaion is largely responsible for variaion of srains and impac force during collision. He aribued he variaion of srain wih ime o he difference in local deformaion occurring a differen pars of he pager. Low e al [4] sudied he impac effec on mini Hi-Fi audio producs He used Pro-E o creae his model and used PAM- 3

23 CRASH o sudy he effec of maerial properies on he resuls. However, using FEA as a mehod o simulae he impac even has a few disadvanages. A major difficuly is in meshing some of he smaller componens. Some of hese subsrucures in elecronic componens need eremely fine meshes and his causes oo much ime for he analysis and requires high end sofware ha could be very epensive [4-6].Anoher problem is seing up he simulaion model accuraely including he geomery of he componen being sudied, selecion of he righ maerial for differen componens, boundary condiions ec [5]. Some of he limiaions of using sofware o simulae he impac even can be overcome by sudying he analyical or mahemaical models of impac. Goyal e al [6] used a linear spring-mass sysem o analyically model he drop impac of a cellular phone. He concluded ha he geomery of he componen is an imporan facor in analyzing he failure of he componen. He suggesed ha improving he rigidiy of he case may preven slipping of he cellular phone from is cover upon drop. Goyal [5] also sudied he claering effecs ha occur when a componen his he ground a an angle. He sudied he jumps in velociy of he ends of a bar for successive impacs. He found ha when a wo dimensional bar was dropped on he floor he second impac could be wice as large as he firs. He concluded ha he number of imes ha he bar would impac he ground depends on he co efficien of resiuion and he mass disribuion. Shan e al [7] also did a comprehensive sudy on he effec of claering on hree dimensional bars and found ha his resuls were comparable wih Goyal. Xiang [7] did a sudy on he effec of a coninual shock loads on packaged producs during ransporaion. Coninuous shock loads cause undesirable sresses on packaged maerial during ranspor. Xiang sudied he effec of acceleraion ampliude on he number of 4

24 shocks ha lead o failure by performing a series of ess in he laboraory. He esablished a relaion beween fragiliy of he produc and he number of coninuous shock loads o failure. Wang e al [9] in his sudy of wo dimensional rigid body collisions found some mehods of rigid body impac could violae he energy conservaion principles. He classified he differen modes of impac using Rouh s graphical mehod o deermine he impulse. He blamed he violaion of energy conservaion o he use of Newon s hypohesis for he coefficien of resiuion. Wang proved ha using Poisson s hypohesis over Newon s for he co efficien of resiuion yielded beer resuls He also saed ha idenifying he correc mode of conac is vial in order o saisfy he law of conservaion of energy in an impac process. However Brach [8] blamed he violaion of he principle of energy conservaion o he definiion of fricion. He adoped lower values of coefficien of fricion o saisfy he law of conservaion of energy. Keller [8] eended he Rouh s mehod o solve hree dimensional problems. Barbulescu e al [] used Kane s mehod o sudy spaial impac of a slender beam. In his sudies he deal wih boh slip a he conac poin and no slip. In he former case he assumed angenial velociy a separaion o be zero. He deduced a relaionship beween he loss of kineic energy and he angle of impac. He concluded ha for condiions of no slip he energy loss increased wih increase in he angle of impac. Younis e al [3] used Galerkin s approach o sudy he dynamic response of beams o mechanical shock. He invesigaed he nonlineariy of he response and aribued i o he effec of mid-plane sreching. He suggesed improving he hickness of he beam o improve shock resisance. He saed ha mechanical shock when combined wih elecrosaic force could cause a dynamic pull-in in MEMS devices causing insabiliy. He also sudied he effec of packaging on MEMS 5

25 devices. One of his imporan findings was ha neglecing he effec of packaging could lead o failure of he device. 6

26 CHAPTER III MATHEMATICAL MODELING 3. Inroducion In his Chaper we esablish an impac model ha calculaes he impulse and hence he change in velociies a he ends of he beam upon impac. A rigid frame modeled as he housing of an elecronic componen wih a fleible beam aached o i modeled as an inernal componen is subjeced o drop. The impac produces a change in velociy of he housing ha changes he boundary condiions of he beam. The response of he beam is solved for he resuling boundary condiions. An Euler-Bernoulli beam is used for analysis. Iniially he problem is homogeneous wih non homogeneous boundary condiions. Using a linear funcion in we ransfer i ino a non homogeneous problem ha calculaes he change in velociy upon impac. The impulse produced upon impac produces boundary condiions on he beam. The relaion beween he impulse and boundary condiions is shown. Finally he response of he beam wih he maimum ampliude, sress and srain is presened. 3. Problem Formulaion The iniial configuraion of he sysem is shown in Figure 3., illusraing he collision beween a rigid frame wih an aached fleible beam and he ground. 7

27 Q ê ê B A P y) G, R ĵ î S Figure 3.: Rigid frame wih fleible beam Le PQRS represen he rigid frame wih G being he mass cener. AB represens a fleible beam aached o he frame wih he help of wo suppors as shown in he figure. The rigid frame represens he housing of an elecronic device while he fleible beam models an inernal componen such as a prined circui board. Le he poin of conac S be he origin. î and ĵ are uni vecors in he global coordinae sysem fied o he ground, ê and ê represen uni vecors in he local coordinae sysem of he beam relaive o he frame. The housing makes direc conac wih he ground producing an impulsive load a he conac poin. The impulse produces a change in velociy of he housing. This change in velociy impulsively loads he beam. Using Hamilon s principle, he ransverse displacemen of he beam w, ) wih respec o he housing is derived as 4 w w w ρ A EI P = 3.) 4 8

28 where E is he elasic modulus of he beam,ρ is is mass densiy, A) is is cross secional area, I) is is cross secion momen of ineria and P is he componen of he aial load in he ransverse direcion. Equaion 3.) represens he model of an Euler- Bernoulli beam. I is convenien o nondimensionalize he parial differenial equaions governing he vibraions of coninuous sysems hrough he inroducion of nondimensional dependen and independen variables. Nondimensional variables are inroduced such as * w w = L * = * = L T 3.) 3.3) 3.4) Where he variable wih an * is a nondimensional variable, L is a characerisic lengh in he sysem such as he lengh of a beam, and T is a characerisic ime scale. Equaion 3. is nondimensionalized using he nondimensional variables of equaion The beam is considered o be of uniform cross secion and herefore spaial dependence is no considered. ge he nondimensional form of equaion 3.) as: 9 ρal β = 3.5) PT EI α = 3.6) PL The effec of roary ineria and shear deflecion is ignored and as a resul he load in he aial direcion is P =, dropping he * s and considering α and β o be consans we

29 4 w w w α β = 3.7) 4 Equaion 3.7 represens a homogenous equaion governing he vibraions of he beam AB. The equaion is homogeneous because he forcing erm is assumed o be equal o zero. 3.3 Boundary Condiions The boundary condiions are he displacemen and velociies a he wo ends of he w beam a all imes. If w, ) is he displacemen of a paricle a ime and, is is velociy, he boundary condiions of he non dimensional beam are given by: w, ) = 3.8) w, ) = 3.9) w, ) = v A ) w, ) = vb ) 3.) 3.) 3.3. Iniial Condiions The iniial condiions represen he displacemen and velociies before impac. A ime = he displacemen and velociy of any paricle over he enire span of he beam relaive o he rigid frame is zero. This would mean ha v A and vb are zero before impac, so ha he iniial condiions are: w,) = 3.) w, ) = = 3.3)

30 I is seen ha he differenial equaion governing he moion of he beam is homogeneous whereas he boundary condiions are non-homogeneous. We find a soluion by ransforming i o a problem wih a non-homogeneous differenial equaion having homogeneous boundary condiions Transformaion of boundary condiions We firs choose a funcion b, ) ha renders he boundary condiions homogeneous. The soluion o he problem is hen assumed o be he sum of a variable u, ) and he funcion b, ) w, ) = u, ) b, ) 3.4) Here b, ) is chosen o be a funcion linear in. b, ) = [ v ) v ) v ) ] A B A ) 3.5) This funcion saisfies he boundary condiions of equaions 3.) and 3.). In his manner, we ransform he boundary value problem for he variable w, ) ino a boundary value problem for he variable u, ). Taking he parial derivaive of equaion 3.4), using values of = and = we ge: w u, ) =, ) v A ) w u, ) =, ) vb ) 3.6) 3.7) Subsiuing equaions 3.6) and 3.7) in 3.) and 3.) we ge: u, ) = 3.8)

31 u, ) = 3.9) Therefore he non homogeneous boundary condiions are made homogeneous by he choice of a funcion ha saisfies he boundary condiions. The new homogeneous boundary condiions are wrien as: u, ) = 3.) u, ) = 3.) u, ) = v A ) u, ) = vb ) 3.) 3.3) 3.4 The dynamics of impac When he objec his he ground here is an impulse produced a he conac poin. Due o he sudden impac upon conac wih he ground he acceleraion a he ends of he beam are epressed as: dv d A ) v δ ) = 3.4) A dv d B ) v δ ) = 3.5) B In he above wo equaions δ ) is a uni impulse ha has an infinie value when = and zero a all oher places. v A and vb are he changes in velociies a he ends of he beam. Therefore now subsiuing equaion 3.4) in equaion 3.7) and using equaion 3.5) we ge he governing equaion as: 4 u u u α β 4 = [ v v v ) ] δ ) A B A 3.6)

32 Equaion 3.6) represens a non homogeneous boundary value problem for he variable u, ) wih equaions 3.) - 3.3) represening he homogeneous boundary condiions. 3.5 Impac Velociies Upon impac here is a change of velociy a he wo ends of he beam. These velociies denoed by v A and vb depend upon he impulse produced upon impac a he inerface beween he ground and he rigid block. Using Linear momenum balance, we ge: Equaion 3.7) can also be wrien as, Where, F = force acing on he body m = mass of he body, a = acceleraion of he mass cener of he body. v = velociy of he mass cener of he body Equaion 3.8) is wrien as, v v F = ma 3.7) dv F = m 3.8) d G G ) G ) v ) = m F τ dτ 3.9) 3 ) ) v ) = m P 3.3) G Equaion 3.3) gives he change in linear velociy a he mass cener G upon impac. Using angular momenum balance we ge, G M = I α 3.3) G

33 Where, M G = momen acing on he body. G I = momen of ineria abou he mass cener α = acceleraion abou he mass cener of he body. Using equaion 3.3) we ge, I G [ ) ω ) ] = r P ) ω 3.3) BG G ) ω ) = I [ r P ) ] ω 3.33) Equaion 3.33) gives he change in angular velociy a he mass cener upon impac. Here r is he posiion vecor of B wih respec o he mass cener G and ) BG BG P is he impulse acing on he body a he poin of conac. B Q A P r B P y r BG r G G R S P Figure 3.: Illusraion of Impac Using basic vecor mechanics o epress he velociies a he ends of beam wih respec o heir mass ceners we ge, ) r ) ω ) r ) ω ) v A = m P AG AG 3.34) v ) r ) [ ω ) ω) ] A = m P AG 3.35) 4

34 Similarly we ge, v 3.6 Rouh s mehod o find he impulse ) r ) [ ω ) ω) ] B = m P BG 3.36) There are several mehods o calculae he impulse produced upon impac of a rigid body. In his secion we briefly describe Rouh s graphical procedure for wo dimensional collisions [9]. Rouh s mehod is a graphical echnique for analyzing planar fricional impac using Coulomb s law of fricion. According o his mehod an impulse is made of wo pars, a compression impulse, and a resiuion impulse. The compression impulse or he compression phase is from he beginning of collision o he ime ha heir relaive velociy is zero. The resiuion impulse or he resiuion phase is measured from he ime he relaive velociy is zero o he ime ha objecs begin o separae. Accordingly impulse P is epressed as: P ) = P ˆ P nˆ 3.37) In equaion 3.37) subscrips and n denoe he angenial and normal direcions respecively, which are in he î and ĵ direcions in he given conac plane. This model uses he Poisson s hypohesis o define he coefficien of resiuion. Accordingly he coefficien of resiuion denoed by e is defined by: Pn ) r P n ) c n e= 3.38) In equaion 3.38) P n ) r is he normal impulse during resiuion and n ) c P is he normal impulse during compression. By using he Poisson s hypohesis o define he coefficien of resiuion his mehod saisfies he basic energy conservaion principles as here is no 5

35 increase in kineic energy. The Rouh-Poisson analysis gives an impulse in accordance wih Coulomb s law, wihou an increase in oal energy. Also i considers a new ype of impac called he angenial impac, an impac wih zero iniial approach velociy. I can be used o disinguish beween several ypes of conac and o idenify when sliding ceases or reverses. In his hesis he Rouh s mehod is used o find he impulse a he conac poin of he rigid body wih he ground. Once he impulse is known he velociies a he wo ends of he beam are calculaed. Finally he response of he beam is presened knowing he velociies Procedure o calculae impulse Once he objec his he ground an impulse P ) is produced. This impulse can be divided ino wo componens, an impulse in he normal direcion Pn and an impulse in he angenial direcion P.Le he body wih mass cener, y) have iniial ranslaional and roaional velociies &, y& and & θ.using linear and angular impulse-momenum laws we wrie he following equaions [9]: m & & ) = 3.39) P m y& y& ) = 3.4) P n mρ & θ & θ ) = P y P 3.4) Here m is he mass,ρ is he radius of gyraion of ineria and &, y& are he velociies upon collision. The kineic energy is given by: & y& ) mρ & θ n T = m 3.4a) 6

36 7 = θ ρ ρ & & & m P P y m y m P m P m T n n 3.4b) Epanding and rearranging we ge: ) ) [ ] ) = ρ θ θ ρ m P P y m P m P P P y P y P m y m T n n n n & & & & & & 3.4c) The isoenergeic ellipse saisfies he law of conservaion of energy. Hence for his condiion o be saisfied P and n P assume values such ha: ) [ ] ) = ρ θ m P P y m P m P P P y y P P n n n n & & &. Figure 3.3: Posiion of Mass cener The velociy of he objec when i comes in conac wih he ground is called conac velociy and is given by: y c θ & & & = 3.4) y y c θ & & & = 3.43) G G r j y i r G ˆ ˆ =

37 B A y) G, &, & ) c y c Figure 3.4: Conac velociies Equaion 3.4) denoes he angenial componen of he relaive velociy of he poins in conac. This velociy represened by S is called he sliding velociy. Similarly equaion 3.43) denoes he normal componen of relaive velociy and is denoed by C. In oher words ge: S = & and C = y& c.subsiuing equaions 3.39)-3.4) ino 3.4) and 3.43) we c S S B P B3 P n = 3.44) C C B3 P B P n = 3.45) Equaions 3.44) and 3.45) represen he line of sicking and line of maimum compression respecively. S andc represen he iniial values of sliding and compression respecively. These depend on he values of he angenial and normal componen of he iniial velociy and he iniial roaional velociy. They are epressed as: = & & θ y 3.46) S = y& & θ 3.47) C B, B and B 3 are consans ha depend on he geomery and mass properies of he sysem. They are given by: 8

38 y B = 3.48) m mρ B = 3.49) m mρ y B 3 = 3.5) mρ The Rouh mehod deermines he wo componens of impulse P and Pn using equaions 3.44)-3.5).This is done graphically and eplained in deail in he ne secion The impac process diagram: When an objec deforms, i happens in wo phases, compression and resiuion. When compression ends, he normal componen of he relaive velociy of he poins in conac is zero C = ).Therefore from equaion 3.45) a linear relaionship beween he impulse componens a maimum compression is obained as C B P B P 3.5) 3 n = Similarly in he sicking case, he angenial componen of relaive velociy becomes zero S = ). Therefore a linear relaionship beween he impulse componens a slip is obained as S B P B P 3.5) 3 n = Now, equaions 3.5) and 3.5) represen he line of maimum compression and line of no sliding respecively. These wo lines represen sraigh lines since hey are linear and are ploed on he impulse plane wih P represening he horizonal ais and P n he verical ais. Therefore using equaions 3.5) and 3.5) we ge: 9

39 P B P C = n 3.53) B3 B3 P B P S 3 = n 3.54) B B We also define he line of limiing fricion as P = µ 3.55) sp n In equaion 3.55)µ is he coefficien of fricion and is a consan and s is he sign of he iniial sliding velociy S, S s= if eplanaion of he impac process diagram. S S is no zero. Figures 3.5 and 3.6 give a deailed P n P n f ) Fricion Compression Sick P n c ) P Figure 3.5: Impac process diagram illusraing Slip-Sick When impac begins he poin P which is he oal impulse is a he origin and lies on he line of limiing fricion. Assuming iniial sliding i increases along he line of limiing fricion. I proceeds along his line unil i reaches line sicking or line of maimum compression. If i reaches he line of maimum compression firs, he value of normal 3

40 impulse P n c )) a ha insan is noed. The process of impac will end when he value of P is e imes he value of )) n P.This value is denoed by P ) n c n ) e) P ) f n c n f P = 3.56) This is called he erminaion condiion.then he poin coninues along he line of limiing fricion unil erminaion is me or P reaches he line of sicking. If P reaches he line of sicking before erminaion i coninues along his line ill erminaion. This process of slip-sick is illusraed in Figure 3.5. Alernaively if i reaches he line of sick firs hen slipping ends and if he limiing fricion is more han he fricion necessary o preven sliding P will coninue o sick ill he process erminaes as illusraed in figure 3.6.I will evenually cross he line of maimum compression and erminae according o he erminaion condiion as saed earlier. On he oher hand if he limiing fricion is less han he fricion necessary o preven sliding P will cross he line of sicking and will ravel along he line of reversed limiing fricion and will coninue ill erminaion. In such a scenario he objec will slip hrough ou he impac process ill erminaion condiion is reached. 3

41 P n Fricion Compression Sick P Figure 3.6: Impac process diagram illusraing Sick The isoenergeic ellipse depiced in he above figures should saisfy he energy conservaion principles.acccordingly if T is he change in kineic energy hen according o he law of conservaion of energy T =. T [ P V V )] T = C CO 3.56a) P =, P P n Where [ ] T C = [ S C] T and = [ S C ] T V, VC, The energy change is given by: T T [ P BP V P] T = C 3.56b) Where B B = B 3 B B 3 The energy ellipse requires ha T =. 3

42 3.7 Galerkin Reducion Galerkin s mehod is a mehod used o find an approimae soluion o coninuous sysems. Ofen eac soluions for higher order equaions do no eis. Even if hey do hey are cumbersome o use, requiring soluions o higher order equaions. In such a scenario Galerkin s mehod is a convenien mehod o use alhough i yields only an approimae soluion. I is a means for convering a parial differenial equaion PDE) o a sysem of ordinary differenial equaions ODEs), which become easier o handle. I works on he principle of resricing he possible soluions o a smaller space han he original. These smaller sysems are easier o solve and less ime consuming. Galerkin s mehod approimaes he soluion o a boundary value problem by using a linear combinaion of rial funcions. In order o solve our problem we choose a rial funcions ha saisfy he boundary condiions Galerkin s mehod as applied o he beam model. Assume he approimae soluion o he problem o be: N ) u, ) = A ) φ 3.57) i= Where he rial funcions φ ) φ ),... φ ) i i, n are he independen comparison funcions from a complee se and A A,... A, n are undeermined coefficiens. Comparison funcions are rial funcions which are differeniable as many imes as he order of he sysem and saisfy all he boundary condiions. The above soluion may no saisfy he eac differenial equaion defining he eigenvalue problem, so ha some error is incurred. The ) error is denoed by R u n ),, known as he residual, and because 33 n) u is a linear combinaion of comparison funcions, he boundary condiions are saisfied eacly. To

43 34 deermine he coefficiens n A A A,.., we muliply he residual R ) ) u n, by ) ) ) n φ φ φ,...,, in sequence, inegrae he resul over he domain of he sysem, and se equal o zero. ) ) ),, = d R u n l i φ i =,..n 3.58) Now consider equaion 3.6), subsiuing for ) ) ) ), ), A u u i N i i N φ = = where ) A i is he ih generalized coordinae and ) i φ is he ih linear undamped mode shape of he sraigh beam we ge: ) [ ] ) ), ) ) ) ) ) ) ) 4 4 R u v v v A A A n A B A N i i i N i i i N i i i = = = = δ φ β φ φ α 3.59) Equaion 3.59) represens he residual. I has been shown in previous sudies [3] ha four modes are sufficien o capure he dynamic response of a beam pinned a boh ends. Therefore in equaion 3.7) subsiuing N = 4 we ge: ) ) ) ) ) ) ) ) ) ), A A A A u N φ φ φ φ = 3.6) We choose he firs four mode shapes o be: l π φ sin ) =, l π φ sin ) =, l π φ 3 sin ) 3 =, l π φ 4 sin ) 4 = Therefore equaion 3.6) becomes: ) ) 4 sin ) ) 3 sin ) ) sin ) sin ), ), 4 3 A l A l A l A l u u N π π π π = 3.6)

44 Equaion 3.6) represens he oal deflecion of he beam. To find he generalized coordinaes in equaion 3.6) we subsiueφ ) = sin kπ. Using values of k = o 4 we ge four ses of equaions from which A, A, A3 and A4 are deermined. Therefore we ge: For N = 4, k = we ge: Similarly for k =, 3 and 4 we ge: l i sin ) N kπ R Ai ) φi ), d= 3.6) i= 4 vb v A [.5απ.5π ] A ) = δ ) β A & ) π π 3.63) 4 vb v A [ 8απ π ] A ) = δ ) β A & ) π π 3.64) 4 vb v A [ 4.5απ 4.5π ] A ) = δ ) β A & 3 ) 3 3π 3π 3.65) 4 vb v A [ 8απ 8π ] A ) = δ ) β A & 4 ) 4 4π 4π 3.66) Equaions 3.63)-3.66) represen he firs four mode shape equaions. δ ) is a uni impulse due o an insananeous collision wih he ground. For an undamped sysem of he form & &ω n = δ ), he response o a uni impulse is denoed by h ) and is given by: h ) = n ω sinω 3.67) n In he above equaions impac. v A and vb are he changes in velociies of he beam upon 35

45 3.8 Sresses and srains If M ) is he bending momen and y is coordinae measured from he neural ais in he cross secion of he beam, using elemenary beam heory he normal sress a a poin in he cross-secion due o bending is epressed as M ) y σ = 3.68) I u M ) = EI 3.69) Equaions 3.68) and 3.69) are used o ge he sress response of he beam. Poenial energy in he form of srain energy is sored in all deformable sysems. For a loaded member he normal sress σ ) and normal srain ε ) follow he Hooke s law. The sress -srain curve is linear. Therefore we have σ = ε E.Using his we find he maimum srain. 3.9 Review The previous secions deal wih calculaing he value of he impulse produced upon drop. Once he impulse is known i is subsiued back in equaions 3.35 and 3.36) o ge he values of changes in velociies a A and B respecively. Before his he geomeric dimensions of he problem including he angle of impac is used o calculae he posiion vecors of he mass cener of he body and he ends of he beam. Once he change in velociies are known hese are subsiued back in he mode shape equaions ).This gives he values of he consans which are hen subsiued back in he response on he beam i.e. equaion 3.6) o ge he deflecion of he beam. Upon 36

46 knowing he response of he beam upon impac, he locaions of maimum sress, srain, deflecion can be go. 37

47 CHAPTER V THE DYNAMICS OF IMPACT AND IMPACT VELOCITIES 4.: Overview In his chaper we apply he basic equaions for he dynamics of impac in Chaper 3 o a specific impac problem. We firs find he impulse developed because of impac and hence he change in velociy due o impac. A final soluion is hen presened using he Galerkin s approach. Finally he sress, srain and ampliude responses illusraing heir maimum values are presened. 4. An impac problem Consider a rigid frame PQRS) of mass m = kg, widh W =.5m, heigh H =.3m and ρ = wih a fleible beam AB of lengh.5m falling from a heigh and o hiing he ground aθ = 45. Assume ha he angular velociy ω of he block is zero. Q ĵ A P r AS r BS P y B r BG G r GS R ê ê î S P Figure 4.: An Impac Problem 38

48 Le î and ĵ be uni vecors of he coordinae sysem fied o he ground. Le ê and ê be uni vecors of he coordinae sysem of he moion of he frame relaive o he ground. Since here is boh ranslaion and roaion of he frame relaive o he ground heir uni vecors are epressed as: e e ˆ ˆ = cosθ iˆ sinθˆj 4.) = sinθ iˆ cosθˆj 4.) Le r AS, r BS and GS r be he posiion vecors of poins A, B, and G relaive o he ground and r AG, r BG be he posiion vecors of A relaive o G and B relaive o G respecively. Using equaions 4.) and 4.) and he dimensions of he frame and beam we ge he following equaions for figure 4. r GS =.89ˆ e.5e 4.3) ˆ r AS = eˆ.398e 4.4) ˆ r BS =.469ˆ e e 4.5) 3 ˆ.9 Also using basic vecor mechanics we ge: r = r r 4.6) AS GS AG r = r r 4.7) BS GS BG In equaions 4.6) and 4.7) r AG and r BG are he posiion vecors of A relaive o G and B relaive o G respecively. From equaions 4.6) and 4.7) we ge: r AG =.89ˆ e.398e 4.8) ˆ r BG =.8ˆ e e 4.9) 4 ˆ

49 o The frame makes conac wih he ground a an angle θ = 45.This produces an impulse a he conac poin which affecs he velociies a he ends of he beam. We use linear and angular momenum balance equaions o epress he relaion beween he change in velociy and he impulse produced. Equaions 3.35) and 3.36) give he relaionship beween he change in velociies and he impulse. We now proceed o find he impulse in order o calculae he change in velociies Rouh s graphical mehod We use Rouh s mehod for wo dimensional impacs o find he value of he impac P) a he poin of conac. Using equaion 4.3), we find he mass cener, y) of he frame o be.89, ). Considering he geomery of our problem and using equaions 3.48) 3.5) we ge:.5 B = =.5 4.) ).89 B = =.54 4.) ) =.7 B 3 = 4.) ) A ime = here is an iniial angenial and normal componen of velociy ha is assumed. These are denoed by S and C respecively. The angular velociy is zero, herefore he iniial values of S and C are given by he iniial ranslaional velociies & and y&.they are also called he iniial sick and compression velociies and heir values are assumed o be and - respecively. Using equaions 3.53) and 3.54), values of B, B, B 3 and he iniial sick and compression velociies we plo he line of sick and 4

50 line of maimum compression on an impulse plane wih P represening he horizonal ais and Pn he verical ais. The coefficien of fricion µ is assumed o be.6. Figure 4. illusraes he impac process. Fricion Sick Pn) Impulse Pah Line of Compression P) Figure 4.: Impac process diagram. When impac begins P is a he origin, as i progresses Pn increases as i begins o slip along he line of fricion, P also accumulaes according o he relaionship beween he wo as epressed in equaions 3.53) and 3.54).When P n reaches he line of maimum compression he value of he normal impulse is noed. Terminaion occurs when he value of Pn reaches e imes he value of Pn obained a maimum compression. In his case impac erminaes when P n =.55.The objec coninues o slip along he line of limiing fricion ill erminaion is reached. 4