Semi-Analytical Solution For Elastic Impact Of Two Beams

2y ago
32 Views
2 Downloads
1.27 MB
10 Pages
Last View : 18d ago
Last Download : 2m ago
Upload by : Evelyn Loftin
Transcription

Strojarstvo 54 (5) 341-350 (2012)I. ĆATIPOVIĆ et. al., Semi-analytical Solution for Elastic. 341CODEN STJSAOZX470/1581ISSN 0562-1887UDK 624.072.23:519.62/.63:519.85Semi-Analytical Solution for Elastic Impact of Two BeamsIvan ĆATIPOVIĆ, Nikola VLADIMIR andSmiljko RUDANFakultet strojarstva i brodogradnje, Sveučilište uZagrebu (Faculty of Mechanical Engineering andNaval Architecture, University of Zagreb),Ivana Lučića 5, HR-10000 Zagreb,Republic of on methodElastic impactFEMSemi-analytical solutionKljučne riječiElastični sudarMetoda dijagonalizacijeMKEPoluanalitičko rješenjeReceived (primljeno): 2011-04-26Accepted (prihvaćeno): 2011-09-15Original scientific paperThis paper presents semi-analytical solution for the problem of elasticimpact of two beams. The solution is based on the finite elementdiscretization of the structure and equation of motion solution usingdiagonalization method for solving a system of differential equations. Thisprocedure avoids temporal discretization typical for numerical methods,and in this way eliminates influence of the time-step size on the solutionand numerical stability problems that can occur if time step isn’t definedproperly. Computer code is developed for calculation, and application ofthe procedure is illustrated with the case of collision of communicationbridge between two floating objects and safety structure called arrestor.The results are compared to those obtained by numerical procedure usingthe same discretization of the structure and applying energy conservationprinciple. Very good agreement of the results obtained by three methodsis achieved.Poluanalitičko rješenje elastičnog sudara dviju gredaIzvornoznanstveni članakU članku je prikazano poluanalitičko rješenje problema elastičnog sudaradviju greda. Rješenje se temelji na fizičkoj diskretizaciji konstrukcijemetodom konačnih elemenata i rješavanju jednadžbe gibanja metodomdijagonalizacije za rješavanje sustava diferencijalnih jednadžbi. Ovajpostupak omogućuje izbjegavanje vremenske diskretizacije koja jetipična za numeričke metode, i na taj način se eliminira utjecaj veličinevremenskog koraka na rješenje kao i probleme s numeričkom stabilnošću,koji se mogu pojaviti ako vremenski korak nije pogodno odabran.Razvijen je kod za provedbu proračuna, a primjena postupka je ilustriranaslučajem sudara komunikacijskog mosta između dvaju plovnih objekata izaštitne konstrukcije zvane arestor. Rezultati su uspoređeni s rezultatimadobivenim pomoću numeričkog postupka koristeći identičnu diskretizacijukonstrukcije i primjenom principa očuvanja energije, te je uočeno njihovojako dobro podudaranje.1. IntroductionDevelopment of the procedure for solving elasticimpact problem of two beams is inspired by practicalengineering problem of analyzing impact response ofcommunication bridge between two floating objectsand a safety structure called arrestor. The problem isdescribed in details in [1], where numerical procedurebased on the FEM technique and modified linearacceleration method is presented. Impact problem of thistype can be also solved by applying energy conservationprinciple [1, 2]. The former method is more suitable fordetailed structural analysis, and the latter one is moreconvenient for standard engineering practice. It shouldbe noticed that energy approach, which is very fast andespecially appropriate in early design stage, gives lowerstress level than numerical procedure since it can notcapture local deformations and stresses, and it doesn’tdistinguish natural vibration modes and their influenceon stress shape. Numerical procedure captures theseeffects, but it requires discretization of the structure andtemporal discretization, which both influence the results.Moreover, the time step duration influences the stabilityof the solution. Concerning discretization of the structureinto number of finite elements, it should be emphasizedthat longer finite elements close to contact location givelower values of contact force, so the topology of thefinite elements should be properly adjusted to describephysical background of the problem in reliable way.Temporal discretization can be avoided by analyticalsolving differential equation of motion, and in this wayone can get more reliable results. The semi-analyticalprocedure presented in this paper combines finiteelement technique for continuum discretization, and

342I. ĆATIPOVIĆ et. al., Semi-analytical Solution for Elastic.Strojarstvo 54 (5) 341-350 (2012)Symbols/OznakeAc- cross-section area of contact element- površina presjeka kontaktnog elementac- integration constant- konstanta integracijed- horizontal distance between lower beam supportand end- horizontalna udaljenost između oslonca i krajadonje gredeyk,u- degree of freedom of upper beamwhere contact is realized- stupanj slobode gornje grede namjestu kontaktaA- auxiliary matrix- pomoćna matricaC- damping matrix- matrica prigušenjaE- Young’s modulus- Youngov modul elastičnostiCl- lower beam global damping matrix- globalna matrica prigušenja donje gredeFc- contact element load- opterećenje kontaktnog elementaCu- upper beam global damping matrix- globalna matrica prigušenja gornje gredeFdin - dynamic contact force- dinamička kontaktna silaD- auxiliary matrix- pomoćna matricag- acceleration of gravity- gravitacijsko ubrzanjeK- stiffness matrix- matrica krutostih- vertical distance between the upper beam centreof gravity at initial and at impact time instant- vertikalna udaljenost između težišta gornje gredeu početnom trenutku i trenutku udaraKc- contact element stiffness matrix- matrica krutosti kontaktnog elementaKlhdin - vertical distance between the upper beam centreof gravity at impact time instant and at maximumdeflection instant- vertikalna udaljenost između težišta gornje gredeu trenutku udara i maksimalnog progiba- lower beam global stiffness matrix- globalna matrica krutosti donje gredeKu- upper beam global stiffness matrix- globalna matrica krutosti gornje gredeM- mass matrix- matrica masei- time step- vremenski korakMl- lower beam global mass matrix- globalna matrica masa donje gredekf- flexion coefficient- koeficijent fleksijeMu- upper beam global mass matrix- globalna matrica masa gornje gredelc- contact element length- duljina kontaktnog elementaP, Q, R- auxiliary matrices- pomoćne matricell- lower beam length- duljina donje gredeS- integration matrix- matrica integracijelu- upper beam length- duljina gornje gredeΦ- natural vibration modes matrix- matrica prirodnih oblika vibriranjami- modal mass for certain vibration mode- modalna masa za određeni oblik vibriranja{F(t)}- force vector- vektor sileml- lower beam mass- masa donje grede{Fl}- lower beam nodal forces vector- vektor čvornih sila donje gredemu- upper beam mass- masa gornje grede{Fu}- upper beam nodal forces vector- vektor čvornih sila gornje gredeql- lower beam distributed load- distribuirano opterećenje donje grede{f}i 1 {gp} {hp} {p} - auxiliary vectors- pomoćni vektoriqu- upper beam distributed load- distribuirano opterećenje gornje grede{δ}- displacement vector- vektor pomakat- time variable- vrijeme{δl}- lower beam displacement vector- vektor pomaka donje gredewdin - dynamic deflection- dinamički progib{δu}- upper beam displacement vector- vektor pomaka gornje gredeyk,l{ }- velocity vector- vektor brzine{ l}- lower beam velocity vector- vektor brzine donje grede- degree of freedom of lower beam where contact isrealized- stupanj slobode donje grede na mjestu kontakta

Strojarstvo 54 (5) 341-350 (2012){uI. ĆATIPOVIĆ et. al., Semi-analytical Solution for Elastic. 343} - upper beam velocity vector- vektor brzine gornje grede{ } - acceleration vector- vektor ubrzanja{ l} - lower beam acceleration vector- vektor ubrzanja donje grede{u} - upper beam acceleration vector- vektor ubrzanja gornje gredeγi- non-dimensional damping coefficient for certainvibration mode- bezdimenzijski koeficijent prigušenja za određenioblik vibriranjaΔt- interval duration- vremenski intervalεu- upper beam angular acceleration- kutno ubrzanje gornje gredeωi- natural frequency of certain vibration mode- prirodna frekvencija određenog oblika vibriranjadiagonalization method for solving differential equationof motion. The commercial software package [3] is usedto develop computer code for calculation, and numericalexample of collision of communication bridge betweentwo floating objects and arrestor is solved. Verification ofthe developed semi-analytical procedure is checked bycorrelating stress level obtained by numerical procedureand by applying energy conservation principle.Figure 1. Disposition of upper and lower beam2. Formulation of the problemSlika 1. Dispozicija gornje i donje gredeTwo elastic beams, i.e. upper and lower beam areconsidered, Figure 1. One end of the upper beam issliding supported, and the other one is hinge jointed. Ifsliding support is moved off, upper beam is pitching andstrikes the free end of the second beam. Another end ofthe lower beam is fixed. It is assumed that pitching of theupper beam is influenced only by gravitational force, butother excitation can also be taken into account. Dynamicinteraction of the beams starts when upper beams strikesthe lower one. The interaction can be treated as alterationof contact and non-contact stages, and basic equation ofdynamic equilibrium (equation of motion) yields [4]:(1)Upper and lower beam are being discretized into anumber of finite elements. Global stiffness and massmatrices for beams are defined according to [4], andglobal damping matrices for upper and lower beam aredefined according to [5]. Constitution of damping matrixis rather complex, and study of damping influence onsystem dynamic response, conducted and presented in[1], showed that damping matrix can be derived by takinginto account critical damping for all vibration modes. So,the global damping matrix can be calculated according tothe following formula [5]:(2)For standard vibration analyses, recommended valueof non-dimensional damping coefficient is between 0,02and 0,05.Beam 4DOF finite element is used for beam modeling,and bar 2DOF finite element is used for contact elementmodeling. The role of the contact element is descriptionof local behavior of both beams close to the impact place.Since the procedure intends to give global response,the influence of considered contact element on globaldeformations of the rest of the construction should beminimized. This can be achieved by choosing largestiffness of the contact element so that its deformation iswithin elastic domain.During the pitching of the upper one, the consideredbeams are two independent bodies. Only during thecontact these beams can be treated as one structure. It isalso obvious that the beams are not in contact from theimpact time instant till the resting time instant. Moreover,the interaction of considered beams consists of a numberof contact and non-contact periods. In other words,we can say that the contact of beams can sustain onlycompressive but not the tensile force. This fact should betaken into account when defining the system of equationsof motion. Therefore, by means of Eq. (1) two systemswith two possible stages of beams interaction shouldbe defined, i.e. the first one without the contact and thesecond one with the contact. Governing equation for thefirst stage, where beams act as two independent bodies,yields:

344I. ĆATIPOVIĆ et. al., Semi-analytical Solution for Elastic.Strojarstvo 54 (5) 341-350 (2012)(7)(3)When considered beams are in contact, inertial anddamping forces take effects on both beams independently,but restoring forces are coupled. So, the coupling shouldbe introduced in Eq. (3) through new stiffness matrix, andit is achieved by imposing the 2DOF bar finite elementstiffness matrix into the old one. The stiffness matrix of2DOF bar finite element yields [4]:(4)and it is added up together with matching terms in theold stiffness matrix. Addition of matching terms into oldstiffness matrix can be described in the following form:Thus, if the condition in Eq. (7) is satisfied, there iscontact between the beams and vice-versa.As mentioned before, external load is caused only bygravitational force, and taking into account that crosssection of beams is assumed to be uniform, the weightis uniformly distributed per length. In that case the loadvector can easily be defined, and for distributed load ofthe upper and the lower beam one can write:(8)3. Solution of differential equation ofmotionEquation (1) is a nonhomogenous second-orderdifferential equation (system of differential equations)which can be reduced to a first-order system. If Eq. (1) ismultiplied by M-1 from the left we can write:(5)For the purposes of the calculation execution andbecause of code development requirements it is necessaryto describe alteration of contact and non-contact stagesby appropriate condition which expresses appearance anddisappearance of compressive load in contact element,and it is given by the following formula:(6)It is more suitable to replace previous equation byusing the following one:(9)To reduce Eq. (9) to a first-order differential equationwe introduce the following substitution:(10)Thus, one can write:(11)

Strojarstvo 54 (5) 341-350 (2012)I. ĆATIPOVIĆ et. al., Semi-analytical Solution for Elastic. 345Having in mind Eqs. (10) and (11) one can obtain:Now, we have to multiply (20) by [X]-1 from left,obtaining:(12)(21)whereBecause of simplicity we introduce the followingsubstitutions:(22)Because of (17) we can write this:(13)(23)and in components:(14)(15)(24)where j 1, , n. We can solve each of these n linearequations and then we have:(25)Now, after the reduction of equation of motion to afirst-order nonhomogenous system we write:(16)There are several methods on disposal for solving Eq.(16), as for example method of undetermined coefficients,method of the variation of parameters or method ofdiagonalization [6]. Here, the method of diagonalizationis applied, since this kind of solving of equations requireseigenvectors and eigenvalues calculation, which is verysimilar to ordinary engineering problem of analyzing freevibrations, i.e. natural modes and natural frequenciescalculation. The idea of this method is to “decouple” the nequations of a linear system, so that each equation containsonly one of the unknown functions, i.e. deflections p1, , pn and thus can be solved independently of the otherequations. Matrix [A] has a basis of eigenvectors {x}(1), , {x}(n). Then, one can write:These are the components of {z(t)}, and from themwe obtain the solution of (19).It is already mentioned that execution of thecalculation consists of solving differential equation ofmotion in coupled and uncoupled form, since contactand non-contact stages are altering. This fact enables usdetermination of integration constants cj, which are foreach stage determined from the antecedent one. Whenstarting the calculation, the displacement vector is known,and bearing in mind Eqs. (19) and (25), we can write:(26)(27)and finally:(28)(17)where [D] is diagonal matrix with eigenvalues λ1, , λnof [A] on the main diagonal, and [X] is the n x n matrixwith columns {x}(1), , {x}(n). It should be noted that [X]1exists since these columns are linearly independent.To apply diagonalization to (16) we define the newunknown function:4. Outline of numerical procedure andenergy conservation principle approachNumerical procedure is detailed described in [1], anddescription of energy conservation principle is given in[2]. Here, only basic remarks and formulae are given.(18)4.1. Numerical procedureThen we can write:(19)Substituting (19) into (16) we obtain:(20)Solution of the Eq. (1) can be obtained by assuminglinear acceleration in specified time interval [3]. Velocityand displacement vector at time instant ti 1 are beingcalculated by using the following formula [1, 3]:

346I. ĆATIPOVIĆ et. al., Semi-analytical Solution for Elastic.Strojarstvo 54 (5) 341-350 (2012)5. Numerical example(29)Acceleration vector in (29) is obtained from:(30)where:(31)Matrix [S] in (30) is usually called integration matrix,and it is defined by the expression:(32)The expressions for auxiliary matrices yield:(33)The load is defined by (8), and definition of initialconditions is described in [1] in details.4.2. Energy conservation principleFor the stress determination of the upper and lowerbeam, contact dynamic force has to be calculated.Potential energy of the upper beam transforms due togravity force into the kinetic energies and strain energiesof the upper and lower beam. If one assumes that upperbeam has several times higher stiffness than lower beamand that the lower beam has zero velocity at highestdeflection (the proof is given in [1]), we can write:The application of semi-analytical solution isillustrated with the case of collision of communicationbridge between two offshore units and safety structurecalled arrestor, Figure 2. When offshore units arebeing installed on the exploitation location, seakeapingand anchoring calculations are required. Sinceexternal forces, which are assessed by stochastic typecalculations, acting on the offshore objects and causetheir movement, it is obvious that the case when distancebetween objects exceeds the bridge length can occur. Inthat case, the bridge is pitching and strikes the end ofarrestor. The bridge is braced structure and it is madeof high-tensile steel [1, 7], Figure 3. Arrestor consistsof two longitudinal beams and one transverse beam [1],Figure 4. Both, the bridge and the arrestor are modelledas uniform beams and their properties are given in Table1. Maximum bending stress of material of the bridge andarrestor is σal 533 MPa, maximum shear stress is τal 355MPa, and Young’s modulus equals E 2,1·105 MPa.Vertical distance between the bridge centre of gravityat initial and at impact time instant is equal to 0,57 m.The discretization of the bridge structure is done with 14finite elements, and arrestor is discretized into 7 finiteelements, Figure 5. One finite element is used for contactelement simulation. The length of contact element ischosen to be 0,3 m, and it’s cross-section area is equal to0,04 m2. Non-dimensional damping coefficient is takento be 0,3.(34)The contact dynamic force is largest at maximumdeflection of the lower beam, and it is calculated accordingto simple expression:(35)Flexural stiffness in previous equation is equal to:(36)Bearing in mind equation (34) and geometricalrelations shown in Figure 1, deflection is obtained from:(37)Figure 2. Bridge and arrestor – schematic viewSlika 2. Most i arestor – shematski prikaz

Strojarstvo 54 (5) 341-350 (2012)I. ĆATIPOVIĆ et. al., Semi-analytical Solution for Elastic. 347Figure 3. Bridge structureSlika 3. Struktura mostagood agreement between the results obtained by both,semi-analytical and numerical procedure in arrestor andbridge deflections is achieved, Figures 10 and 11. Thesame can be also concluded for time histories of arrestorand bridge bending stresses, shear stresses in the arrestorcritical cross-section as well as for time history of contactelement stresses, Figures 12, 13, 14 and 15. The valueof deflection, obtained by semi-analytical approachequals 0,658 m, which is higher than value obtained byapplying energy conservation principle, but lower thenthe value obtained by the numerical procedure, Table 2.This is also valid for bending stress levels of the arrestor,while semi-analytical procedure gives slightly highervalues of the bridge bending stresses and arrestor shearstresses comparing to numerical procedure and energyconservation principle approach.Figure 4. Arrestor structureSlika 4. ArestorFigure 6. Deflections of arrestor and bridge at contact locationSlika 6. Progibi arestora i mosta

This paper presents semi-analytical solution for the problem of elastic impact of two beams. The solution is based on the finite element . discretization of the structure and equation of motion solution using diagonalization method for solving a system of differential equations. This

Related Documents:

Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original

10 tips och tricks för att lyckas med ert sap-projekt 20 SAPSANYTT 2/2015 De flesta projektledare känner säkert till Cobb’s paradox. Martin Cobb verkade som CIO för sekretariatet för Treasury Board of Canada 1995 då han ställde frågan

service i Norge och Finland drivs inom ramen för ett enskilt företag (NRK. 1 och Yleisradio), fin ns det i Sverige tre: Ett för tv (Sveriges Television , SVT ), ett för radio (Sveriges Radio , SR ) och ett för utbildnings program (Sveriges Utbildningsradio, UR, vilket till följd av sin begränsade storlek inte återfinns bland de 25 största

Hotell För hotell anges de tre klasserna A/B, C och D. Det betyder att den "normala" standarden C är acceptabel men att motiven för en högre standard är starka. Ljudklass C motsvarar de tidigare normkraven för hotell, ljudklass A/B motsvarar kraven för moderna hotell med hög standard och ljudklass D kan användas vid

LÄS NOGGRANT FÖLJANDE VILLKOR FÖR APPLE DEVELOPER PROGRAM LICENCE . Apple Developer Program License Agreement Syfte Du vill använda Apple-mjukvara (enligt definitionen nedan) för att utveckla en eller flera Applikationer (enligt definitionen nedan) för Apple-märkta produkter. . Applikationer som utvecklas för iOS-produkter, Apple .

o Le 17 mai : semi-marathon du Dreilaenderlauf (Courses des trois pays, Bâle) o Le 14 juin : semi-marathon des foulées epfigeoises o Le 21 juin : semi-marathon du vignoble d’Alsace (Molsheim) o Le 13 septembre : semi-marathon de Colmar o Le 27 septembre : semi-marathon des F4P (Rosheim) o Le 4 octobre : semi-marathon de Sélestat

Magic Quadrant Vendor Strengths and Cautions Elastic Elastic is a Niche Pla yer in this Magic Quadrant. Elastic is based in Mountain View, California, U.S., the Netherlands and Singapor e. It has customers worldwide. Its SIEM platform is Elastic Security, which offers endpoint security, following Elastic 's acquisition of Endgame in 2019. Its .

Organization 67 SECTION 2: ORGANIZATIONAL BEHAVIOR IN GROUP LEVEL Chapter 6 Organizational Communication in Islamic Management 91 Chapter 7 Organizational Conflict Management in Islamic Management 111. SECTION 3: ORGANIZATIONAL BEHAVIOR IN ORGANIZATION LEVEL Chapter 8 Influence and Leader–Follower Relations in Hereafter-oriented Organizations 137 Chapter 9 Leadership Styles in Islamic .