Analysis of Linear Structures With Non Linear Dampers

Transcription

1 Analysis of inear Srucures Wih Non inear Dampers V.Denoël Deparmen of Mechanics of maerials and Srucures, Universiy of iège, Belgium H.Degée Deparmen of Mechanics of maerials and Srucures, Universiy of iège, Belgium ABSTRACT: This paper provides informaion abou he numerical simulaion of he dynamic behaviour of linear srucures including non linear dampers, for which he relaion beween he damping force and he velociy is described by a power law. A firs par of he paper deals wih single degree of freedom (SDOF sysems, wih a paricular emphasis on he resoluion of he non linear equaion allowing o compue he damping force corresponding o a given velociy. A second par deals wih muli degree of freedom (MDOF sysems, and presens a special algorihm o sudy he behaviour of srucures wih a very small number of non linear componens. 1 INTRODUCTION In many cases, special devices such as viscous dampers mus be added o srucures, in order o compensae for heir bad dynamic behaviour. Viscous dampers are nowadays commonly used in wind and seismic engineering. Some of hem can someimes be assumed o exhibi a linear behaviour, whereas oher damping devices are characerised by a non linear behaviour, mos of he ime described by a power law (F = C.V α - α<1. Compared o linear dampers, such devices presen he advanage o dissipae a significanly greaer amoun of energy for idenical maximum forces and displacemens. However hey are obviously more complicaed o manage in he conex of numerical simulaions. F ( = C v( (1 Hence, a harmonic imposed displacemen would give : d( = A sin( ω ( v( = Aω cos( ω F( = C Aω cos( ω (3 Removing he ime parameer beween Equaions and 3 provides he equaion of he curve in he FD diagram : d A F CAω = 1 ( This well-known ellipic equaion characerizing a linear dash-po is ploed on figure 1. NON INEAR DAMPERS The mos basic es ha can be realized o deermine he behavior of a dash-po consiss in forcing he ends of he device o move wih a harmonic relaive displacemen. A firs characerizaion can hen be obained by varying he frequency of he periodic imposed displacemen. Afer sabilizaion of he response, he dash-po describes a characerisic curve in a Force- Displacemen (FD diagram. For example, for a perfecly linear dash-po, he force is proporional o he relaive velociy : Figure 1. Characerisic curve of a linear damper in he Force- Displacemen diagram e us consider now a non linear damper characerized by he power law : [ ( ] α F ( = C v (5

2 The equaion of he characerisic curve in he FD diagram is hen given by : d A F C 1/ α Aω 1/ α = 1 (6 As an illusraion, Figure represens he consiuive law and he FD curve of a non linear damper. 3 ANAYSIS OF SDOF SYSTEMS The equaion of moion corresponding o a single degree of freedom sysem wih consan mass and siffness and wih power law damping is represened by his non linear second order differenial equaion: α [ u& ( ] k u( p( m u& ( c = (8 The wo main characers of his equaion (differenial and non linear are generally considered he one afer each oher. Figure. Represenaion of a power consiuive law ( α = 0.5 ; C = 50 kn / (mm/s 0.5 and he corresponding FD curve. In equaion (6, he linear behavior can be obained by seing α = 1. When he parameer α approaches 0, he behavior becomes rigid perfecly viscous and he shape of he FD curve ends owards a recangle. Here comes he firs advanage of non linear dampers : as he area enclosed inside he FD curve represens he amoun of energy dissipaed per cycle, a non linear damper can dissipae a larger amoun of energy han a linear one for idenical maximum force and displacemen. The maximum benefi corresponding o a rigid perfecly viscous damper can be easily compued by comparing he area of an ellipse wih he area of he recangle circumscribing his ellipse : π Q = = 7,3% (7 π Anoher reason for which non linear dampers are used is no based on energy consideraions bu raher on a securiy design. As his kind of device is generally designed o improve he behavior of a srucure under probabilisic acions (earhquake, wind,, he maximum level of he soliciaion canno be deermined exacly. As a consequence, he maximum velociy of any poin of he srucure canno be deermined eiher. The maximum force applied by a linear damper is hen unknown whereas he maximum force applied by a non linear damper can be limied as much as waned by choosing a sufficienly low value for parameer α. So, in case of unexpeced increase of he exernal forces, a non linear damper won apply inolerable forces on he srucure. 3.1 The differenial characer The large number of mehods allowing o solve he well-known linear equaion is of course beyond he scope of his paper. This paragraph presens however a shor summary of he resoluion mehods in he ime domain. Basically, his firs mehod considers he exernal force as a succession of shor impulses. For each of hese, he response can be compued and he oal response a ime is hen obained as a sum of all he conribuions associaed o he effecs of each impulse. Analyically, he duraion of each impulse ends owards zero and he sum becomes an inegral. The mehod explained here above leads o he well known Duhamel inegral : u 1 ξωτ ( = p( τ e sin( ω dτ. dτ mω d 0 (9 whereξ is he damping coefficien and ω d is he damped pulsaion of he sysem. Regarding he kind of problem ha has o be reaed here, his mehod has he grea disadvanage of being based on he superposiion principle, which is only valid for linear srucures. Numerically, i is possible o develop mehods which are no based on he superposiion principle. The so-called sep by sep mehods are no based on a superposiion of differen conribuions anymore, bu raher on a discreizaion of ime in small ime seps. Assumpions are hen made regarding displacemen, velociy and acceleraion a he end of each ime sep. Togeher wih he discreized equaion of moion, i is hen possible o compue he response a he end of he ime sep. A huge number of echniques have been developed amongs which he mos famous are : consan or linear acceleraion mehod, cenral difference mehod, Newmark mehod, Houbol s mehod, Wilson-θ mehod, HHT mehod. Since he purpose of he paper deals wih solving a non linear equaion, a sep by sep mehod has been chosen. The developmens are made wih Newmark mehod.

3 Classical linear Newmark mehod can be adaped for solving he non linear Equaion 8. The displacemen a he end of a ime sep is hen given by : R f 10 precision (15 m k. u = p m u& u&& u C [ ] α & (10 u where he subscrips and denoe respecively he response a he beginning and a he end of he ime sep. As he velociy a he end of he ime sep can be expressed in erms of he displacemen a he end of he sep, Equaion 10 can be rewrien in he form : F [ u ] m u u& u& (11 = p The second par of he equaion is a known quaniy and he funcion F is defined by : F m [ u ] = k u C[ u& ( u ] α 3. The non linear characer (1 The firs consideraions relaed o he differenial characer of he equaion allow o ransform he original problem o he resoluion of a series of equaions like Equaion 11. This laer is he non linear equaion ha has o be solved. The nex paragraphs illusrae differen mehods o solve i. For convenience, Equaion 11 will be rewried : F [ x] = f (13 where x represens he unknown u The Newon-Raphson mehod The mos famous mehod for solving such problems is he Newon-Raphson (NR mehod summarized in figure 3. Saring from an approximaion x (0 of he soluion, he mehod consiss in considering he inersecion x (1 of he angen o he funcion F(x a poin x (0 wih he horizonal F(x = f as a beer approximaion of he soluion. Mahemaically, his can be expressed by : ( i 1 R( x x = x (1 F ( x A each ieraion he remainder, difference beween he arge value f and he funcion F(x (i, should decrease unil a convergence crierion is verified : Figure 3 : Illusraion of he Newon-Raphson mehod for solving non linear equaions This mehod encouners several problems wih Equaion 11 which has o be solved. They resul mainly from he verical slope inflexion poin in he funcion F. Indeed, he firs erm in his funcion (see Eq. 1 is linear whereas he second one has he same shape as he consiuive law. The funcion F exhibis herefore a verical slope inflexion poin. Despie is very fas convergence ( nd order, he NR mehod misbehaves in he viciniy of inflexion poins. This is illusraed in figure. Figure : Convergence problems of he Newon-Raphson mehod around inflexion poins The original mehod is hen in general replaced by he modified-nr mehod which, in case of non convergence, coninues he ieraions wih a fixed slope larger han he slope a he inflexion poin; his enables hen o reach he convergence. 3.. The Regula Falsi mehod As in he equaion o solve, he slope a he inflexion poin is verical, i is impossible o find a larger slope and o proceed o he modified-nr mehod. I is hen necessary o use anoher mehod. Only firs order mehods can succeed in crossing inflexion poins. To his scope, he regula falsi mehod can be used. The basic idea is o urn around a reference poin a each ieraion and, during he ieraive procedure, o modify he posiion of his poin in order o opimise he convergence. The mehod works always

4 wih he reference poin and wih a moving poin obained by he inersecion of he horizonal F = f wih he line going hrough he reference poin and he previous poin. Mahemaically, i is expressed by : x ( i 1 = x ( x f x R( x ( i R( x R( x f (16 Figure 5 shows an example where he reference poin x f is no modified, whereas Figure 6 shows anoher one where he reference poin x f is modified a each ieraion. The condiion for changing he reference poin is simple : if he new remainder has he same sign as he remainder a he reference poin, hen i should be changed o he previous poin, else i mus be kep unchanged. 3.3 Example Here is he response of a single degree of freedom characerized by he following parameers : Mass : m = 1 Siffness : k = 1 Damping : C = 10 ; α = 0.1 Exernal Force : Harmonic wih a pulse ω = 0,5 rad/s during 0 s. = 0.01 s [ Newmark mehod : α = ¼ ; δ = ½ ] Figure 7 : Displacemen of he generalized degree of freedom Figure 5 : Regula Falsi mehod. The reference poin is unchanged from beginning ill end of he ieraions Figure 8 : Acceleraion of he generalized degree of freedom Figure 6 : Regula Falsi mehod. The reference poin is changed a each ieraion. The regula falsi mehod is also an ieraive mehod and he convergence crierion used o sop can be expressed by Equaion 15 as for he NR mehod Conclusion The regula falsi is a mehod which will always lead o convergence bu slowlier han he NR mehod. So, combining hese wo mehods, he bes is o use NR where he convergence can be reached by his mehod and o use he regula falsi elsewhere. In pracice, his can be achieved by always beginning wih he NR mehod; once he soluions are going back and forh a limi value, we swich o he regula falsi mehod. Figure 9 : Phase diagram and FD diagram of he dash-po 3. Observaions In Figure 7, he free displacemen looks classic beween 0 s and 30 s. Afer his, he oscillaor seems o be frozen bu his is of course no a residual displacemen. The sysem doesn have enough energy anymore o go beyond he elbow of he consiuive law of he dash-po. For he res of he compuaion he viscosiy is hen very high and i akes a very long ime o reach a zero displacemen. In Figure 8, he acceleraion seems o be disconinuous. In fac, his is he resul of passing rough he elbow : he acceleraion is of course perfecly coninuous bu varying very fas. Furher invesigaion, using a non consan ime sep, allows o

5 compue more precisely he poins in he fas varying zone. In Figure 9, i is possible o recognize he classical recangular shape of a non linear dash-po in he FD diagram. Several curves are covered, because he response is no harmonic. Furhermore, because of he consan ime sep, he response in he fas varying zone is less precise han for higher velociies. Wih a precision facor (See Eq. 15 of 8, he mean number of ieraions was 3 ieraions per ime sep for which he NR mehod converged and 15 o 0 ieraions oherwise (in hese are included he NR ieraions which did no converge. ANAYSIS OF MDOF SYSTEMS Basic developmens concerning he differenial and non linear characers of Equaion 8 can be adaped in order o solve a muli-degree of freedom sysem. Obviously, he main difference lies in he grea number of degrees of freedom, implying a large sysem o solve. For complex srucures, he size of he sysem can ofen reach 10 or 10 5 DOF. Amongs hese degrees of freedom, only a small par is concerned wih he non linear dampers. The res of he srucure can generally be considered o behave linearly. This could be for example he case in he design of damping devices for a srucure encounering roubles due o wind..1 The equaion of moion Similarly o he SDOF sysem, a sep-by-sep mehod is firs used o ransform he non linear differenial sysem o several sysems of non linear equaions. Wih he Newmark mehod, his sysem can be wrien in he following form (o compare o Equaion 11 : { CD nonlin } { b} [ ]{ u} F ({ u ({ u }} A, = (17 & where [ A ] = [ M ] [ K] mass and siffness marices of he srucure and ([ M ] and [ K ] are he { b } { P} [ M ] {} u { u& } { u& } =. in which [ C ] and [ S ([ C ] [ C ]. { u} { u& } S D lin C, (18, D lin ] represen respecively he srucural Rayleigh damping marix and he concenraed linear damping marix (linear dash-pos. Equaion 16 represens he non linear sysem of equaions o be solved wih he NR mehod. The order of his sysem is he oal number of degrees of freedom in he srucure. Wrien like his, he addiional ieraions required by he non lineariy s imply o ierae on he full sysem. To avoid his, i is ineresing o reduce he number of equaions o handle in he NR ieraions hrough a saic condensaion.. Saic condensaion e us qualify by he erm non linear a degree of freedom which is direcly relaed o a non linear damper (i.e. a ranslaion of a node a which a non linear dash-po is aached. For example, in figure 10, he non linear degrees of freedom are he DOF 1,,7 and 8. Figure 10. Illusraion of he definiion of he non linear DOF We can now sor he equaions in he sysem (Eq. 17 by placing he non linear DOF above he ohers : [ ANN ] [ AN ] [ A ] [ A ] N { un } { u } F CD, nonlin 0 ({ un } { b } N = { b} (19 The subscrips and N are used respecively for linear and non linear DOF. In Equaion 19, only he group of equaions corresponding o he firs line is non linear. The oher equaions remain linear since he non linear erm is zero (he damper do no apply any direc force on hese nodes. The second line of his x sysem gives : 1 { u } = [ A ] ({ b } [ A ]{. u } N N (0 Afer subsiuion of Equaion 0 in Equaion 19, we obain a new reduced sysem: [ A ]{. u } { F } { f } * (1 N CD, nonlin = 1 where [ A *] [ ANN ] [ AN ][. A ].[ AN ] 1 { f } = { b } [ A ][. A ].{ b }. N = and N The order of he new sysem (Eq. 1 is hus reduced o is minimum value i.e. he number of non linear degrees of freedom. The NR (or regula falsi mehod can hen be applied o compue he response of he non linear degrees of freedom a he end of he ime sep. Then, equaion 0 can be used o compue he response of he linear degrees of freedom. This huge

6 amoun of values are compued only afer convergence and no a each ieraion of he NR mehod! Going furher ino he developmens is of course beyond he scope of his paper. Ineresed readers can found he full developmens and discussions abou his mehod in Reference 1..3 Example The example consiss in deermining he response of a 37 meer high bridge pier modelled by 11 beam elemens (A=1.3 m² ; I=38.3 m ; E = 3600 MPa This pier is subjeced o a very paricular ground moion shaking characerized by his acceleraion : u& g = sin ( 8. Two differen dampers are placed a he head of he pier in order o reduce he effecs of he ground moion : A firs one wih a bilinear consiuive law (C 1 =.5E8 N.s/m ; C =E6 N.s/m ; v lim = 8E- m/s A second one wih he classic power law (C = 1E6 N.(s/m 0. ; α = 0. In order o invesigae he possibiliy of replacing a power law damper by a bilinear law damper, he wo dash-pos have been chosen in such a way ha heir behavior is he mos similar. Displacemen [m] 0,00 0,0015 0,001 0, , ,5 1 1,5,5 3 3,5-0,001-0,0015-0,00 Time [s] Bilinear law Power law Figure 11 : Displacemen a he head of he pier (for boh dampers Force [kn] ,00-0,0015-0,001-0, ,0005 0,001 0,0015 0, Displacemen [m] Bilinear law Figure 1 : FD diagrams of he dash-pos. Power law Figure 11 represens he displacemen a he head of he pier (for boh dampers, whereas Figure 13 represens heir FD curve.. Observaions The FD diagram has he classical shape, exceped in he corners where inerial effecs (paricipaion of he second vibraion mode modify slighly he shape which would be obained wih a SDOF sysem ; Resuls relaed o he power law damper can be approached quie precisely wih a bilinear damper which is consuming.8 imes less ieraions han he power law dash-po. Advanage could hen be aken of he bilinear law, provided a good equivalence is ensured. 5 CONCUSIONS This documen firs presened soluions o he numerical problems encounered during he analysis of srucures including power law dampers, problems relaed o he verical inflexion poin in he consiuive law. In a second sep, by considering SDOF sysems, he paper poins ou he characerisics of he response of srucures consiued of non linear dampers (pseudo-residual displacemen; pseudodisconinuous acceleraion. Furhermore, hrough he analysis of MDOF sysems, i inroduces and jusifies he use of he saic condensaion mehod for linear srucures wih concenraed non lineariy s. This mehod reduces he size of he sysem o is minimum, which enables a huge decrease of he compuaion ime. Finally, hrough an example, he paper shows ha, provided good equivalence is chosen, a power law dash-po can be efficienly replaced by a bilinear law dash-po. In his case he numerical approach is much easier since he verical inflexion poin disappears. 6 REFERENCES 1 V. Denoël. Calcul sismique des ouvrages d ar, Graduaion projec, Universiy of iège, (001. R.W. Clough and J. Penzien, Dynamics of he srucures, Mc Graw-Hill, ondon, Second Ediion, ( J.M. Orega and W.C. Rheinbold, Ieraive soluion of non-linear equaions in several variables, Academic press, New York, (1970