2C09 Desig for seismic ad climate chages Lecture 02: Dyamic respose of sigle-degree-of-freedom systems I Daiel Grecea, Politehica Uiversity of Timisoara 10/03/2014 Europea Erasmus Mudus Master Course Sustaiable Costructios uder Natural Hazards ad Catastrophic Evets 520121-1-2011-1-CZ-ERA MUNDUS-EMMC
Europea Erasmus Mudus Master Course Sustaiable Costructios uder Natural Hazards ad Catastrophic Evets 2C09-L2 Dyamic respose of sigle-degree-offreedom systems I L2.1 Itroductio to dyamics of structures. L2.2 Solutios methods for the equatio of motio. L2.3 Free vibratio aalysis of SDOF systems. L2 Dyamic respose of sigle-degree-of-freedom systems I
Dyamics of structures Dyamics of structures determiatio of respose of structures uder the effect of dyamic loadig Dyamic load is oe whose magitude, directio, sese ad poit of applicatio chages i time p(t) p(t) p(t) ü g (t) t t t t
Sigle degree of freedom systems Simple structures: mass m stiffess k Objective: fid out respose of SDOF system uder the effect of: a dyamic load actig o the mass a seismic motio of the base of the structure The umber of degree of freedom (DOF) ecessary for dyamic aalysis of a structure is the umber of idepedet displacemets ecessary to defie the displaced positio of masses with respect to their iitial positio Sigle degree of freedom systems (SDOF) k m
Sigle degree of freedom systems Oe-storey frame = mass compoet stiffess compoet dampig compoet Number of dyamic degrees of freedom = 1 Number of static degrees of freedom =?
Force-displacemet relatioship
Force-displacemet relatioship Liear elastic system: elastic material fs k u first order aalysis Ielastic system: plastic material First-order or secod-order aalysis f f u, u S S
Dampig force Dampig: decreasig with time of amplitude of vibratios of a system let to oscillate freely Cause: thermal effect of elastic cyclic deformatios of the material ad iteral frictio
Dampig i real structures: Dampig frictio i steel coectios opeig ad closig of microcracks i r.c. elemets frictio betwee structural ad o-structural elemets Mathematical descriptio of these compoets impossible Modellig of dampig i real structures equivalet viscous dampig
Dampig Relatioship betwee dampig force ad velocity: c - viscous dampig coefficiet uits: (Force x Time / Legth) Determiatio of viscous dampig: free vibratio tests forced vibratio tests Equivalet viscous dampig modellig of the eergy dissipated by the structure i the elastic rage f D c u
Equatio of motio for a exteral force Newto s secod law of motio D'Alambert priciple Stiffess, dampig ad mass compoets
Equatio of motio: Newto s 2d law of motio Forces actig o mass m: exteral force p(t) elastic (or ielastic) resistig force f S dampig force f D Exteral force p(t), displacemet u(t), velocity ad acceleratio u ( t) are positive i the positive directio of the x axis Newto s secod law of motio: p f f mu S S D mu f f p D mu cu ku p( t) u ( t)
Equatio of motio: D'Alambert priciple Iertial force equal to the product betwee force ad acceleratio acts i a directio opposite to acceleratio D'Alambert priciple: a system is i equilibrium at each time istat if al forces actig o it (icludig the iertia force) are i equilibrium f I fs fd p mu f I mu f f p S D mu cu ku p( t)
Equatio of motio: stiffess, dampig ad mass compoets Uder the exteral force p(t), the system state is described by displacemet u(t) velocity u ( t) acceleratio u ( t) System = combiatio of three pure compoets: stiffess compoet fs k u dampig compoet f D c u mu mass compoet f I f I fs fd p Exteral force p(t) distributed to the three compoets
SDOF systems: classical represetatio
Equatio of motio: seismic excitatio Dyamics of structures i the case of seismic motio determiatio of structural respose uder the effect of seismic motio applied at the base of the structure Groud displacemet u g Total (or absolute) displacemet of the mass u t Relative displacemet betwee mass ad groud u t u ( t) u( t) u ( t) g
Equatio of motio: seismic excitatio D'Alambert priciple of dyamic equilibrium Elastic forces relative displacemet u f S k u Dampig forces relative displacemet u f D c u Iertia force total displacemet u t f I mu t fi fs fd t mu cu ku 0 0 t u ( t) u( t) u ( t) mu cu ku mu g g
Equatio of motio: seismic excitatio Equatio of motio i the case of a exteral force mu cu ku p( t) Equatio of motio i the case of seismic excitatio mu cu ku mu g Equatio of motio for a system subjected to seismic motio described by groud acceleratio u g is idetical to that of a system subjected to a exteral force Effective seismic force p ( t) mu ( t) eff g mu g
Problem formulatio Fudametal problem i dyamics of structures: determiatio of the respose of a (SDOF) system uder a dyamic excitatio a exteral force groud acceleratio applied to the base of the structure "Respose" ay quatity that characterizes behaviour of the structure displacemet velocity mass acceleratio forces ad stresses i structural members
Determiatio of elemet forces Solutio of the equatio of motio of the SDOF system displacemet time history u( t) Displacemets forces i structural elemets Imposed displacemets forces i structural elemets Equivalet static force: a exteral static force f S that produces displacemets u determied from dyamic aalysis f ( t) ku( t) s Forces i structural elemets by static aalysis of the structure subjected to equivalet seismic forces f S
Combiatio of static ad dyamic respose Liear elastic systems: superpositio of effects possible total respose ca be determied through the superpositio of the results obtaied from: static aalysis of the structure uder permaet ad live loads, temperature effects, etc. dyamic respose of the structure Ielastic systems: superpositio of effects NOT possible dyamic respose must take accout of deformatios ad forces existig i the structure before applicatio of dyamic excitatio
Solutio of the equatio of motio Equatio of motio of a SDOF system mu ( t) cu ( t) ku( t) p( t) differetial liear o-homogeeous equatio of secod order I order to completely defie the problem: iitial displacemet u(0) u(0) iitial velocity Solutio methods: Classical solutio Duhamel itegral Numerical techiques
Classical solutio Complete solutio u(t) of a liear o-homogeeous differetial equatio of secod order is composed of complemetary solutio u c (t) ad particular solutio u p (t) u(t) = u c (t) +u p (t) Secod order equatio 2 itegratio costats iitial coditios Classical solutio useful i the case of free vibratios forces vibratios, whe dyamic excitatio is defied aalytically
Classical solutio: example Equatio of motio of a udamped (c=0) SDOF system excited by a step force p(t)=p 0, t 0: mu ku p 0 Particular solutio: Complemetary solutio: where A ad B are itegratio costats ad The complete solutio u ( ) p t Iitial coditios: for t=0 we have u(0) 0 ad u (0) 0 A p 0 0 k 0 B the eq. of motio p k 0 uc ( t) Acost B sit u( t) Acos t Bsi t p k m k p u( t) (1 cos t) k 0
Duhamel itegral Basis: represetatio of the dyamic excitatio as a sequece of ifiitesimal impulses Respose of a system excited by the force p(t) at time t sum of respose of all impulses up to that time 1 u( t) p( )si[ ( t )] d m Applicable oly to "at rest" iitial coditios u(0) 0 u (0) 0 Useful whe the force p(t) is defied aalytically t 0 is simple eough to evaluate aalytically the itegral
Duhamel itegral: example Equatio of motio of a udamped (c=0) SDOF system, excited by a ramp force p(t)=p 0, t 0: mu ku p 0 t t 1 p 0 cos ( t ) 0 m m 0 0 u( t) p si[ ( t )] d p 0 (1 cos t) k Equatio of motio u t p 0 ( ) (1 cos t) k
Udamped free vibratios Geeral form of the equatio of motio: Equatio of motio i the case of udamped free vibratios: mu ku Vibratios the system disturbed from the static equilibrium positio by iitial displacemet u(0) iitial velocity u(0) Classical solutio where 0 u(0) u( t) u(0)cost sit k m mu cu ku p( t)
Simple harmoic motio Udamped free vibratios u(0) u( t) u(0)cost sit
Udamped free vibratios Natural period of vibratio - T - time eeded for a udamped SDOF system to perform a complete cycle of free vibratios T Natural circular frequecy k m 2 Natural frequecy of vibratio f represets the umber of complete cycles performed by the system i oe secod 1 k m f T f 2 mass stiffess "Natural" - depeds oly o the properties of the SDOF system
Udamped free vibratios Alterative expressios for, f, T : g 1 g f T 2 st st 2 st g mg k st elastic deformatio of a SDOF system uder a static force equal to mg Amplitude: magitude of oscillatios u 0 2 2 u 0 u 0
Damped free vibratios Geeral form of the eq. of motio: Equatio of motio i the case of damped free vibratios: mu cu ku Dividig eq. by m we obtai with the otatios: Critical dampig coefficiet 0 k m mu cu ku p( t) u u u 2 2 0 2m ccr Dampig coefficiet c - a measure of the eergy dissipated i a complete cycle - critical dampig ratio: a o-dimesioal measure of dampig, which depeds o the stiffess ad mass as well c c c 2m 2 km cr 2k
Types of motio c=c cr or = 1 the system returs to the positio of equilibrium without oscillatio c>c cr or > 1 the system returs to the positio of equilibrium without oscillatio, but slower c<c cr or < 1 the system oscillates with respect to the equilibrium positio with progressively decreasig amplitudes
Types of motio c cr - the smallest value of the dampig coefficiet that completely prevets oscillatios Most egieerig structures - uderdamped (c<c cr ) Few reasos to study: critically damped systems (c=c cr ) overdamped systems (c>c cr )
Solutio of the eq. c<c cr or < 1: with the otatio: Uderdamped systems mu cu ku 0 for systems with u (0) (0) ( ) t u u t e u(0)cosdt sidt D 1 D 2
The effect of dampig i uderdamped systems Evelope of damped vibratios 2 u 0 u(0) D 0 u e t Lowerig of the circular frequecy from to D Legtheig of he period of vibratio form T to T D 2
Atteuatio of motio Ratio betwee displacemet at time t ad the oe after a period T D is idepedet of t: u( t) expt u( t TD ) 2 T Usig ad u( t) 2 exp u( t TD ) 1 D T D 2 2 T 1 u u i i1 exp 2 1 2
Natural logarithm of the ratio Atteuatio of motio u i exp u i1 2 1 2 is called logarithmic decremet ad is deoted with : l u u i i1 2 1 2 For small values of the dampig 2 2 1 1 Determiatio of logarithmic decremet based o peaks several cycles apart u1 u1 u u 2 u3 j j e 1 jl u 1 u1 j 2 u u u u u 1 j 2 3 4 1 j
Free vibratio tests Determiatio of dampig i structures: free vibratio tests 1 ui 1 u i l sau l 2 j u 2 j u i j i j
Refereces / additioal readig Ail Chopra, "Dyamics of Structures: Theory ad Applicatios to Earthquake Egieerig", Pretice-Hall, Upper Saddle River, New Jersey, 2001. Clough, R.W. şi Pezie, J. (2003). "Dyammics of structures", Third editio, Computers & Structures, Ic., Berkeley, USA
daiel.grecea@upt.ro http://steel.fsv.cvut.cz/suscos