C9 Desig for seismic ad climate chages Lecture 3: Dyamic respose of sigle-degree-of-freedom systems II Daiel Grecea, Politehica Uiversity of Timisoara 11/3/14 Europea Erasmus Mudus Master Course Sustaiable Costructios uder Natural Hazards ad Catastrophic Evets 511-1-11-1-CZ-ERA MUNDUS-EMMC
Europea Erasmus Mudus Master Course Sustaiable Costructios uder Natural Hazards ad Catastrophic Evets C9-L3 Dyamic respose of sigle-degree-offreedom systems II L3.1 Respose of SDOF systems to step, ramp ad harmoic forces. L3 Dyamic respose of sigle-degree-of-freedom systems II
Respose to step force Step force: p t p t Duhamel itegral p u( t) ust 1 cos 1 cos t k t T
Respose to step force Maximum displacemet (udamped system): The system vibrates with a period T about the static positio Effect of dampig: a smaller overshoot over the static respose a more rapid decay of motio u ust
Respose to ramp force t Ramp force p t p t tr Respose of a udamped system: t sit t T si t T u( t) ust ust tr tr T tr tr T The system vibrates with a period T about the static positio
Respose to step force with fiite rise time Force (ramp phase ad costat phase): p t tr t tr pt p t tr Respose of a udamped system: ramp phase costat phase t sit u( t) ust t t tr tr 1 u( t) u 1 si t si t t t t st r r tr r
Respose to step force with fiite rise time Ramp phase: system vibrates with a period T about the static positio Costat phase: idem u t r the system does ot vibrate for t>t r Small t r /T respose similar to the oe uder a step force Large t r /T respose similar to the static oe
Harmoic vibratios of udamped systems Harmoic force: amplitude p circular frequecy p( t) p si t or p( t) p cost
Harmoic vibratios of udamped systems Equatio of motio: Iitial coditios Particular solutio Complemetary solutio mu ku p si t u u() u u () p 1 u p ( t) sit k 1 u ( t) Acos t B si t c Complete solutios p 1 u( t) Acost Bsit sit k 1 Fial solutio u p / p 1 u( t) u cost si si t t k 1 k 1 trasiet respose steady-state respose
Harmoic vibratios of udamped systems. u() u () p / k o
Harmoic vibratios of udamped systems Steady-state respose: due to applied force; is ot iflueced by the iitial coditios Trasiet respose: depeds o iitial displacemet ad velocity, as well as properties of SDOF ad excitig force u p / p 1 u( t) u cost si si t t k 1 k 1 trasiet respose steady-state respose Neglectig dyamic respose static respose p u st t sit k u Steady-state respose: st p k 1 u( t) u st sit 1
Harmoic vibratios of udamped systems < displacemet u(t) ad excitig force p(t) have the same algebraic sig. Displacemet is i phase with the applied force. 1 u( t) u st sit 1 > displacemet u(t) ad excitig force p(t) have differet algebraic sigs. Displacemet is out of phase with the applied force.
Harmoic vibratios of udamped systems Steady-state respose: 1 u( t) u st sit 1 Alterative represetatio of steady-state respose: u( t) u si t u R si t st d R d u 1 ad ust 1
Displacemet respose factors Displacemet respose factor u R small < : amplitude of dyamic respose close to the static deformatio / >: amplitude of dyamic respose smaller the the static deformatio / 1: amplitude of dyamic respose much larger tha static deformatio Resoat frequecy - frequecy for which the respose factor R d is maximum (= ) d u st
Resoace Solutio for the equatio of motio whe = : particular solutio total solutio u p t p t cos t k 1 p u( t) t cos t si t k u() u ()
Harmoic vibratios of damped systems Equatio of motio mu cu ku p si t Iitial coditios u u() u u () Particular solutio C p k Complemetary solutio Complete solutio u ( t) C sit Dcost p 1 p D 1 k 1 1 D u ( t) e t Acos t Bsi t c D D u( t) e Acos t Bsi t C sit D cost t D D trasiet respose steady-state respose
Harmoic vibratios of damped systems u( t) e Acos t Bsi t C sit Dcost t D D trasiet respose steady-state respose..5 u() u () p / k o
Harmoic vibratios of damped systems: = For = respose of a damped SDOF system is: 1 u( t) u t st e cosdt sidt cost 1 1 ( ) 1 u t u t st e cos t.5 u() u ()
Harmoic vibratios of damped systems: = Small dampig: Larger amplitude More cycles to attaimet of a certai ratio of the steady-state respose u() u ()
Harmoic vibratios of damped systems: R d ad Steady-state respose ca be writte as: u t u si t u R si t st d Displacemet respose factor R d R d u u st 1 1 / / / 1 ta 1 /
Harmoic vibratios of damped systems: R d ad.
Harmoic vibratios of damped systems: R d ad 1 : amplitude of dyamic respose close to the static deformatio (R d 1) ad almost idepedet of dampig. Respose cotrolled by stiffess of the system. 1 : amplitude of dyamic respose approaches (R d ) ad almost idepedet of dampig. Respose cotrolled by mass of the system. 1 : amplitude of dyamic respose larger tha the static deformatio (R d max) ad sesible to dampig. Respose cotrolled by dampig of the system. u u u st p k p ust u u m st p c
Harmoic vibratios of damped systems: R d ad 1 : phase agle close to, displacemet i phase with the applied force. 1 : phase agle close to, displacemet out of phase with the applied force. 1 : phase agle equal to / for ay value of, displacemet maximum whe force equals.
Resoat frequecy: frequecy for which the maximum respose i terms of displacemet (or velocity or acceleratio) is obtaied Displacemet resoat frequecy: 1 Maximum respose: Resoace R 1 1 d
Differece betwee circular frequecies for which the displacemet respose factor is 1 times smaller tha the resoat respose Half-power badwidth b a
Dampig for egieerig structures stress level structural type (%) stress level below.5 times the yield stregth welded steel structures, prestressed cocrete structures, strogly reiforced cocrete structures (limited cracks) -3 reiforced cocrete structures with sigificat crackig 3-5 steel structures with bolted or riveted coectios, wood structures coected with screws or ails 5-7 welded steel structures, prestressed cocrete structures (without total loss of prestress) 5-7 stresses close to the yield stregth prestressed cocrete structures with total loss of prestress 7-1 reiforced cocrete structures 7-1 steel structures with bolted or riveted coectios, wood structures coected with screws 1-15 wood structures coected with ails 15-
Refereces / additioal readig Ail Chopra, "Dyamics of Structures: Theory ad Applicatios to Earthquake Egieerig", Pretice-Hall, Upper Saddle River, New Jersey, 1. Clough, R.W. şi Pezie, J. (3). "Dyammics of structures", Third editio, Computers & Structures, Ic., Berkeley, USA
daiel.grecea@upt.ro http://steel.fsv.cvut.cz/suscos