2012/10/25 Chapter Dynamic Response of MDOF System 3.1 Introduction 4 1 Introduction 3.2 Free Vibration of SDOF Systems 4.2 Modal Analysis of MDOF Systems AA 3.7Response of Nonlinear SDOF Systems ( MDOF Systems Response to SDoF☐ Base Excitation Single-degree-of-freedom oscillators 自→2 DOF] OF SYSTEMS Mulli-degree-of-free dom system y g n degrees of freedom will hav ca1espeeeCanteewenemsondu 且 na 窗目目目 1
2012/10/25 1 Chapter 3 Dynamic Response of Structure 3.1 Introduction 3.2 Free Vibration of SDOF Systems 3.3 Impulsive Vibration of SDOF Systems 3.4 Harmonic Vibration of SDOF Systems 3.5 Response of Linear SDOF Systems 3.6 Spectrum of Seismic Response of SDOF 3.7 Response of Nonlinear SDOF Systems （*） No. 1 Chapter 4 Dynamic Response of MDOF System 4.1 Introduction 4.2 Modal Analysis of MDOF Systems 4.3 Mode Superposition Spectrum Analysis of MDOF Systems 4.4 No. 2 Instructor: Dr. C.E. Ventura No. 3 Instructor: Dr. C.E. Ventura No. 4 time, sec 0.0 0.2 0.4 0.6 0.8 Vibration Period T W g K 2 SDOF • Single-degree-of-freedom oscillators W K T Instructor: Dr. C.E. Ventura No. 5 FirstMode Shape ThirdMode Shape SecondMode Shape • Linear response can be viewed in terms of individual modal responses. MDOF Idealized Model Actual Building • Multi-story buildings can be idealized and analyzed as multi-degree-of-freedom systems. Instructor: Dr. C.E. Ventura No. 6 MDOF SYSTEMS • Multi-degree-of-freedom systems – Systems having n degrees of freedom will have: • n modes of vibration; • n natural frequencies. – Response will, in general, be some linear combination of response in these modes – Common approach is to treat the n-DOF system as n SDOF systems instead
2012/10/25 Natural Vibration Period Fundamental Vibration Period for Buildings 40 DOF Systems Example-3-storey building 4.2 Modal Analysis of MDOF Systems Newton's Second Law states: ndnaorsrngd .Axial deformations of the beam and columns neglected e the the floor levels en6-明明g+g0 nfo:8al份}份}-a
2012/10/25 2 Instructor: Dr. C.E. Ventura No. 7 Natural Vibration Period Every body has a vibration natural period that depends on its mass and stiffness. If the body has several degrees of freedom, it has many vibration periods, and the larger period is called the Fundamental Period of Vibration Instructor: Dr. C.E. Ventura No. 8 Fundamental Vibration Period for Buildings Instructor: Dr. C.E. Ventura 4.2 Modal Analysis of MDOF Systems No. 9 Instructor: Dr. C.E. Ventura No. 10 MDOF Systems • Example – 3-storey building Instructor: Dr. C.E. Ventura No. 11 Formulation of the Equation of Motion and Selection of the Dynamic Degrees of Freedom Idealized : • Beams and floors rigid • Axial deformations of the beams and columns neglected • Axial load on columns neglected • Concentrate the mass at the floor levels Instructor: Dr. C.E. Ventura No. 12 Formulation of the Equation of Motion and Selection of the Dynamic Degrees of Freedom •Newton’s Second Law states: If a force acts upon a body, the rate of change of the quantity of motion of the body (momentum = m*v) is equal to the applied force. pj - fSj - fDj = mjüj or mjüj + fDj + fSj = pj (t) ( ) ( ) 0 2 1 0 2 1 2 1 2 1 2 1 p t p t f f f f u u m m S S D D •For our Example: •In matrix form: mu dt dv f m
2012/10/25 …与Σ要 +For Damping fn=c盒，+G(位-，) ·石=+0 fm=c(,-) 。·=k1u1+k(u-u2） ·=kuu) 中份-好的 in matrix form: 份生 Equation of Motion: mutcu+ku=p(r) System:Translational Ground Motion 0n .ForN c+ku=-mlo( with ++6-0 ma+c0+ku =p(r) system: faaotou eneral Overview:MDOF Discrete systems dal Analysis Procedres of odal analysis Solutions to the equations of motions This method is applicable when: cally damped Response spectramethod 3
2012/10/25 3 Instructor: Dr. C.E. Ventura No. 13 Formulation of the Equation of Motion and Selection of the Dynamic Degrees of Freedom • fs1 = fs1 b + fs1 a • fs1 = k1u1 + k2 (u1 -u2 ) • fs2 = k2 (u2 -u1 ) columns c j h EI k 3 12 2 1 2 2 1 2 2 2 1 u u k k k k k f f s s •In matrix form: •Story Stiffness = Instructor: Dr. C.E. Ventura No. 14 Formulation of the Equation of Motion and Selection of the Dynamic Degrees of Freedom ( ) 1 1 1 2 1 2 f c u c u u D ( ) 2 2 u2 u1 f c D 2 1 2 2 1 2 2 2 1 u u c c c c c f f D D Equation of Motion: •In matrix form: •For Damping: m ( ) .. . u cu ku p t Instructor: Dr. C.E. Ventura No. 15 Planar Systems: Translational Ground Motion •uj t (t) = uj (t) + ug (t) •u t (t) = u(t) + ug (t)1 •Displacements related by: •For N masses: •Equation of Dynamic Equilibrium fI + fD + fS = 0 •For a linear system: fS = ku fD = ců •Inertia forces related to total acceleration of mass fI = müt •1= vector of order N with each element = 1 Instructor: Dr. C.E. Ventura No. 16 Planar Systems: Translational Ground Motion The new equations of motion have now become: mü+ ců + ku = -m1üg (t) Comparing: mü+ ců + ku = -m1üg (t) with mü+ ců + ku = p(t) The ground motion can be replaced by the effective earthquake forces: peff(t) = -m1üg (t) Instructor: Dr. C.E. Ventura No. 17 General Overview: MDOF Discrete systems • Procedures to formulate equations of motions • Solutions to the equations of motions – Numerical solution of eigenproblems – Modal superposition method • Classically damped systems • Nonclassically damped systems – Solution in the frequency domain – Rayleigh-Ritz method – Direct integration of equation of motion – Response spectra method Instructor: Dr. C.E. Ventura No. 18 Modal Analysis • Dynamic analysis using modal superposition method for classically damped systems also called: – Modal analysis – Classical modal analysis – Classical mode superposition method – Classical mode displacement superposition method • This method is applicable when: – The system is classically damped – The system is linear
2012/10/25 Modal Analyais dal Analysis Overview of the method mii+ci+ku=plt) .The re se in each mode can be c 0=立a.0=pq Substitute u(t)in equation of motion: 立m.0+立c40+4g.0= 之'ma0+立'e.o)+立4.e)=0 Modal Analysis Modal Analysis .Because of orthogonality of modes We define: -Generalined modal mass: M,=m响 Generalized modal stiffness: K.■kp Since the system is elassically damped c=02r Modal Analysis Modal Analysis beunoupdde .Divide the uncoupled equations by M..+C.A.+K.9.-P.0) Or in matrix form: Mu+Ca+Ku=P( Psdiagomalmtrofrthegerenledmoalarce
2012/10/25 4 Instructor: Dr. C.E. Ventura No. 19 Modal Analysis • Overview of the method: – The equations of motions, when transformed to modal coordinates, become uncoupled. – The response in each mode can be computed independently of the other modes by solving an SDOF system with the vibration properties of that mode. – Modal responses are combined to obtain the total response. Instructor: Dr. C.E. Ventura No. 20 Modal Analysis • Equation of motion: • Transfer to modal equations: – Displacements in terms of modal coordinates: – Substitute u(t) in equation of motion: – Premultiplying each term by : mu cu ku p(t) ( ) ( ) ( ) 1 t q t t r N r u r Φq ( ) ( ) ( ) ( ) 1 1 1 q t q t q t t r N r r r N r r r N r m r c k p ( ) ( ) ( ) ( ) 1 1 1 q t q t q t t r N r r T r n N r r T r n N r r T n m c k p T n Instructor: Dr. C.E. Ventura No. 21 Modal Analysis • Because of orthogonality of modes: • Since the system is classically damped: 0 0 r T n r T n k m r 0 T n c n r n r n r Instructor: Dr. C.E. Ventura No. 24 Modal Analysis • We define: – Generalized modal mass: – Generalized modal stiffness: – Generalized modal damping: – Generalized modal forces: n T Mn n m n T Kn n k n T Cn n c P (t) (t) T n n p Instructor: Dr. C.E. Ventura No. 25 Modal Analysis • The modal equations will be uncoupled and reduces to: • Or in matrix form: M is the diagonal matrix of the generalized modal mass K is the diagonal matrix of the generalized modal stiffness C is diagonal matrix of the generalized modal damping P(t) is diagonal matrix of the generalized modal forces M q C q K q P (t) n n n n n n n Mu Cu Ku P(t) Instructor: Dr. C.E. Ventura No. 26 Modal Analysis • Divide the uncoupled equations by : • Solve the above equation for the modal coordinate similar to an SDF system. Mn n n n n n n n n M P t q q q ( ) 2 2 q (t) n
2012/10/25 Modal Analyeis Modal Analysis Element forees: is:u(r) m.0) 0=立a.=立a.0 0=2x.0 Modal Analysis:Summary (cont'd) d the stiffnes Compute n-modal displacements) l displacements r .Determine the natural frequenciesand mode shapes Combine the contributions of all the modes to determine the total response. solving for modal coordinates( itl-Story Structures moda 4.3 The Response Spectrum Method of Analysis for MDOF Systems 5
2012/10/25 5 Instructor: Dr. C.E. Ventura No. 27 Modal Analysis • Displacements: – Contribution of the nth mode to the displacement is: – Combining these modal contributions gives the total displacement: N n N n n n n t u t q t 1 1 u( ) ( ) ( ) (t) q (t) un n n u(t) Instructor: Dr. C.E. Ventura No. 28 Modal Analysis • Element forces: – Determine the contributions of the individual modes to the element force : • From modal displacements using element stiffness properties or, • By static analysis of the structure subjected to the equivalent static forces associated with the nth-mode response defined as: – The total element force considering contributions of all modes is: r (t) n (t) un ( ) ( ) ( ) 2 t t q t n kun mn n f N n n r t r t 1 ( ) ( ) Instructor: Dr. C.E. Ventura No. 29 Modal Analysis: Summary • Define the structural properties: – Determine the mass matrix m and the stiffness matrix k. – Determine the modal damping ratios • Determine the natural frequencies and mode shapes . • Compute the response in each mode by uncoupling the equations of motion and solving for modal coordinates . n n q (t) n Instructor: Dr. C.E. Ventura No. 30 Modal Analysis: Summary (cont’d) • Compute n-modal displacements un (t) . • Compute the element forces associated with the n-modal displacements rn (t) . • Combine the contributions of all the modes to determine the total response. Instructor: Dr. C.E. Ventura No. 31 • Individual modal responses can be analyzed separately. Reference: A. K. Chopra, Dynamics of Structures: A Primer, Earthquake Engineering Research Institute • For typical low-rise and moderate-rise construction, first-mode dominates displacement response. • Total response is a combination of individual modes. Total Roof Displ. Resp. Multi-Story Structures Instructor: Dr. C.E. Ventura 4.3 The Response Spectrum Method of Analysis for MDOF Systems No. 32