The Hidden Beauty of Structural Dynamics

Transcription

1 BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS DEPARTMENT OF STRUCTURAL MECHANICS Róbert K. NÉMETH Attia KOCSIS The Hidden Beauty of Structura Dynamics Budapest, 213 ISBN

2 CONTENTS Contents Contents List of figures List of tabes Preface i iv v vii 1 Dynamics of singe- and muti-dof systems Vibration of SDOF systems Derivation of the equation of motion Genera soution of the homogeneous ODE Particuar soution of the non-homogeneous ODE with harmonic forcing Support vibration of SDOF systems Genera forcing of SDOF systems Duhame s integra Numerica soution of the differentia equation Vibration of MDOF systems Equation of motion of MDOF systems Free vibration of MDOF systems Harmonic forcing of MDOF systems (direct soution and moda anaysis) Approximate soution of the generaized eigenvaue probem (Ritz- Rayeigh s method) Summation theorems Dunkerey theorem Southwe theorem Föpp Papkovich theorem Dynamics of sender continua Longitudina vibration of prismatic bars Differentia equation of motion Free ongitudina vibration Free torsiona vibration of prismatic shafts Shear vibration of prismatic beams i

3 CONTENTS Differentia equation of motion Transverse vibration of prismatic beams The equation of transverse vibration Free vibration of prismatic beams Forced vibration of prismatic beams Dynamics of panar frame structures Statica matrix dispacement method Noda decomposition of panar frames Goba and oca reference systems, transformations Eementary statica stiffness matrix in the oca reference system Transformation of the eementary stiffness matrix Compiation of the tota stiffness matrix Boundary conditions Dynamica stiffness matrix of frame structures Diagonay umped mass matrix Dynamica stiffness matrix Consistent mass matrix Accuracy with the consistent mass matrix Additiona masses Equivaent dynamica noda oads Support vibration of MDOF systems Prescribed motion of DOFs Harmonic support vibration Support motion due to earthquake Rea moda anaysis, interna forces Rea moda anaysis Interna forces Partia soution of the generaized eigenvaue probem Second order effects Rotationa inertia Static norma force Damping in structura dynamics Steady-state vibration of viscousy damped systems Harmonic excitation of damped MDOF systems Proportiona damping The Kevin-Voigt materia Stiffness of a damped beam made of Kevin-Voigt materia Rea moda anaysis of proportionay damped systems Rate-independent damping Rea moda anaysis in case of rate-independent damping Direct soution of rate-independenty damped systems Equivaent rate-independent damping ii

4 CONTENTS Quasi-moda anaysis for equivaent rate-independent damping Damping effects of soi Differentia equation of the equivaent soi bar Numerica soution of the matrix differentia equation Newmark method Earthquake anaysis Introduction to earthquakes Response spectrum of SDOF systems Response functions Response spectrum Design spectrum Response spectrum of MDOF systems Moda anaysis Earthquake anaysis Ineastic response of structure Time history anaysis A Derivations 18 A.1 Static shape functions A.2 Stiffness matrices of beam members A.3 Eementary dynamica stiffness matrix using purey dynamica shape functions 182 A.4 Consistent mass matrices of beam members A.5 Few trigonometric identities A.6 Damped SDOF system soved with a different approach A.6.1 Sine A.6.2 Cosine A.6.3 Sine and cosine A.6.4 Quasi-periodic oading A.7 Damped MDOF systems soved using compex agebra A.7.1 Inverse of a compex square matrix A.7.2 Appication forced MDOF systems A.8 Fourier series and the Fourier Transform A.8.1 Fourier series iii

5 LIST OF FIGURES List of Figures 1.1 Common exampes of singe-degree-of-freedom structures A mass-spring-damper mode Typica time-dispacement diagrams of free vibration of a damped, eastic supported SDOF system Responses of a damped SDOF system to a harmonic excitation Support vibration of an undamped mechanica system Response factor of the eongation of the spring as a function of the ratio of the forcing and natura frequencies due to a harmonic support vibration Genera time-dependent forcing Expanation of the Cauchy-Euer method and the second order Runge-Kutta method Expanation of the finite difference approximation of veocity and acceeration using secant ines Exampes of muti-degree-of-freedom structures Free body diagrams of the mode shown in Figure 1.1 (c) Two-storey frame structure with rigid foors Vibration of a three-storey frame structure with rigid interstorey girders and fexibe coumns Mode shapes of the three-storey structure of Fig corresponding to the natura circuar frequencies Mode of a 1-storey frame structure with rigid interstorey girders and eastic coumns Rayeigh quotient of Probem A camped rod with its mass concentrated at three points of equa distances Mode of a rigid roof supported by a inear spring and by two camped, massess rods of equa ength A straight, massess rod carrying a umped mass at its free top end, connected to a fixed hinge and a rotationa spring at the bottom Sketch of (a) a prismatic bar subjected to a ongitudina distributed oadq n (x,t) Rod with fixed-free ends with an initia dispacement and the traveing waves Sketch of an inextensiona, unbendabe prismatic beam subjected to a transverse, distributed oadq t (x,t) iv

6 LIST OF FIGURES 2.4 Mode of an inextensiona and unbendabe beam which is easticay camped at one end and free at the other end Sketch of a prismatic beam subjected to a transverse, distributed oad q t (x,t) Common types of supporting modes Sketch of a prismatic beam subjected to an axia compressive force Prismatic beam subjected to a harmonic exciting force F sin(ωt) Comparing discrete modes of beam structures Loca and goba reference systems Mode of a frame structure Sketch of the deformed shape of a fixed-fixed beam ij due to a unit transation of end i aong axis y, and the corresponding bending moment diagram Sketch of the deformed shape of a fixed-fixed beam ij due to a unit rotation of endi, and the corresponding bending moment diagram Easticay supported node Sketch of the deformed shape of beam ij due to a harmonic transation of unit ampitude of endiaong axis y, and the corresponding bending moment diagram Sketch of a simpe panar frame and the mechanica mode for the matrix dispacement method The change of the bock structure of the matrix equation of motion during the eimination of the prescribed motion of supports Sketch of a fixed-fixed beam and the mechanica mode for the matrix dispacement method (fixed support mode) Sketch of a fixed-fixed beam and the mechanica mode for the matrix dispacement method (spring mode) Sketch of the deformed shape of beam ij due to a harmonic transation of unit ampitude of endiaong axis y, and the corresponding bending moment diagram Demonstration of the moment caused by the norma force S on a rotated eementary segment of the beam Sketch of the deformed shape of a damped beam and the bending moment diagram due to a dynamica vibration of endiaong axis y Assumed stress propagation in the equivaent soi bar Soid and surface waves Response functions Concept of response spectrum The function β(t ) of the pseudo-acceeration response vs. the natura period Force-deformation diagram of the inear eastic-pastic materia mode and structure Typica enveope functions of artificia earthquake records A.1 The first four partia sums of the Fourier series for a square wave. Source: Wikimedia Commons v

7 LIST OF TABLES List of Tabes 1.1 Harmonic forcing of a three-storey structure. Moda oads, coefficients of resonance, and other parameters The first few natura circuar frequencies of the frame and a the six natura circuar frequencies of the approximate modes (with the consistent the diagonay umped mass matrices). The dimension of the frequencies is rad/s The first six natura circuar frequencies of the frame, the projections of the oad vector to the moda shape vectors, and the moda participation factors The natura circuar frequencies of the beam with eastic supports for various spring stiffnesses vi

8 PREFACE Preface This ecture notes of the MSc course Structura Dynamics is devoted for the civi engineering students of the Budapest University of Technoogy and Economics. The objective of the course is to introduce the basic concepts of the dynamica anaysis of engineering structures. The topics that are covered in this course are equations of motion of singe- and muti-degreeof-freedom systems, free and steady-state vibrations, anaytica and numerica soution techniques, and earthquake oads on structures. Both continuum and discrete mechanica systems are considered. In civi engineering practice structures are aimed to be in equiibrium. However, due to continuous disturbances (effects of wind, heat, traffic, movement of the foundations, etc.), the structures undergo vibrations. Some of these motions are sow, aowing us to treat them as a quasi-static kinematic oad, and to negect the inertia effect of the mass of the structure. But some of them happen fast enough to exert a significant dynamica impact on the structure. Many of these cases are sti handed as a quasi-static oad with a proper dynamica factor, but other cases reay require the engineers to accompish dynamica anaysis. The goa of the semester is to prepare our students for these tasks. Dynamics pay an important roe in many fieds of structura engineering. Earthquakes, fast moving trains on bridges, urban traffic generated or machine induced vibrations, etc. Modern materias enabe the fabrication of ighter, more fexibe structures, where the effects of vibrations can be significanty high. Additionay, investment companies desire cost effective structures, which aso tends the engineers towards more accurate computations, which impies dynamica anaysis, too. Not ony theory is given in this notes, but there are aso many probems soved. The authors hope that these exampes hep our students to comprehend a the introduced concepts. In these probems the cacuations are done foowing the care your units approach. It means that we use a consistent system of units, which does not require us to carry the units during the operations. Every number is substituted in the formuae in a common system of units, in SI (Internationa System of Units), hence the resuts are aso obtained in SI. We offer these notes to our readers under a Creative Commons Attribution- NonCommercia-NoDerivs 3. Unported License, in the hope it wi hep them understand the basics of structura dynamics. Pease fee free to share your thoughts about it with us. Budapest, 29 th August, 213 vii

9 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Chapter 1 Dynamics of singe- and muti-degree-of-freedom systems In this chapter first we repeat the basics of vibration of eastic structures. The motion of continuous structures is often approximated by the dispacements of some of its points. In these modes the mass of the structure is concentrated into discrete points. The concentrated masses are assumed to be rigid bodies, and the easticity and the viscoeasticity of the structure is modeed by massess springs and damping eements, respectivey. These modes are caed mass-spring-damper systems. We introduce the degree-of-freedom (DOF) as the number of independent variabes required to define the dispaced positions of a the masses. If there is ony one mass, with one direction of dispacement, then we tak about a singe-degree-of-freedom (SDOF) system. If there are more than one masses, or one mass with more than one directions of dispacement, then we have a muti-degree-of-freedom (MDOF) system. If we try to describe the deformed shape of a continuous structure with the dispacements of a of (infinitey many of) its points, then we use a continuum approach, where there are infinitey many degrees of freedom. In Section 1.1 we start with the free vibration of SDOF systems, then harmonic forced vibration of SDOF systems, and support vibration of SDOF systems are discussed. Then SDOF systems excited by a genera force are studied in Section 1.2. Section 1.3 is devoted for the free vibration of MDOF systems. We aso present an approximate method capabe to sove the generaized eigenvaue probem occurring in the anaysis of MDOF systems. At the end of the chapter, in Section 1.4 we present a few summation theorems usefu to approximate the first natura frequency of a structure. 1.1 Vibration of singe-degree-of-freedom systems Civi engineering structures are intended in genera to be in equiibrium. Despite the common requirements, many of the oading situations resut in motion of the structures. The most simpe motion occurs when we can describe it by one singe space variabe. Exampes for these type of dynamica systems are horizonta girders with a significant mass (e.g. a machine, where the mass of the girder can be negected with respect to the mass of the 1

10 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS machine) (Figure 1.1 (a-b)), frame structures with significant mass on the rooftop (Figure 1.1 (c)), chimneys and water towers (Figure 1.1 (d)), etc.. Figure 1.1: Common exampes of singe-degree-of-freedom structures: (a) fixed girder, (b) hinged-hinged girder, (c) singe frame with mass on rooftop, (d) chimney or water-tower, and (e) common mechanica mode. The common in the above exampes is that any dispacement from the equiibrium state resuts a force puing the DOF back to the initia state. The simpest mechanica mode of this behaviour is the materia partice (umped mass) connected by a inear spring to a rigid wa (Figure 1.1 (e)) Derivation of the equation of motion If we anayse the motion of a structure caused by a sma disturbance, then we can see that in the absence of externa forcing the ampitude of the vibration around the origina state decreases with the time. This is caused by interna friction in the materia and at the connections. Effect of externa dampers can be considered as we. The mathematicay easiest way to dea with damping is the viscous damping. (In this case the damping force is proportiona to the veocity.) The mechanica mode of the viscous damping is a dashpot. Figure 1.2 (a) shows a damped, easticay supported system with a dashpot of damping coefficient c, a inear eastic spring of stiffness k, and a time dependent exciting force F(t). Our goa is in genera one of the foowings: to find the dispacement function as a function of time to find the eongation of the spring as a function of time to find the force in the spring or in the dashpot as a function of time to find the possibe maxima of the above functions 2

11 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Figure 1.2: (a) A mass-spring-damper mode: a umped massmis connected to a support through a massess inear springk and a massess viscous damper c. The mass excited by the time dependent force F(t) undergoing a singe-degree-of-freedom vibration. (b) Free body diagram of the mass-spring-damper mode. The free body diagram (FBD) of the massmcan be seen in Figure 1.2 (b). Newton s second aw of motion can be written for the body: F(t) f s (t) f d (t) = ma(t), (1.1) where F(t) is the externa force, f s (t) is the eastic force from the massess spring, f d (t) is the damping force from the massess dashpot, m is the mass and a(t) is the acceeration. Assuming a inear spring f s (t) = ku(t), where k is the spring stiffness, and u(t) is the eongation of the spring. Assuming a viscous damping f d (t) = c u(t), where c is the damping coefficient, and u(t) is the derivative of the eongation u(t) with respect to time (i.e. it is the eongation-veocity). (The dot over a variabe denotes differentiation with respect to time.) The acceeration a(t) is the second derivative of the dispacement of the body with respect to time: a(t) = ẍ(t). So the equation of motion is: F(t) ku(t) c u(t) = mẍ(t). (1.2) (Note: in many textbook authors write a so caed kinetic equiibrium equation using the principe of d Aembert with an inertia forcef I = ma(t). Then, Eq. (1.1) woud have the form: F(t) f s (t) f d (t) + f I (t) =. In forma cacuation it eads to the same resut, but during cacuations by hand the correct interpretation of the minus sign in the definition of f I requires a deep understanding of the concept, at which eve writing the cassic formua makes no probem. Because of that we wi avoid writing kinetic equiibrium equations. ) In most cases we are interested in the interna deformations and the corresponding interna forces of the structures. These are represented in this mode by the eongation of the spring, so we have to write the dispacement of the body as a function of eongation. If the support is fixed, then these two vaues are equa (x(t) = u(t)) and the same appies to their derivatives (ẍ(t) = ü(t)). Substituting these into Eq. (1.2) we get: mü(t)+c u(t)+ku(t) = F(t). (1.3) This non-homogeneous, inear, second order ordinary differentia equation of constant coefficients describes the motion of the forced vibration of the damped SDOF-system. 3

12 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS For the soution of the differentia equation (1.3) we introduce its compementary differentia equation: mü(t)+c u(t)+ku(t) =. (1.4) which is a homogeneous differentia equation. The compete soution of Eq. (1.3) can be written in the form: u(t) = u (t)+u f (t), where u (t) is the soution of the compementary equation (the index refers to the right hand side of the homogeneous equation), whie u f (t) is a particuar soution of the origina, nonhomogeneous equation (the index f refers to the forcing). If initia conditions are given (e.g. the dispacement and the veocity at a given time), then they must be fufied for the sum ofu (t) andu f (t) with the free parameters occurring inu (t) Genera soution of the homogeneous ODE Eq. (1.4) describes the free vibration of the mechanica system. Since it is a inear, homogeneous ODE with constant coefficients, the soution can be obtained with an ansatz function u(t) = e λt, which is substituted back in Eq. (1.4) aongside with is derivatives. The resut is the quadratic poynomia equation The roots of the above equation are: mλ 2 +cλ+k =. (1.5) λ 1,2 = c± c 2 4mk. (1.6) 2m These roots might be either rea or compex vaued, depending on the ratio of the system parameters. If c 2 km, the discriminant in Eq. (1.6) is non-negative, thus both λ 1,2 are negative rea numbers, and the soution of Eq. (1.4) is the sum of two exponentia function asymptoticay approaching zero. (Figure 1.3 (a) shows some typica graphs of this vibration.) We ca this damping as heavy damping, the system is an overdamped system. The imit vaue 2 km is the critica damping c cr. If c < 2 km (or c < c cr ), the discriminant is negative, the soution of Eq. (1.5) is a conjugate pair of compex numbers. Using Euer s formua (e ix = cosx + isinx) the soution of Eq. (1.4) can be rewritten in the form: u (t) = e ξω t (Acos(ω t)+bsin(ω t)), (1.7) where is the reative damping coefficient ξ = c 2 km = c c cr ω = ω 1 ξ 2 4

13 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS is the natura circuar frequency of the (under)damped system, ω = k/m is the natura circuar frequency of the undamped system with the same mass and stiffness. The parameters A and B are two free parameters depending on the initia conditions. (Figure 1.3 (b) shows some typica graphs of this vibration.) We ca this case as underdamped system. The soution Eq. (1.7) is a harmonic term (Acos(ω t)+bsin(ω t)) mutipied by an exponentia term ( e ξω t ). The atter one indicates an exponentia decay in the osciatory motion of the body, which can be seen as an exponentia enveope of the osciating harmonic function in Figure 1.3 (b). A higher eve of damping has two effect on the motion. First, the exponentia decay wi be more significant, second, the damped natura circuar frequency wi be ower. Figure 1.3: Typica time-dispacement diagrams of free vibration of a damped, eastic supported SDOF system. (a) Overdamped system, no vibration. (b) Underdamped system: harmonic osciation with the ampitude decaying exponentiay. There are further quantities in use, to describe the vibration of a SDOF system. Natura cycic frequency f is the number of tota osciations done by the body in a unit time: f = ω /(2π). The natura period T is the time required to make a fu cyce of vibration, i.e. T = 1/f = 2π/ω. Both of the above vaues can be written for the damped system as we, caed the damped natura cycic frequency f D and the damped natura period T D. They are interreated to each other with: f D = ω /(2π) and T D = 1/f D = 2π/ω. Logarithmic decrement Let us anayse the dispacements of a mass during its damped free vibration. We have seen, that at a given time instant t the dispacement is (Eq. (1.7)): u (t) = e ξω t (Acos(ω t)+bsin(ω t)). 5

14 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Using the previousy introduced damped natura periodt D, we can write the dispacement after a whoe period of motion as we: u (t+t D ) = e ξω (t+t D ) (Acos(ω (t+t D ))+Bsin(ω (t+t D ))). The ratio of the dispacements can be written as: u (t) u (t+t D ) = e ξωt (Acos(ω t)+b(sinω t)) e ξω (t+t D) (Acos(ω (t+t D ))+Bsin(ω (t+t D ))). SinceT D is the damped period of the motion, the harmonic terms in both time instants have the same vaue, so we can simpify the above formua as ( ) u (t) u (t+t D ) = eξω T D 2ξπ/ 1 ξ = e 2. (1.8) This ratio is constant, and depends ony on the dampingξ. Since we did not have any constraint on t, Eq. (1.8) hods for any two dispacements measured in a time distance T D. In practice, the natura ogarithm of Eq. (1.8) is used for the measurement of damping ϑ = n u (t) u (t+t D ) = 2ξπ/ 1 ξ 2. Here ϑ is caed the ogarithmic decrement which is a system property. In typica engineering structures ξ 1, so the 1 ξ 2 1 approximation can be used: ϑ = n Free vibration of undamped systems u (t) u (t+t D ) 2ξπ. (1.9) The vibration of undamped systems can be derived in a simiar way as we did it for the damped system, or we can anayse our damped resuts in the imit c. According to Eq. (1.3) the differentia equation of motion can be written as: mü(t)+ku(t) = F(t). The compementary equation describes the undamped free vibration: mü(t)+ku(t) =. The soution of the free vibration is directy obtained from Eq. (1.7) atc = (andξ = ): u (t) = Acos(ω t)+bsin(ω t). Here ω = k/m is the natura circuar frequency of the undamped system. The parameters A and B can be cacuated from the initia conditions. The purey harmonic motion can be rewritten into the form: u (t) = Csin(ω t+ϕ), with the ampitude of the motion C = A 2 +B 2 and the phase ange ϕ = arctan A B. 6

15 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Particuar soution of the non-homogeneous ODE with harmonic forcing A simpe exampe for a harmonic excitation force is a rigid body (e.g. a machine) rotating with a constant anguar veocity ω around an axis which is not going through its center of gravity (COG). The distance between the axis and the center of gravity is caed the eccentricity and denoted byr C.) The COG of the body undergoes a panar motion on a circuar path with an anguar veocity ω. From kinematics of rigid bodies the acceeration of the COG equas a n = mω 2 r C, its direction varies with the motion, its component parae with an arbitrary chosen, but fixed direction can be written as a harmonic function of time, and the same appies for the net force acting on the rigid body. The opposite of this force acts on the axis of rotation, resuting in a harmonic excitation force on the oad bearing structure. (The orthogona component of the force shoud be taken into account as we, but the vibration can be prevented by structura constraints, or by appying two we-tuned body rotating in the opposite direction.) Without oss of generaity (for harmonic functions one can transate the time scae to have any other harmonic function with the same frequency and ampitude), we wi write the harmonic excitation force in the form: F(t) = F sin(ωt). HereF is the ampitude of the force, andω is the circuar frequency of the forcing. Substituting this forcing in the right hand side of (1.3) yieds: mü(t)+c u(t)+ku(t) = F sin(ωt). (1.1) To sove Eq. (1.1) we assume that the particuar soution is of the form: u f (t) = u f sin(ωt ϕ), i.e. it is a harmonic function with the same frequency as the forcing, but with a phase shift of ϕ. We substitute our ansatz into Eq. (1.1): mω 2 u f sin(ωt ϕ)+cωu f cos(ωt ϕ)+ku f sin(ωt ϕ) = F sin(ωt). We appy trigonometrica identities for the sums in the sine and cosine functions: mω 2 u f sin(ωt)cos( ϕ) mω 2 u f cos(ωt)sin( ϕ)+cωu f cos(ωt)cos( ϕ) cωu f sin(ωt)sin( ϕ)+ku f sin(ωt)cos( ϕ)+ku f cos(ωt)sin( ϕ) = F sin(ωt). Now we separate the sinusodia and cosinusoida parts: u f cos(ωt) ( mω 2 sinϕ+cωcosϕ ksinϕ ) +u f sin(ωt) ( mω 2 cosϕ+cωsinϕ+kcosϕ ) = F sin(ωt). This equation must hod for any time t. 7

16 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS When sin(ωt) =, then cos(ωt), so must hod, which is true, when mω 2 sinϕ+cωcosϕ ksinϕ = cotϕ = k mω2 cω = m c ω 2 ω 2, (1.11) ω with ϕ π. (See Figure 1.4 (a) for the dependence of phase ange on the ratio of the forcing and natura frequency.) When cos(ωt) =, then sin(ωt), so must hod. u f ( mω 2 cosϕ+cωsinϕ+kcosϕ ) = F We use the identities cosϕ = cotϕ/ 1+cot 2 ϕ andsinϕ = 1/ 1+cot 2 ϕ to get u f mω 2 cotϕ+cω +kcotϕ 1+cot 2 ϕ = F and sove the above equation for u f using Eq. (1.11): u f = F 1+ (k mω2 ) 2 c 2 ω 2 (k mω 2 ) k mω2 +cω. cω Mutipying both the nominator and the denominator with cω eads to 1 u f = F = F (k mω 2 ) 2 +c 2 ω 2 k 1 (1 m ω2) 2 k + c 2 ω k 2 2 Using the natura circuar frequency and the fraction of critica damping coefficient (ω = k/m,ξ = c/(2 km)) the soution for uf is u f = F k 1 ( ). (1.12) 2 1 ω2 +4ξ 2ω 2 ω 2 ω 2 From the above resuts the particuar soution of the differentia equation (1.1) of the harmonicay forced vibration is: u f (t) = F ω2 1 1 ω k ( ) sin ωt arccot ξ ω (1.13) 1 ω2 +4ξ 2ω 2 ω ω 2 ω 2 8

17 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS The compete soution of Eq. (1.1) is the sum of Eq. (1.13) and (1.7): u(t) = F ω2 1 1 ω k ( ) sin ωt arccot 2 2 2ξ ω 1 ω2 +4ξ 2ω 2 ω ω 2 +e ξω t (Acos(ω t)+bsin(ω t)). ω 2 (1.14) The second part of Eq. (1.14) becomes very sma after a sufficienty ong time for any sma damping. That part is caed the transient vibration. The first part, which is equivaent to the particuar soution Eq. (1.13), is caed the steady-state soution of the probem. Since the transient vibration decays exponentiay with time, on a ong time scae the steady-state vibration determines the dynamics. Usuay we are not interested in the phase of the motion, but in the ampitude of the vibrationu f, given by Eq. (1.12). In that formua the quotientf /k can be regarded as the static dispacement under a static force F (which is the ampitude of the harmonic forcing). We wi refer to it as the static dispacementu st. The static dispacement u st = F /k is mutipied by a coefficient in Eq. (1.12), which depends on the damping and on the ratio of the circuar frequency of the forcing to the natura circuar frequency of the system. We ca this quantity as the response factor, and denote it by µ. Figure 1.4 (b) shows the dependence of the response factor on the ratio of frequencies. Figure 1.4: Responses of a damped SDOF system to a harmonic excitation: (a) phase angeϕas a function of the forcing frequency ω, (b) response factor µ as a function of the ratio of the forcing and natura frequencies ω/ω. where and In short, the ampitude of the steady-state vibration can be written as: u f = u st µ, u st = F k (1.15) 1 µ = ( ). (1.16) 2 1 ω2 +4ξ 2ω 2 ω 2 ω 2 9

18 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Now we further anayse the response factor function µ. For sma ω/ω it is sma, but bigger than 1. As ω/ω approaches 1 it reaches a maximum. One can derive, that the maximum occurs atω/ω = 1 2ξ 2, but in the practica range of dampingξ of engineering structures, the difference can be negected, so in genera we can say, that the maxima ampitude is approximatey at ω = ω with the magnitude µ max = 1/(2ξ). The state when µ is maxima is caed the resonance. For the case, when ω > ω, the response factor decreases asymptoticay to zero. The spring force from the steady-state part of the motion can be cacuated from the eongation of the spring: F S = ku st µ = F µ, i.e. the ampitude of the excitation force mutipied by the response factor, thus for fast excitation with arge ω or fexibe structure with ow ω the spring force wi be sma due to the decaying response factorµ. But if we are ooking for the force transmitted to the base, we aso have to take into account the force f D in the damping eement, which may resut higher base forces. Effect of zero damping on the phase ange and response factor The vibration of undamped systems can be derived in a simiar way as for the damped system, or we can anayse our damped resuts in the imit c. In the atter case we can concude, that the particuar soution of the non-homogeneous differentia equation (1.1) is a harmonic vibration. The ampitude of the vibration can be cacuated from Eq. (1.12) with ξ : u f = F k 1 ( 1 ω2 ω 2 ) 2 = F k 1 1 ω2. ω 2 It is the product of the static dispacement and the (undamped) response factor (see Fig. 1.4 (b)). In contrast to the damped case, this response factor has an infinite maximum in the state of resonance (ω = ω ). For the phase ange ϕ we can concude from Eq. (1.11) that it is zero when ω < ω, and it is π when ω > ω (see Fig. 1.4 (a)). In the first case the mass moves in-the-phase with the excitation force, in the second case the mass moves out-of-the-phase with the excitation force. At the resonance stateω = ω the phase ange is ϕ = π/2. Idea damping Anaysis of the damped response factor Eq. (1.16)) and its derivative with respect to ω ω resuts that an increasing damping coefficient ξ decreases the ocation and the vaue of the maximum ofµ(see Fig. 1.4 (b)). Ifξ reaches1/ 2, then the ocation of the maximum reaches ω =, and the vaue of the maximum reaches1. Further increase of the damping decreases the response factor, but the maximum wi be aways 1 at ω =. This damping vaue ξ id = 1/ 2 (or c id = 2km) is caed the idea damping. 1

19 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Support vibration of SDOF systems In many cases the support of the structure is not in rest. During an earthquake or because of the noise of traffic the base (which was assumed unti now to be in rest) might move, making the structure to vibrate. In this subsection we wi show how to hande the support motion for undamped systems. The steps of the soution woud be the same for a damped system as we. In the case of support vibration we have to modify our mechanica mode shown in Fig. 1.2 (a) such that we set the damping to zero (c = ) and appy a support motion u g (t) (where the index g refers to the ground motion). Figure 1.5 (a) shows this mode. If we draw the free body diagram, there is ony one force acting on the body from the spring, so we can write Newton s second aw of motion based on Figure 1.5 (b) as f S (t) = ma(t), or by substituting the spring force f S (t) = ku(t) and the acceerationa(t) = ü(t) as ku(t) = mẍ(t). (1.17) Figure 1.5: Support vibration of an undamped system (a) mechanica mode, (b) free body diagram The eongation of the spring is now and the second derivative of the Eq. (1.18) resuts: One can foow two different approaches. Substitution of u(t) from Eq. (1.18) in Eq. (1.17) eads to u(t) = x(t) u g (t), (1.18) ü(t) = ẍ(t) ü g (t). (1.19) kx(t)+ku g (t) = mẍ(t), which is a differentia equation for the dispacement x(t) of the body. If we write it in a canonica form mẍ(t)+kx(t) = ku g (t) (1.2) one can see, that it is a simpe forced vibration. 11

20 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Substitution of ẍ(t) from Eq. (1.19) in Eq. (1.17) impies ku(t) = mü(t)+mü g (t), which is a differentia equation for the eongation u(t) of the spring. If we write it in a canonica form mü(t)+ku(t) = mü g (t), (1.21) we obtain a simpe forced vibration again. In the next subsections we wi show the soutions of the derived differentia equations for a harmonic support vibration, i.e. u g (t) = u g sin(ωt). Steady-state soution of the eongation of the spring due to a harmonic support motion To find the soution of Eq. (1.21) we have to substitute the second derivative ofu g (t) into Eq. (1.21): ü g (t) = ω 2 u g sin(ωt) mü(t)+ku(t) = mω 2 u g sin(ωt). This is the same equation as Eq. (1.1) withc = andf = mω 2 u g. Therefore, the ampitude of the steady-state soution wi be (see Eq. (1.12)): u f = mω2 u g k 1 ω 2 ( ) = u g 2 ω 2 1 ω2 ω ω2. ω 2 The ampitude of the eongation u(t) wi be the ampitude of the support vibration mutipied by a response factor and by the square of the ratio of the forcing and natura frequencies. The spring force f S (t) is reated to the eongation u(t) of the spring so its ampitude wi be: f max S = ku g ω 2 ω ω2 ω 2 = ω 2 fst S ω ω2. ω 2 Here f st S is the static force, which woud cause an eongation u g in the spring. Figure 1.6 shows the product of two mutipiers(ω 2 /ω 2 and1/ 1 ω 2 /ω 2 ) as the function of the ratio of the forcing and natura frequencies. Steady-state soution of the dispacementx(t) for harmonic support vibration To find the soution of Eq. (1.2) we have to substitute u g (t) into Eq. (1.2): mẍ(t)+kx(t) = ku g sin(ωt). This is the same equation as Eq. (1.1) with c = and F = ku g. Thus, the ampitude of the steady-state soution is (see Eq. (1.12)): x f = ku g k 1 1 ( ) = u g 2 1 ω2. 1 ω2 ω 2 ω 2 12

21 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Figure 1.6: Response factor of the eongation of the spring as a function of the ratio of the forcing and natura frequencies due to a harmonic support vibration 1.2 Genera forcing of SDOF systems Duhame s integra Static (or quasi-static) oads and harmonic forcing represent ony a sma segment of the possibe oads acting on a structure. Athough many of the time-dependent oads can be treated as a quasi-static, or a sum of harmonic oads, there are important excitation forms (impact, support vibration due to earthquakes, etc.) where the transient behavior of the structure must be anayzed. For this type of probem the equation of motion Eq. (1.3) mü(t)+c u(t)+ku(t) = q(t) (1.22) contains an arbitrary function q(t) on the right hand side (see Figure 1.7 (a)). We are ooking for the particuar soution u f (t) of Eq. (1.22) for the t > interva, with the assumption that we know the initia dispacement and veocity in the time instant t =. We denote these two initia conditions with u f () = u and u f () = v. We remind the reader that the soution of a non-homogeneous differentia equation aways consists of the soution of the compementary equation (the free vibrationa part) with free parameters, and a particuar soution of the non-homogeneous equation. The free vibration foows the cassica scheme we presented in Subsection We assumed inear response of the eastic and damping eements (k and c are constants), so the differentia equation is inear, and the rue of superposition hods. If the excitation force can be written in the form q(t) = N i=1 q i(t), then the particuar soution can be expressed as u f (t) = N i=1 u fi(t), where eachu fi is a particuar soution of the differentia equation mü(t)+c u(t)+ku(t) = q i (t). Let us choose a sufficienty sma time interva τ at the time instant t = τ, as shown in Figure 1.7 (a), and et us examine the effect of the force q(τ) during the interva τ on the dispacement u f (t). This specific part of the forcing is shown in Figure 1.7 (b). We denote 13

22 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Figure 1.7: (a) Genera time-dependent forcing. (b) Sma impuseq(τ) τ of the forcing. (c) Increment of dispacement function from the impuseq(τ) τ. the effect of q(τ) on u f (t) by u(t,τ). Since τ is sma, the change of the force during the interva can be negected, so the impuse transmitted from the force to the mass is q(τ) τ. From the theorem of change of inear momentum the impuse resuts a sudden v(τ) change in the veocity: m v(τ) = q(τ) τ v(τ) = q(τ) τ. (1.23) m After this sudden change the force q(τ) wi be zero, so the mass-damper-spring system starts a free vibration with initia veocity v(τ). It is reasonabe to assume that the force q(τ) does not have any effects on the dispacements backwards in time, so we can say that the dispacement of the mass before the force is appied is zero: u(t,τ) =, t τ. (1.24) The time evoution of the increment of dispacement u(t, τ) is obtained from the previousy derived soution (1.7) of the free vibration of a mass-damper-spring system. For this specific case the initia conditions of Eq. (1.22) come from Eqs. (1.23) and (1.24): u(τ,τ) =, u(τ,τ) = q(τ) τ. (1.25) m The exponentiay decaying increment of the dispacement u(t, τ) comes from Eq. (1.7) with initia conditions (1.25) fufiing the differentia equation (1.22) and the initia conditions (1.25) wi be: ( ) u(t,τ) = e ξω (t τ) q(τ) τ sin(ω mω (t τ)). (Note thatξ = c/(2 km) andω = k/m 1 ξ 2.) This resut is shown in Figure 1.7 (c). If τ tends to, then u(t, τ) becomes an eementary increment du(t, τ). For any time t we have to integrate these eementary changes for a the past forces, i.e. for τ < t: u(t) = t q(τ) e ξω(t τ) sin(ω mω (t τ)) dτ. (1.26) 14

23 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS The above formua is the Duhame s integra Numerica soution of the differentia equation For many types of excitation forces Duhame s integra (1.26) can be computed ony numericay. Instead of numerica integration of the formua (1.26) the step-by-step cacuation of the dispacements and veocities directy from the differentia equation (1.22) is possibe. In the numerica cacuations it is a quite usua step to reformuate the second order differentia equation into two, first order equations. For that, first we introduce a new variabe function, the veocity: v(t) = du(t), dt and put it and its derivative with respect to time in the origina, second order differentia equation (1.22). The resuting system of first order differentia equations is: du(t) = v(t), dt dv(t) = c dt m v(t) k q(t) u(t)+ m m, with initia conditions u(t ) = u andv(t ) = v. Cauchy-Euer method (1.27) Let us assume, that we know the dispacement and the veocity at a given time instant t i, and we want to cacuate them at the time instant t i+1. (Let the difference between t i+1 and t i be a chosen constant t = t i+1 t i.) We denote the dispacement and the veocity at t i by u i andv i. From Eq. (1.27) we can cacuate the differences u i / t and v i / t: u i t = u i+1 u i = v i, t v i t = v i+1 v i = t ( c m v i k m u i + q(t i) m The estimated vaues of both variabes u i+1 andv i+1, are u i+1 = u i +v i t, ( v i+1 = v i + c m v i k m u i + q(t ) i) t. m We can iterate the above map starting with i =, i.e. with the given initia vaues u,v. Figure 1.8 (a) shows the concept of the agorithm, and one can see the main probem of this method as we. Using the Cauchy-Euer method invoves a sma error in every step, accumuating during the cacuation. The error depends on the step-size ( t). Smaer stepsize causes smaer error, but it requires more steps to reach the same time. The most important question of numerica methods is the convergence and the stabiity, but the discussion of these properties are beyond the scope of this ecture notes. In order to avoid fase soutions and crash of the procedure, one has to set the time step t sufficienty sma. 15 ).

24 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Figure 1.8: Expanation of (a) the Cauchy-Euer method and (b) the second order Runge-Kutta method. The continuous is the exact soution, the arrows represent tangents and increments. Higher order methods The key idea behind the higher order methods is to use a better approximation for the increments u i, v i, than we had from the tangents cacuated at the end-point of step i. It seems to be reasonabe, that we rather cacuate the tangent somewhere aong the current segment (based on one, or more points). These methods are caed the Runge-Kutta methods. In the second order Runge-Kutta method we cacuate the tangent at the midde of the current segment. So, we go forward with a haf step-size, cacuate the tangents there, and use those vaues to make the actua step-size. It means, that we have to cacuate the derivatives twice as much, but we get a higher precision. The agorithm is of the foowing steps. First we compute the differences just as before: u i t = u i+1 u i = v i, t vi t = v i+1 v i = t Next we step forward with a haf step-size: u 1/2 i v 1/2 i ( c m v i k m u i + q(t i) m = u i + u i/2, = v i + v i/2. Then we compute the differences at the mid-point (this wi be the direction of the actua step): u i+1 u i = v 1/2 i, t ( v i+1 v i = c t m v1/2 i k m u1/2 i + q(t ) i + t/2). m Finay, the map of the iteration is u i+1 = u i + u i = u i +v 1/2 i t, v i+1 = v i + v i = v i + ). ( c m v1/2 i k m u1/2 i + q(t i + t/2) m 16 ) t.

25 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Figure 1.9: Expanation of the finite difference approximation of veocity and acceeration using secant ines Centra difference method Let us assume, that we know the dispacement and the veocity at the given time instances t i 1 and t i, and we want to cacuate them at the time instant t i+1. (Let the difference between two time instants be constant: t = t i+1 t i = t i t i 1.) We denote the dispacement and the veocity att i byu i andv i, att i 1 byu i 1 andv i 1, respectivey. We can write the approximation for the veocity (see Figure 1.9): whie the approximation of the acceeration is: v i = u i u i+1 u i 1 =, (1.28) 2 t a i = ü i = u i+.5 u i.5 t The equation of motion is (1.22): = (u i+1 u i ) (u i u i 1 ) t 2 = u i+1 2u i +u i 1 t 2. (1.29) mü i +c u i +ku i = q(t i ) = q i. Let us substitute the veocity (Eq. (1.28)) and the acceeration (Eq. (1.29)) into the above equation: m u i+1 2u i +u i 1 +c u i+1 u i 1 +ku t 2 i = q i. (1.3) 2 t One can sove Eq. (1.3) for u i+1 : u i+1 = ( ) 2m q i +u i t k 2 +u i 1 ( c 2 t m t 2 ) m t 2 + c 2 t. (1.31) Eq. (1.31) is the map of the iteration containing ony the dispacements of the previous two steps (but no veocities). Therefore, not ony the dispacementu in the initia time instant, but aso the dispacementu 1 is needed to start the iteration. This atter condition can be computed from u andv as u 1 = u v t. 17

26 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS 1.3 Vibration of muti-degree-of-freedom systems Behaviour of rea ife engineering structures usuay cannot be described by the dispacement of ony one point of the structure. In fact, the exact description of the motion requires an approach considering the structure as a continuum. In many cases however, the motion of the continua can be reduced to the motion of a finite-degree-of-freedom system. In a mutistorey buiding with rigid sabs the dispacements of the ends of the coumns depend ony on the dispacements of the foors. In a spatia structure this woud be three degree-of-freedom on each eve (two transations in the horizonta pane and a rotation around a vertica axis, see Figure 1.1 (a) for a foor pate of one eve). If the buiding is reduced to a panar probem, the transation of each eve can be regarded as a degree of freedom. (see Figure 1.1 (b)). Even numerica methods appied in Finite Eement programs do the same: they approximate the dispacements by interpoating from the dispacements of the degrees of freedom. Figure 1.1 (c) shows a simpe mechanica mode for a two-degree-of-freedom system: two bodies are connected to each other by a spring, one of the bodies is supported by another spring, the other body has an excitation force F(t). Figure 1.1: Exampes of muti-degree-of-freedom structures (a) three degrees of freedom of one eve of a spatia muti-storey buiding (u and v are the transations, ϕ is the rotation), (b) mechanica mode of a three-storey frame structure (panar frame with three degrees of freedom) (c) mechanica mode of an undamped two-degrees-of-freedom system excited at its second degree of freedom Equation of motion of MDOF systems There are severa ways to derive the equations of motion for a MDOF system. Here we show one for the system on Figure 1.1 (c). The FBD of the system is shown in Figure The ony dispacement which is not constrained is the horizonta transation of the masses m 1 and m 2. Variabes x 1 (t) and x 2 (t) denote the transations of these masses, respectivey. The number of degrees of freedom is therefore two. Newton s second aw of motion is written for the two masses: f S1 (t)+f S2 = m 1 a 1 (t), f S2 (t)+f(t) = m 2 a 2 (t). (1.32) The forces in the inear springs depend on the eongation of each spring: f S1 (t) = k 1 1 (t), f S2 (t) = k 2 2 (t). For the first spring 1 (t) = x 1 (t) (assuming a fixed support) and for the 18

27 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Figure 1.11: Free body diagrams of the mode shown in Figure 1.1 (c) second spring 2 (t) = x 2 (t) x 1 (t). So the spring forces are: f S1 (t) = k 1 x 1 (t), f S2 (t) = k 2 (x 2 (t) x 1 (t)). The acceeration of each body is the second derivative of its transation with respect to time, i.e.: a 1 (t) = ẍ 1 (t), a 2 (t) = ẍ 2 (t). Substituting these resuts into Eq. (1.32) we get k 1 x 1 (t)+k 2 x 2 (t) k 2 x 1 (t) = m 1 ẍ 1 (t), k 2 x 2 (t)+k 2 x 1 (t)+f(t) = m 2 ẍ 2 (t), (1.33) which can be written in the foowing form: m 1 ẍ 1 (t)+k 1 x 1 (t)+k 2 x 1 (t) k 2 x 2 (t) =, m 2 ẍ 2 (t) k 2 x 1 (t)+k 2 x 2 (t) = F(t). (1.34) What we obtained is a couped system of second order ordinary differentia equations. Is it worth noting that each equation corresponds to one body (the ith) with the externa force acting on that body (or zero when there is none) on the right hand side of the current equation. On the eft hand sides there is aways the corresponding m i ẍ i (t) term (inertia term), and the spring force. The springs appearing in each equation such that the spring stiffness mutipied by the dispacement of the degree of freedom is added to the equation of the corresponding DOF (k 1 x 1 (t) for the first spring in the first equation, k 2 x 1 (t) and k 2 x 2 (t) for the second spring in the first and second equation respectivey). If a spring connects two degrees of freedom, then it coupes the equations of the connected DOFs ( k 2 x 2 term in the first and k 2 x 1 term in the second equation). The sign of the couping terms depends on the sense of the couped DOFs, but is aways the same in both equations. If two DOFs are not connected directy, their equations are not couped directy. Equation (1.34) can be written in a short form: Mü(t) + Ku(t) = q(t) (1.35) as a matrix differentia equation. Here vector u(t) contains the dispacement variabes, the quadratic matrices M and K are the mass and stiffness matrices, respectivey, whie vector q(t) contains the externa forces acting on each degree of freedom. (For an N-degree-of-freedom system the vectors have N entries, whie the size of the matrices is N by N). Properties expained after Eq. (1.34) yieds that the matrices are symmetric matrices. 19

28 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS For the exampe shown in Figure 1.1 (c) the eements are: [ ] [ ] [ ] m1 k1 +k M =,K = 2 k 2 x1 (t),u(t) =,ü(t) = m 2 k 2 k 2 x 2 (t) ] [ ] [ẍ1 (t),q(t) =. ẍ 2 (t) F(t) Simiary to the singe-degree-of-freedom vibrations, we divide the probems described by Eq. (1.35) in two groups: if q(t) =, then the system of differentia equations is homogeneous, and the resuting motion is the free vibration. if q(t), then the system of differentia equations is non-homogeneous, and it is caed a forced vibration. Equations of motion of a two-storey frame Let us anayse the equations of motion for a two-storey frame structure with a machine exerting a force on the upper eve. The foors are rigid, so we ony have two degrees of freedom. Figure 1.12 (a) shows the structure and one possibe dispacement system. Figure 1.12 (b) shows the free body diagrams for the same structure. The interna forces f S1 (from the coumns 1 and 1 ) and f S2 (from the coumns 2 and 2 ) depend on the inter-storey drifts x 1 and x 2 x 1, respectivey. Assuming inear eastic coumns one can cacuate the equivaent stiffness coefficients k 1 and k 2 for the coumns on each eve. Writing the equations of motion and the eements of the mass and stiffness matrices are eft for the reader as an exercise. Figure 1.12: Two-storey frame structure with rigid foors. (a) Mechanica mode, (b) free body diagram. Equations of motion with different variabes The deformed state of the structure in Figure 1.12 can be described not ony with the goba coordinates of each eve, but with the inter-storey drifts as we. (In accordance with the earier 2

29 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS notation we wi denote them by 1 and 2.) Then we have to substitute x 1 (t) = 1 (t), x 2 (t) = 1 (t)+ 2 (t) and their derivatives into Eq. (1.32), and we get 1 (t)+k 2 2 (t) = m 1 1 (t), k 2 2 (t)+f(t) = m 2 1 (t)+m 2 2 (t) instead of Eq. (1.33). One can see, that using this description of the probem resuts nonsymmetric mass- and stiffness matrices. This is due to the fact, that the equations sti beong to the goba x 1 and x 2 transations, whie our variabes are the reative dispacements 1 and 2. Symmetry of the system matrices is often used during the cacuations, so we can concude, that this hybrid approach shoud be avoided if possibe Free vibration of MDOF systems During the anaysis of a muti-degree-of-freedom system the soution of Eq. (1.35) foows the same steps as for SDOF systems. The free vibration of the system is anaysed using the compementary equation of Eq. (1.35). That is the homogeneous matrix differentia equation We search for the soution of Eq. (1.36) in the form: Mü(t)+Ku(t) =. (1.36) u(t) = u (acos(ω t)+bsin(ω t)), (1.37) i.e. the dispacement functionu(t) is assumed to be a product of a constant vectoru describing the ratio of the degrees of freedom to each other and a harmonic function depending on time, natura frequency ω and two parameters a and b. The cases when u = or a = b = woud ead to the trivia soution of the Eq. (1.36). We are ooking for the nontrivia soutions. The second derivative of the dispacement vectoru(t) is ü(t) = u ( ω 2 )(acos(ω t)+bsin(ω t)). We substitute u(t) andü(t) into the homogeneous differentia equation (1.36): Mu ( ω 2 )(acos(ω t)+bsin(ω t))+ku (acos(ω t)+bsin(ω t)) =. (1.38) This equation must hod for any time t, thus either (acos(ω t)+bsin(ω t)) =, or Mu ( ω) 2 + Ku =. The equation (acos(ω t)+bsin(ω t)) = hods for a t ony with the trivia soution a = b =, therefore the time-independent matrix equation Mu ( ω)+ku 2 = must be fufied, so it is rewritten in the more cassica form ( K ω 2 M ) u =. (1.39) The above equation is a system of a homogeneous, inear equations, which is caed a generaized eigenvaue probem in mathematics. It has nontrivia soutions if and ony if the matrix of coefficients is singuar, or equivaenty if and ony if its determinant is zero. The equation: det ( K ω 2 M ) = 21

30 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS eads to a poynomia of degree N for ω 2 (where N is the degree of freedom of the system). Typicay it has N rea soutions, denoted by ω1 2 ω2 2,..., ωn 2 (i.e. the first one is the smaest), and their positive square roots ω 1 ω 2,..., ω N are the natura circuar frequencies of the system. In the foowing steps we assume that a the natura circuar frequencies are different. We can definen natura period of the system as: T 1 = 2π ω 1 > T 2 = 2π ω 2 >,...,> T N = 2π ω N. In the next step we have to find the eements of vector u of Eq. (1.37). Since we have N natura circuar frequencies, we wi have N different vectors. We wi denote the vector corresponding toω j by u j. The vectoru j must fufi Eq. (1.39): ( K ω 2 j M ) u j =. (1.4) Because of the matrix ( K ω 2 jm ) is singuar,u j has onyn 1 independent rows, i.e. it has not a uniqueu j soution. Ifu j is a soution, then the vectorαu j wi be a soution for any reavaued α. These vectors are the (generaized) eigenvectors of the system. The meaning of the jth eigenvector u j is that if we dispace the degrees-of-freedom in the same proportion as the eements of the eigenvector, then it wi move such a way that the ratios of the dispacements wi be the same during the motion with frequency ω j. In this case the structure vibrates in its jth mode. The shape of the vibration (the moda shape) is described by the eigenvector (or mode vector). Normaized eigenvectors For further cacuations we have to make the eigenvector unique. It can be done in different ways: making the first eement of the vector be equa to 1, making the argest (in absoute vaue) eement of the vector be equa to 1, making the ength of the vector be equa to 1 (i.e. u T j u j = 1), making the vector be normaized to the mass matrix (i.e. u T j Mu j = 1). The first method is usefu when the cacuations are done by hand. The second method has an important roe in numerica soution of the eigenvaue probem. The third method woud resut in possibe sma numbers in the case of a arge system. The ast method has positive consequences on further resuts so we assume that the eigenvectors are normaized to the mass matrix. (If we have a non-normaized eigenvector u j, we can sti cacuate the product u T j Mu j = α j. It foows from the rues of matrix operations that the vector (1/ α j )u j wi be normaized to the mass matrix.) 22

31 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS If we substitute the jth normaized eigenvector into Eq. (1.39), and mutipy it with the transpose of the same vector from the eft we get: u T j Ku j ω 2 ju T j Mu j =. Because of the eigenvector is normaized, the vector-matrix-vector product on the eft hand side equas 1, resuting in: u T j Ku j = ω 2 j. (1.41) Orthogonaity of eigenvectors Let us take two different natura circuar frequencies ω i ω j, and the corresponding eigenvectors u i and u j. Then it hods from Eq. (1.4) that Ku i = ω 2 imu i, (1.42) Ku j = ω 2 jmu j. (1.43) Mutipying Eq. (1.42) by u T j and Eq. (1.43) by u T i from the eft and subtracting the resutant equations ead to: u T j Ku i u T i Ku j = ω 2 iu T j Mu i ω 2 ju T i Mu j. Due to the symmetry of matrices K andm so we have: u T j Ku i = u T i Ku j, u T j Mu i = u T i Mu j, (1.44) = ( ω 2 i ω 2 j) u T j Mu i. The above equaity ony hods for different ω i andω j if: u T j Mu i =. (1.45) Dividing both side of Eq. (1.42) by ω 2 i, then mutipying the resut by u T j from the eft, dividing both side of Eq. (1.43) by ω 2 j, then mutipying the resut by u T i from the eft, finay subtracting the resutant equations ead to: 1 ω 2 i u T j Ku i 1 u T ωj 2 i Ku j = u T j Mu i u T i Mu j. Due to Eq. (1.44) ( 1 1 ) u T ωi 2 ωj 2 j Ku i =, which hods for different nonzero ω i andω j ony when: u T j Ku i =. (1.46) We refer to this atter properties as the orthogonaity of the eigenvectors u i and u j to the mass matrix (Eq. (1.45)) and to the stiffness matrix (Eq. (1.46)). 23

32 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Genera soution of the homogeneous differentia equation The genera soution of the homogeneous differentia equation Eq. (1.36) is constructed from the sum of the soutions corresponding to the eigenmodes: u(t) = N u j (a j cos(ω j t)+b j sin(ω j t)). (1.47) j=1 To find the parameters a j andb j we need the vector of veocities: u(t) = N u j ω j ( a j sin(ω j t)+b j cos(ω j t)). (1.48) j=1 Initia conditions of a muti-degree-of-freedom system are dispacements and veocities of the degrees of freedom at a given time instant t : u(t ) = u, u(t ) = v. By substituting Eq. (1.47) and (1.48) into the above formua 2N constraints are obtained which can be used to find the parameters a j and b j in the genera soution Eq. (1.47). Using the orthogona properties of the eigenvectors one can avoid the soution of a system of 2N inear equations. If we mutipy these 2N equations from the eft byu T j M, then we get a j cos(ω j t )+b j sin(ω j t ) = u T j Mu, ω j ( a j sin(ω j t )+b j cos(ω j t )) = u T j Mv, so, varying j from 1 to N we have to sove N system of 2 inear equations instead of a system of 2N equations for the coefficients a j and b j. The resutant motion wi be the sum of harmonic vibrations, which is not necessariy a periodic motion! Harmonic forcing of MDOF systems (direct soution and moda anaysis) The soution of forced vibration probems of muti-degrees-of-freedom systems foows a simiar schema as we experienced with the SDOF systems. The compete soution is the sum of the genera soution of the compementary differentia equation and a particuar soution of the nonhomogeneous differentia equation. So the soution of Eq. (1.35) constructed from Eq. (1.47) and a particuar soutionu f (t), which is the answer of the system to the forcing (that is the subscript f is for). In this subsection we wi give a possibe soution of the probem, where the excitation force is harmonic, i.e. q(t) can be written in the form: q(t) = q (t)sin(ωt). (1.49) 24

33 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Here ω is the circuar frequency of the forcing, the vector q is the ampitude of the forcing. Thus each DOF is excited with the same frequency ω. Is it worth mentioning, that a zero extarna force acting on a degree-of-freedom can be treated as a harmonic force with zero ampitude and arbitrary circuar frequency. Soving the nonhomogeneous equation we are ooking ony for the steady-state part of the vibration. We show two possibe soution method: direct soution, soution with the moda shape vectors and natura circuar frequencies. Direct soution In the case of a direct soution we assume the particuar soution of the form: u f (t) = u f sin(ωt), (1.5) i.e. it is a harmonic response with the same harmonic term as the forcing, with constant ampitudes given in the vectoru f. The second derivative of Eq. (1.5) with respect to time is: ü f (t) = ω 2 u f sin(ωt). (1.51) Substituting the oad, the ansatz and its derivative (Eqs. (1.49), (1.5), and (1.51)) into Eq. (1.35) we get: ω 2 Mu f sin(ωt)+ku f sin(ωt) = q sin(ωt). (1.52) The above equation fufis either if sin(ωt) =, or if Ku f ω 2 Mu f = q. Because the oading is a rea, time dependent harmonic force, the term sin(ωt) cannot be zero for every time instant. So, we can write the atter condition as: ( K ω 2 M ) u f = q. (1.53) The soution of this non-homogeneous matrix differentia equation for the ampitude u f is needed. The matrix of coefficients of Eq. (1.53) is quadratic, so it has a soution if and ony if there exists its inverse matrix, i.e. the matrix is non-singuar, or with other words, its determinant is nonzero. In that case we get the soution by mutipying both sides of Eq. (1.53) by the inverse (K ω 2 M) 1 : u f = ( K ω 2 M ) 1 q. (1.54) The particuar soution then can be written as: u f (t) = ( K ω 2 M ) 1 q sin(ωt). (1.55) This is the steady-state part of the vibration. We can see that each degree of freedom vibrates with the same frequency. Without computing the inverse matrix, we cannot read out directy whether a degree of freedom is in an in-phase or in an out-of-phase vibration. 25

34 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS We have to note that the non-existence of the inverse of the matrix (K ω 2 M) means that the matrix is singuar, i.e. its determinant equas zero. But if det(k ω 2 M) =, then the circuar frequency of forcing is one of the natura circuar frequencies of the system, i.e. the system is in the state of resonance. It means that if the frequency of forcing coincides with one of the natura frequencies of the structure, then the direct method gives an infinite ampitude of the vibration. However, in rea structures it does not occur, because there is aways some damping in the system. Direct soution requires the cacuation of the inverse of ann-by-n matrix. In genera, the required computationa capacity increases proportiona to the second- or third power ofn (the order of the computationa time is O(N 2 N 3 )). Thus for arge systems this method has a very high computationa costs. Moda anaysis Instead of the direct soution one can make use of the soutions of the unforced system, i.e. of the generaized eigenvaue probem Eq. (1.36). These soutions contain the natura circuar frequencies (ω 1, ω 2,..., ω N ) and the corresponding mode shape vectors normaized to the mass matrix (u 1, u 2,..., u N ). These eigenvectors are ineary independent and span an N dimensiona inear space. Dispacements are written as a inear combination of the normaised mode shape vectors: N u f (t) = u j y j (t), (1.56) j=1 where functions y i (t) are the moda dispacements. The moda shape vectors are invariant in time, so the second derivative of the dispacement is: ü f (t) = N u j ÿ i (t). (1.57) j=1 Substituting the oad, the ansatz and its derivative (Eqs. (1.49), (1.56), and (1.57)) into Eq. (1.35) we get: N N M u j ÿ j (t)+k u j y j (t) = q sin(ωt). (1.58) j=1 j=1 Let us mutipy both sides of Eq. (1.58) from the eft byu T i. Using the orthogonaity of the eigenvectors (Eq. (1.45), and (1.46)) we ony have nonzero vaues if j = i: u T i Mu i ÿ i (t)+u T i Ku i y i (t) = u T i q sin(ωt). Moreover, the eigenvectors are normaized to the mass matrix, so: u T i Mu i = 1, u T i Ku i = ω 2 i, which eads to the moda differentia equation of vibration: ÿ i (t)+ω 2 iy i (t) = f i sin(ωt), i = 1,2,...,N. (1.59) 26

35 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Here f i = u T i q is the moda ampitude of the harmonic excitation force. For the particuar soution y if (t) of the non-homogeneous differentia equation (1.59) we assume the soution in the harmonic form: Its second derivative is y if = y if sin(ωt). ÿ if = ω 2 y if sin(ωt). Substitute these equaities into Eq. (1.59), and sove for arbitrary nonzerosin(ωt): ω 2 y if +ω 2 iy if = f i. If ω = ω i and f i, then the frequency ω of the oad and the ith natura frequency ω i coincide, thus the system is in the state of resonance. It resuts in an infinitey arge ith moda dispacement. Otherwise, the unique, finite soution is: 1 y if = f i ω = f 1 i 2 ω2 i ω 2 i 1 1 ω2 ω 2 i The ast term in the above product is a response factor of an undamped osciatory system of natura circuar frequency ω i, excited by a harmonic force of circuar frequency ω. This coefficient is denoted by µ i. Let us summarise our resuts. In the absence of resonance the steady-state part of the motion (the particuar soution) can be written in the form: u f (t) = N i=1 1 ω 2 i. µ i u i u T i q sin(ωt). (1.6) Checking the terms of the above equation from right to eft we can conude that the response is harmonic (sin(ωt)). For each mode the ampitude of the moda oad (u T i q ) is cacuated. It is then mutipied by the response factor µ i depending on the ratio of the forcing and the natura frequency of the corresponding mode. Finay, the ampitude for each mode is divided by the square of the natura circuar frequency ω 2 i of the mode. Due to this ast term the effect of higher modes is usuay much smaer, except for the case when the excitation occurs cose to one of the resonant frequencies of the system. Apparenty, the soution of the probem with moda anaysis seems to need even more computationa effort, than that of the direct soution, beacuse we first have to sove a generaized eigenvaue probem of the free vibration. For arge systems, with many degrees of freedom, the soution of the eigenveue probem has high computationa needs. However, the parts from higher modes typicay pay a ess significant roe in the soution. There are numerica agorithms which do not compute a the eigenvaues and eigenvectors of the generaized eigenvaue probem, nut ony the first few of them. Later in the semester we wi show that a reduced set of mode shape vectors cacuated with these methods can be sufficient to approximate we the motion of the MDOF system. 27

36 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Probem (Exampe on harmonic forcing of a three-storey frame structure). Figure 1.13 shows the dynamica mode of a three-storey buiding with rigid girders. On each eve the stiffness of the coumns is the same and the eves have the same mass. On the top eve a machine exerts a harmonic force on the structure. We are ooking for the ampitudes of each degree-of-freedom in the steady-state motion. Figure 1.13: Vibration of a three-storey frame structure with rigid interstorey girders and eastic coumns. (a) Dynamica mode with system properties k 1 = k 2 = k 3 = 25 MN/m, and m 1 = m 2 = m 3 = 15 t. (b) Degrees of freedom in a dispaced position and the excitation force F = 15 kn, ω = 9 rad/s. Soution. The matrix differentia equation of the motion is: Mü(t)+Ku(t) = q sin(ωt), where u(t) = [x 1 (t),x 2 (t),x 3 (t)] T, K = q = [F,,] T = [15,,] T kn, m 1 15 M = m 2 = 15 m 3 15 k 1 +k 2 k 2 k 2 k 2 +k 3 k 3 k 3 k 3 = t kn/m. Direct soution The system of inear equations of the probem (Eq. (1.53)) with substitution of K,Mand q is: u f = (1.61) The soution of the above equation requires the inverse of the matrix of coefficients: =

37 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS We have to mutipy both sides of Eq. (1.61) from the eft, resuting: u f = u f =.1689 m. (1.62) Moda anaysis This soution requires the soution of the generaized eigenvaue probem: ( K ω 2 M ) u =. (1.63) First we compute the eigenvaues of the probem. The condition we use is that the determinant of the matrix ( K ωm ) 2 must be zero: 5 15ω 2 25 = ω ω 2 = ( )( ( ) ) 5 15ω 2 (5 15) 25 15ω 2 ( 25)( 25) ( 25) ( ( 25) ( 25 15ω 2 ) ) +, which resuts the foowing cubic equation forω 2 : ω ω ω =. There are three rea vaued soution of the above poynomia equation: resuting the natura circuar frequencies in: ω 2 1 = 33.1, ω 2 2 = , ω 2 3 = , ω 1 = rad/s, ω 2 = rad/s, ω 3 = rad/s. (The corresponding natura periods are: T 1 = s, T 2 =.393 s, and T 3 =.279 s.) We show ony the cacuation of the first eigenvector (u 1 ). It must fufi the equation: ( K ω 2 1 M ) u 1 =, which has the form after substitution of previous resuts: u 11 u 12 u 13 =. Here we assume a tria vector in the form û 1 = [c 1,1,c 3 ] T. So, with the first and the ast equation we avoid muti-variabe equations. (It is not aways possibe, in that case we shoud sove a system of inear equations.) From the mentioned rows we have: 4548c = c 1 = c 3 = c 3 = (The second equation is ineary dependent, but it can be used to check our resuts both for the natura circuar frequency and the vector eements.) 29

38 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Now the tria eigenvector û 1 is normaized to the mass matrixm. To do this, first we cacuate: α 1 = û T 1Mû 1 = [ ] [ ] = = , then the normaized shape vector correspondiong to the first natura mode wi be: u 1 = 1 α1 û 1 = [ ] T. (1.64) The steps between Eq. (1.13) and (1.64) must be repeated for ω 2 and for ω 3 as we, to cacuate the corresponding normaized eigenvectors. The fina resuts of that cacuations are: and u 2 = [ ] T u 3 = [ ] T. Figure 1.14 shows the deformed shape of the structure corresponding to the three moda vector. Now we can cacuate the ampitude vector of the steady-state vibration using the formua of Eq. (1.6). The terms are summarized in Tabe 1.1. Figure 1.14: Mode shapes of the three-storey structure of Figure 1.13 corresponding to the natura circuar frequencies (a) ω 1 = rad/s, (b) ω 2 = rad/s, and (c)ω 3 = rad/s. u T i i q µ i ω 2 iµ i ωiµ 2 i u T i q Tabe 1.1: Harmonic forcing of a three-storey structure. Moda oads, coefficients of resonance, this coefficient divided by the square of the ith natura circuar frequency, participation of the mode in the steady-state vibration. 3

39 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS From Tabe 1.1 we can see, that this specific oading has a projection in the same order of magnitude in each mode, and the coefficient of resonance does not change this proportion much. Contrary to this, the whoe participation of the first mode is 4.5 times higher than the participation of the second mode, and 2 times higher, than that of the third mode, due to the division byω 2 i. The ampitude vector of the steady-state vibration wi be: u f =.1881u 1.453u 2.873u 3 = m. (1.65) For both soution methods we can concude, that in the steady-state vibration each eve osciates in a cm range, the ower two eves are out-of-phase, the upper eve is in-phase with the forcing Approximate soution of the generaized eigenvaue probem (Ritz- Rayeigh s method) We have seen aready, that a higher natura frequency of a muti-degrees-of-freedom system pays an important roe ony if the forcing has a frequency cose to that natura frequency. In practica probems, the first few natura modes are sufficient to describe the vibration of the structure. On the mode shape eve, a mode vector of a higher natura frequency resuts more changes in the sense of dispacements of DOFs. So, the simper mode shapes correspond to ower natura frequencies, and an eigenvector (normaized to the mass matrix) can be used as a base for the cacuation of the eigenvaue (see Eq. (1.41)). The Rayeigh quotient Approximate soutions can be obtained by guessing the mode shape vector of the structure, and finding the corresponding natura frequency. This is the opposite of the cassica soution of the generaized eigenvaue probem, where we started with finding the eigenvaues from the poynomia equation defined by the determinant of the matrix of coefficients K ω 2 M of the homogeneous equation, and then the eigenvectors were cacuated. Let us assume, thatvis a vector of N eement. We define the Rayeigh quotient as: R = vt Kv v T Mv. (1.66) Atough we do not know the eigenvectors,vcan be written as a inear combination of them with coefficientsα j : N v = α j u j. j=1 Let us expand the denominator and the numerator of Eq. (1.66). The denominator can be written as: ( N ) T ( N ) v T Mv = α j u j M α i u i = j=1 i=1 N N α j α i u T j Mu i. j=1 i=1 31

40 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Orthogonaity of the normaized eigenvectors (Eq. (1.45)) impies that the quadratic product u T j Mu i = 1 ifj = i and zero otherwise, thus v T Mv = N αj. 2 j=1 The numerator of Eq. (1.66) can be written as: ( N ) T ( N ) v T Kv = α j u j K α i u i = j=1 i=1 N N α j α i u T j Ku i. j=1 i=1 Orthogonaity of the normaized eigenvectors (Eq. (1.46)) impies that the quadratic product u T j Ku i = ω 2 j if j = i and zero otherwise. Therefore v T Kv = N αjω 2 j. 2 j=1 The above formua can be expanded to: v T Kv = N ( ( ) N α 2 j ω 2 j ω1 2 +α 2 h ω1) 2 = ( N ω 2 j ω1) 2 + αjω αj 2 j=1 j=1 j=1 The first summation term is zero if j = 1, so we got finay: v T Kv = N αjω j=1 N j=2 α 2 j ( ) ω 2 j ω1 2. If we write the resut into the definition of the Rayeigh quotient (1.66) we get: R = N αjω N j=1 αj 2 j=2 N αj 2ω2 1 j=1 ( ω 2 j ω 2 1) = ω N αj 2 j=2 N ( ω 2 j ω 2 1) αj 2ω2 1 j=1. (1.67) The sum on the right hand side of Eq. (1.67) contains ony positive numbers (here we remind, thatω 1 is the smaest natura frequency), or zeros (if a specificα j is zero). So we can concude, that the Rayeigh quotient is aways higher than, or equa to the square of the first natura circuar frequency. The accuracy of the resut depends naturay on the initia guess on the mode shape vector (v): the coser the guessed shape vector v is to the exact one u 1, the more precise ω 1 is. Seeding the ω 2 N eement instead of ω2 1 resuts in a proof for the Rayeigh quotient to be smaer than, or equa to the square of the highest natura circuar frequency. 32

41 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Probem (Exampe on finding an approximate soution for a two-storey frame). Find an approximate soution of the first natura circuar frequency of the two-storey structure shown in Figure The masses of the storeys are m 1 = m 2 = 2 t, and the stiffness of the coumns is given by spring stiffnesses k 1 = k 2 = 5 kn/m. The first mode shape shoud be assumed as: v = [ 1 2 ] T Soution. The system has the foowing mass and stiffness matrices: [ ] [ M =,K = ]. The numerator and the denominator of the Rayeigh quotient are: v T Mv = [ 1 2 ][ ][ ] 2 1 = [ 2 4 ][ ] 1 = 1, So, the Rayeigh quotient is: resuting in the approximation: v T Kv = [ 1 2 ][ ][ 1 2 R = vt Kv v T Mv = 1 1 = 1, ω 2 1 1, ] ω rad/s. = [ 5 ][ 1 2 ] = 1. Exercise Find first natura circuar frequency ω 1 of the above probem with the exact first mode shape vector: u 1 = [ ] T. Probem (Exampe on finding an approximate soution for a muti-storey frame). Let us find the first natura circuar frequency of a 1-storey frame. (See Figure 1.15 (a).) The inter-storey stiffnesses and the eve masses are the same on each eve, m = 15 t and k = 25 kn/m respectivey. Soution. The mass matrix of the structure is a 1 1 diagona matrix, where each eement equas m. The stiffness matrix is k +k k.... k k +k k... K =. k k +k k... k k It is a crucia step of the method to find a good assumption of the tria vector. During the drift of the storeys, the rigid girders are staying horizonta, and so the structure foows a pattern of dispacements simiar to a rod with finite shear stiffness. The frame can be treated as a discrete mode of the sheared (continuous) coumn (Figure 1.15 (b)). A sheared rod has a moda shape of a sinusoida function with a zero vaue at the bottom and a zero tangent at the free end. Simiar dispacement vector can be used with: jπ v j = sin 2N +1, j = 1,...,N. Dispacements are shown in Fig (c). The numerator and the denominator of the Rayeigh quotient are: N N v T Mv = v j mv j = m sin 2 jπ +1 = m2n = 787.5, 2N +1 4 j=1 j=1 33

42 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS N N v T Kv = v 1 kv 1 + (v j 1 kv j 1 2v j 1 kv j +v j kv j ) = v 1 kv 1 + (v j v j 1 ) 2 k = The Rayeigh quotient is: j=2 R = vt Kv v T Mv = = This wi be an upper bound for the square of the first natura circuar frequency: ω , ω rad/s. Note: in this specific case the supposed shape vector was the actua first mode vector, so in this probem the accurate soution was obtained. j=2 Figure 1.15: A 1-storey frame structure with rigid interstorey girders and eastic coumns. (a) Dynamica mode of the structure. (b) Equivaent continuous rod with finite shear stiffness. (c) First moda shape of the continuous rod. The Ritz-Rayeigh method We have seen in the previous probems what effect the assumed shape on the accuracy of the resut has. If, instead of guessing one vector, we make our tria vector as a inear combination of fixed base vectors, then the Rayeigh quotient wi be a function of the coefficients of the base vectors. The first natura frequency wi be equa to, or smaer than any Rayeigh quotients, so the minimum of the avaiabe vaues in the space of the base vectors wi give an upper bound for the first natura frequency. This is the theory behind the Ritz-Rayeigh method. We have to choose some ineary independent base vectors Φ i (i =,...,n) in the N- dimensiona space (where N is the number of degree-of-freedom of the system), and make the tria vector as a inear combination of these vectors. (Here we ca the attention, that the 34

43 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS inequaity n + 1 N, must hod, otherwise the vector Φ i can not make a base of the N- space.) We want to make our v unique among a vectors parae with it. We have seen, that normaizing the eigenvector to the mass matrix is a very practica way, but now it gives a noninear constraint to the system, and that makes difficut to use. Instead of that, we set the coefficient of one base vector (Φ i ) equa to one, and write the tria vector as: v(c 1,...,c n ) = Φ + n c i Φ i. i=1 Using this tria vector the Rayeigh quotient wi depend on the variabes c 1,...,c n : R(c 1,...,c n ) = vt (c 1,...,c n )Kv(c 1,...,c n ) v T (c 1,...,c n )Mv(c 1,...,c n ). We are ooking for the possibe smaest R in the space of the vectors Φ 1,...,Φ n : R(c 1,...,c n ) = min! Ifn = 1: The quotient depends on one singe variabe. At the minimum the first derivative vanishes: dr(c 1 ) dc 1 =. (1.68) The soution of the (noninear) equation (1.68) resuts in a possibe best resut for the tria vector coefficientc 1 in the space of the base vectors. Ifn > 1: The quotient depends on mutipe variabes. At the minimum the gradient of the quotient is zero: R(c 1,...,c n ) c i =. i = 1,...,n, (1.69) which is a noninear system of equations for n variabes. This type of equations does not necessariy have a unique soution, thus soution method must be chosen according to this. We mention here, that the Ritz-Rayeigh method is capabe of finding the exact soution if n + 1 = N, i.e., if the base vectors Φ i (i =,...,n) span the whoe N space. Otherwise, for the tria vector the method minimizes the error to the the exact soution, i.e. it finds the projection of the exact soution on the space spanned by the base vectorsφ i (i =,...,n), and gives the corresponding Rayeigh quotient. Probem (Exact soution of a two-storey frame). Find the exact soution for the first natura circuar frequency of the two-storey structure of Probem

44 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Soution. Let us assume the tria vector as v(c 1 ) = [ 1 c 1 ] T, i.e. we chose the base vectors: Φ = [ 1 ] [, Φ 1 = 1 Here n = 1 and N = 2, so1+1 = 2, i.e. we wi get the exact soutions. The numerator and the denominator of the Rayeigh quotient are: v T (c 1 )Mv(c 1 ) = [ ] [ ][ ] c 1 = [ ] [ 1 2 2c 2 c 1 1 c 1 v T (c 1 )Kv(c 1 ) = [ 1 c 1 ] [ = 1 1c 1 +5c 2 1. The resuting Rayeigh quotient is: ][ 1 c1 R(c 1 ) = vt (c 1 )Kv(c 1 ) v T (c 1 )Kv(c 1 ) = 1 1c 1 +5c c 2. 1 ] ] ] = 2+2c 2 1. = [ ] [ ] 1 5c 1 c c 1 The first derivative is: dr(c 1 ) = ( 1+1c ( ) ( ) 1) 2+2c c1 +5c 2 1 (4c1 ) dc 1 = 2( c 2 1 c 1 1 ) 4(c c2 1 +1) = (2+2c 2 1 )2 It is sufficient, if the nominator equas zero, so the coefficient c 1 we are ooking for is the soution of the quadratic equation: c 2 1 c 1 1 =. There are two soutions: ( 1+ 5 ) /2 and ( 1 5 ) /2. If we substitute them back to the Rayeigh quotient, the first one resuts the smaer number, so this wi be the first mode whie for the second soution we get The resutant moda shape vectors are: [ v 1 = R(1.618) = ω 1 = 3.9 rad/s, R(.618) = ω 1 = 8.9 rad/s ] [, v 2 =.618 ]. Probem (Exact soution for a three-storey structure). Find the exact soution for the first natura circuar frequency of the three-storey structure of Probem Soution. First we repeat the matrices of the system from Probem 1.3.1: 15 M = 15, 15 36

45 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS K = Let us assume the tria vector as v(c 1,c 2 ) = [ 1 c 1 ] T c 2, i.e. we chose the base vectors: 1 Φ =, Φ 1 = 1, Φ 2 = 1 Heren = 2 andn = 3, so2+1 = 3, i.e. the Ritz-Rayeigh method eads to the exact soutions of the probem. The numerator and the denominator of the Rayeigh quotient are: v T (c 1,c 2 )Mv(c 1,c 2 ) = [ ] 1 c 1 c c 1 15 c 2 = [ ] 1 c 1 c c 1 15c 2 v T (c 1,c 2 )Kv(c 1,c 2 ) = [ ] 1 c 1 c 2 = [ ] 1 c 1 c 2 = 15 ( 1+c 2 1 +c 2 2), 5 25c c 1 25c 2 25c 1 +25c 2 = 25 ( 2 2c 1 +2c 2 1 2c 1 c 2 +c 2 2) The resuting Rayeigh quotient is R(c 1,c 2 ) = 25( ) 2 2c 1 +2c 2 1 2c 1 c 2 +c (1+c 2 1 +c2 2 ) (1.7) The first partia derivatives are: 1 c 1 c 2 R(c 1,c 2 ) c 1 =, R(c 1,c 2 ) c 2 =. The partia derivatives resut a cumbersome system of two equations. However, the soution can be cacuated numericay, resuting in the foowing soution pairs: c 1 = 1.82, c 2 = 2.247, c 1 =.445, c 2 =.82, c 1 = 1.247, c 2 =.555. These points are on the surface defined by the Rayeigh quotient over the (c 1,c 2 )-pane (see Eq. (1.7)). The shape of the surface is shown in Figure To decide, which one of the above three soution pairs eads to the first natura frequency, we have two options: We decide which one of the soution pairs correspond to the minimum point of the surface given by the function R(c 1,c 2 ). We cacuate the Rayeigh quotient with the soution points and pick the smaest one. 37

46 CHAPTER 1. DYNAMICS OF SINGLE- AND MULTI-DOF SYSTEMS Either way, the smaest possibe Rayeigh quotient of this probem, and so the square of the first natura circuar frequency is R = 33.1, ω 1 = rad/s, which is the same as the anaytica resut. Figure 1.16: Rayeigh quotient of Probem Exercise Find an approximate soution of the above probem for the first natura circuar frequency ω 1 using the base vectors: 1 Φ = 1.5, Φ 1 =. 1 38