An Introduction To Dynamics Of Structures
Dynamics ofStructuresGiacomo BoffiAn Introduction to Dynamics of StructuresGiacomo iacomoDipartimento di Ingegneria Civile Ambientale e TerritorialePolitecnico di MilanoMarch 10, 2017OutlineDynamics ofStructuresGiacomo BoffiPart 1IntroductionCharacteristics of a Dynamical ProblemFormulation of a Dynamical ProblemFormulation of the equations of motionPart 21 DOF SystemFree vibrations of a SDOF systemFree vibrations of a damped systemDefinitionsDynamics ofStructuresGiacomo BoffiLet’s start with some definitionsDynamic adj., constantly changingDynamics noun, the branch of Mechanics concerned withthe motion of bodies under the action of forcesDynamic Loading a Loading that varies over timeDynamic Response the Response (deflections and stresses) of asystem to a dynamic loading; DynamicResponses vary over timeDynamics of Structures all of the above, applied to a structuralsystem, i.e., a system designed to stay inequilibriumIntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotion
Dynamics of StructuresDynamics ofStructuresGiacomo BoffiOur aim is to determine the stresses and deflections that a dynamicloading induces in a structure that remains in the neighborhood of apoint of equilibrium.Methods of dynamic analysis are extensions of the methods ofstandard static analysis, or to say it better, static analysis is indeed aspecial case of dynamic analysis.IntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionIf we restrict ourselves to analysis of linear systems, it is soconvenient to use the principle of superposition to study thecombined effects of static and dynamic loadings that differentmethods, of different character, are applied to these differentloadings.Types of Dynamic AnalysisDynamics ofStructuresGiacomo BoffiTaking into account linear systems only, we must consider twodifferent definitions of the loading to define two types of dynamicanalysisDeterministic Analysis applies when the time variation of the loadingis fully known and we can determine the complete timevariation of all the required response quantitiesNon-deterministic Analysis applies when the time variation of theloading is essentially random and is known only interms of some statisticsIn non-deterministic or stochastic analysis thestructural response can be known only in terms ofsome statistics of the response quantitiesIntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionOur focus will be on deterministic analysisTypes of Dynamic LoadingDynamics ofStructuresGiacomo BoffiDealing with deterministic dynamic loadings we will study, in order ofcomplexity,Harmonic Loadings a force is modulated by a harmonic function of time,characterized by a frequency ω and a phase ϕ:p(t) p0 sin(ωt ϕ)Periodic Loadings a periodic loading repeats itself with a fixed period T :p(t) p0 f (t) with f (t) f (t T )Non Periodic Loadings here we see two sub-cases,I the loading can be described in terms of an analyticalfunction, e.g.,p(t) po exp(αt)Ithe loading is measured experimentally, hence it isknown only in a discrete set of instants; in this case,we say that we know the loading time-history.IntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotion
Characteristics of a Dynamical ProblemDynamics ofStructuresGiacomo BoffiA dynamical problem is essentially characterized by the relevance ofinertial forces, arising from the accelerated motion of structural orserviced masses.A dynamic analysis is required only when the inertial forces representa significant portion of the total load.IntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionOn the other hand, if the loads and the deflections are varying slowly,a static analysis will often provide an acceptable approximation.We will define slowlyFormulation of a Dynamical ProblemDynamics ofStructuresGiacomo BoffiIntroductionIn a structural system the inertial forces depend on the timederivatives of displacements while the elastic forces, equilibrating theinertial ones, depend on the space derivatives of the displacements. the natural statement of the problem is hence in terms of partialdifferential equations.Characteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionIn many situations it is however possible to simplify the formulationof the problem to ordinary differential equations.Lumped MassesDynamics ofStructuresGiacomo BoffiIntroductionIn many structural problems, we can say that the mass isconcentrated in a discrete set of lumped masses.Under this assumption, the analytical problem is greatly simplified:1. the inertial forces are applied only at the lumped masses,2. the only deflections that influence the inertial forces are thedeflections of the lumped masses,3. using methods of static analysis we can determine thosedeflections,thus consenting the formulation of the problem in terms of a set ofordinary differential equations, one for each component of theinertial forces/masses displacements.Characteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotion
Dynamic Degrees of FreedomDynamics ofStructuresGiacomo BoffiIntroductionThe dynamic degrees of freedom (DDOF) in a discretized system arethe displacements components of the lumped masses associated withthe components of the inertial forces.If a lumped mass can be regarded as a point mass then 3translational DDOFs will suffice to represent the associated inertialforce.On the contrary, if a lumped mass has a discrete volume its inertialforce depends also on its rotations (inertial couples) and we need 6DDOFs to represent the mass deflections and the inertial force.Characteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionOf course, a continuous system has an infinite number of degrees offreedom.Generalized DisplacementsDynamics ofStructuresGiacomo BoffiIntroductionThe lumped mass procedure that we have outlined is effective if alarge proportion of the total mass is concentrated in a few points(e.g., in a multi-storey building one can consider a lumped mass foreach storey).When the masses are distributed we can simplify our problemexpressing the deflections in terms of a linear combination of assignedfunctions of position, the coefficients of the linear combination beingthe generalized coordinates (e.g., the deflections of a rectilinearbeam can be expressed in terms of a trigonometric series).Generalized Displacements, cont.Characteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionDynamics ofStructuresGiacomo BoffiIntroductionTo fully describe a displacement field, we need to combine an infinityof linearly independent base functions, but in practice a goodapproximation can be achieved using just a small number offunctions and degrees of freedom.Even if the method of generalized coordinates has itsbeauty, we must recognize that for each different problemwe have to derive an ad hoc formulation, with an evidentloss of generality.Characteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotion
Finite Element MethodDynamics ofStructuresGiacomo BoffiIntroductionCharacteristics ofa DynamicalProblemThe finite elements method (FEM) combines aspects of lumpedmass and generalized coordinates methods, providing a simple andreliable method of analysis, that can be easily programmed on adigital computer.Finite Element MethodFormulation of aDynamicalProblemFormulation ofthe equations ofmotionDynamics ofStructuresGiacomo BoffiIIn the FEM, the structure is subdivided in a number ofnon-overlapping pieces, called the finite elements, delimited bycommon nodes.IThe FEM uses piecewise approximations (i.e., local to eachelement) to the field of displacements.IIn each element the displacement field is derived from thedisplacements of the nodes that surround each particularelement, using interpolating functions.IThe displacement, deformation and stress fields in eachelement, as well as the inertial forces, can thus be expressed interms of the unknown nodal displacements.IThe nodal displacements are the dynamical DOFs of the FEMmodel.Finite Element MethodIntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionDynamics ofStructuresGiacomo BoffiIntroductionSome of the most prominent advantages of the FEM method are1. The desired level of approximation can be achieved by furthersubdividing the structure.2. The resulting equations are only loosely coupled, leading to aneasier computer solution.3. For a particular type of finite element (e.g., beam, solid, etc)the procedure to derive the displacement field and the elementcharacteristics does not depend on the particular geometry ofthe elements, and can easily be implemented in a computerprogram.Characteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotion
Writing the equation of motionDynamics ofStructuresGiacomo BoffiIn a deterministic dynamic analysis, given a prescribed load, we wantto evaluate the displacements in each instant of time.In many cases a limited number of DDOFs gives a sufficientaccuracy; further, the dynamic problem can be reduced to thedetermination of the time-histories of some selected component ofthe response.The mathematical expressions, ordinary or partial differentialequations, that we are going to write express the dynamicequilibrium of the structural system and are known as the Equationsof Motion (EOM).The solution of the EOM gives the requested displacements.The formulation of the EOM is the most important, often the mostdifficult part of a dynamic analysis.Writing the EOM, cont.IntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionDynamics ofStructuresGiacomo BoffiIntroductionCharacteristics ofa DynamicalProblemWe have a choice of techniques to help us in writing the EOM,namely:Ithe D’Alembert Principle,Ithe Principle of Virtual Displacements,Ithe Variational Approach.D’Alembert principleBy Newton’s II law of motion, for any particle the rate of change ofmomentum is equal to the external force,dd u p (t) (m ),dtdtwhere u (t) is the particle displacement.In structural dynamics, we may regard the mass as a constant, andthus writep (t) m u ,where each operation of differentiation with respect to time isdenoted with a dot.If we writep (t) m u 0and interpret the term m u as an Inertial Force that contrasts theacceleration of the particle, we have an equation of equilibrium forthe particle.Formulation of aDynamicalProblemFormulation ofthe equations ofmotionDynamics ofStructuresGiacomo BoffiIntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotion
D’Alembert principle, cont.Dynamics ofStructuresGiacomo BoffiIntroductionThe concept that a mass develops an inertial force opposing itsacceleration is known as the D’Alembert principle, and using thisprinciple we can write the EOM as a simple equation of equilibrium.The term p (t) must comprise each different force acting on theparticle, including the reactions of kinematic or elastic constraints,internal forces and external, autonomous forces.In many simple problems, D’Alembert principle is the most directand convenient method for the formulation of the EOM.Principle of virtual displacementsCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionDynamics ofStructuresGiacomo BoffiIntroductionCharacteristics ofa DynamicalProblemIn a reasonably complex dynamic system, with e.g. articulated rigidbodies and external/internal constraints, the direct formulation ofthe EOM using D’Alembert principle may result difficult.In these cases, application of the Principle of Virtual Displacementsis very convenient, because the reactive forces do not enter theequations of motion, that are directly written in terms of themotions compatible with the restraints/constraints of the system.Principle of Virtual Displacements, cont.Formulation of aDynamicalProblemFormulation ofthe equations ofmotionDynamics ofStructuresGiacomo BoffiFor example, considering an assemblage of rigid bodies, the pvdstates that necessary and sufficient condition for equilibrium is that,for every virtual displacement (i.e., any infinitesimal displacementcompatible with the restraints) the total work done by all theexternal forces is zero.For an assemblage of rigid bodies, writing the EOM requires1. to identify all the external forces, comprising the inertial forces,and to express their values in terms of the ddof;2. to compute the work done by these forces for different virtualdisplacements, one for each ddof;3. to equate to zero all these work expressions.The pvd is particularly convenient also because we have only scalarequations, even if the forces and displacements are of vectorialnature.IntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotion
Variational approachDynamics ofStructuresGiacomo BoffiVariational approaches do not consider directly the forces acting onthe dynamic system, but are concerned with the variations of kineticand potential energy and lead, as well as the pvd, to a set of scalarequations.For example, the equation of motion of a generical system can bederived in terms of the Lagrangian function, L T V, T and Vbeing, respectively, the kinetic and the potential energy of thesystem expressed in terms of a vector q of indipendent coordinates Ld L ,i 1, . . . , N.dt q̇i qiIntroductionCharacteristics ofa DynamicalProblemFormulation of aDynamicalProblemFormulation ofthe equations ofmotionThe method to be used in a particular problem is mainly a matter ofconvenience and, for some measure, of personal taste.1 DOF SystemDynamics ofStructuresGiacomo BoffiStructural dynamics is all about the motion of a system in theneighborhood of a point of equilibrium.We’ll start by studying the most simple of systems, a single degree offreedom system, without external forces, subjected to a perturbationof the equilibrium.If our system has a constant mass m and it’s subjected to a generical,non-linear, internal force F F (y , ẏ ), where y is the displacementand ẏ the velocity of the particle, the equation of motion is1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped system1F (y , ẏ ) f (y , ẏ ).mÿ It is difficult to integrate the above equation in the general case, butit’s easy when the motion occurs in a small neighborhood of theequilibrium position.1 DOF System, cont.Dynamics ofStructuresGiacomo BoffiIn a position of equilibrium, yeq. , the velocity and the accelerationare zero, and hence f (yeq. , 0) 0.The force can be linearized in a neighborhood of yeq. , 0: f f(y yeq. ) (ẏ 0) O(y , ẏ ).f (y , ẏ ) f (yeq. , 0) y ẏAssuming that O(y , ẏ ) is small in a neighborhood of yeq. , we canwrite the equation of motionẍ aẋ bx 0where x y yeq. , a f ẏ ẏ 0and b f y y y .eqIn an infinitesimal neighborhood of yeq. , the equation of motion canbe studied in terms of a linear differential equation of second order.1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped system
1 DOF System, cont.Dynamics ofStructuresA linear constant coefficient differential equation has the integralx A exp(st), that substituted in the equation of motion gives2s as b 0whose solutions ares1,2a 2rGiacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systema2 b.4The general integral isx(t) A1 exp(s1 t) A2 exp(s2 t).Given that for a free vibration problem A1 , A2 are given by the initialconditions, the nature of the solution depends on the sign of the realpart of s1 , s2 , because.1 DOF System, cont.Dynamics ofStructuresGiacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemIf we write si ri ıqi , then we haveexp(si t) exp(ıqi t) exp(ri t).Free vibrations ofa damped systemIf one of the ri 0, the response grows infinitely over time, even foran infinitesimal perturbation of the equilibrium, so that in this casewe have an unstable equilibrium.If both ri 0, the response decrease over time, so we have a stableequilibrium.Finally, if both ri 0 the s’s are imaginary, the response is harmonicwith constant amplitude.1 DOF System, cont.Dynamics ofStructuresGiacomo Boffi1 DOF SystemThe roots beingFree vibrations ofa SDOF systems1,2a 2ra2 b,4Iif a 0 and b 0 both roots are negative or complex conjugatewith negative real part, the system is asymptotically stable,Iif a 0 and b 0, the roots are purely imaginary, theequilibrium is indifferent, the oscillations are harmonic,Iif a 0 or b 0 at least one of the roots has a positive realpart, and the system is unstable.Free vibrations ofa damped system
The famous box carDynamics ofStructuresIn a single degree of freedom (sdof) system each property, m, a andb, can be conveniently represented in a single physical elementI The entire mass, m, is concentrated in a rigid block, its positioncompletely described by the coordinate x(t).I The energy-loss (the a ẋ term) is represented by a masslessdamper, its damping constant being c.I The elastic resistance to displacement (b x) is provided by amassless spring of stiffness kI Eventually we can introduce an external loading too, thetime-varying force p(t).x(t)Giacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systemxkmp(t)cfS (t)fD (t)(a)fI (t)p(t)(b)Equation of motion of the basic dynamic systemDynamics ofStructuresGiacomo Boffix(t)1 DOF Systemxkmcp(t)fS (t)fD (t)(a)Free vibrations ofa SDOF systemfI (t)p(t)Free vibrations ofa damped system(b)The equation of motion can be written using the D’Alembert Principle,expressing the equilibrium of all the forces acting on the mass includingthe inertial force.The forces are the external force, p(t), positive in the direction of motionand the resisting forces, i.e., the inertial force fI (t), the damping forcefD (t) and the elastic force, fS (t), that are opposite to the direction of theacceleration, velocity and displacement.The equation of motion, merely expressing the equilibrium of these forces,writing the resisting forces and the external force across the equal signfI (t) fD (t) fS (t) p(t)EOM of the basic dynamic system, cont.Dynamics ofStructuresGiacomo BoffiAccording to D’Alembert principle, the inertial force is the product ofthe mass and accelerationfI (t) m ẍ(t).Assuming a viscous damping mechanism, the damping force is theproduct of the damping constant c and the velocity,fD (t) c ẋ(t).Finally, the elastic force is the product of the elastic stiffness k andthe displacement,fS (t) k x(t).1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped system
EOM of the basic dynamic system, cont.Dynamics ofStructuresGiacomo BoffiThe differential equation of dynamic equilibrium1 DOF SystemFree vibrations ofa SDOF systemfI (t) fD (t) fS (t) Free vibrations ofa damped systemm ẍ(t) c ẋ(t) k x(t) p(t).The resisting forces in the EoMfI (t) fD (t) fS (t) p(t)are proportional to the deflection x(t) or one of its time derivatives,ẋ(t), ẍ(t).The equation of motion is a linear differential equation of the secondorder, with constant coefficients.The resisting forces are, by convention, positive when opposite to thedirection of motion, i.e., resisting the motion.Influence of static forcesDynamics ofStructuresGiacomo Boffi stx̄(t)1 DOF Systemxx(t)kmcFree vibrations ofa SDOF systemk stp(t) fS (t)fD (t)WfI (t)(a)p(t)Free vibrations ofa damped system(b)Considering the presence of a constant force W , the EOM ism ẍ(t) c ẋ(t) k x(t) p(t) W .Expressing the displacement as the sum of a constant, staticdisplacement and a dynamic displacement,x(t) st x̄(t),and substituting in the EOM we havem ẍ(t) c ẋ(t) k st k x̄(t) p(t) W .Influence of static forces, cont.Dynamics ofStructuresGiacomo BoffiRecognizing that k st W (so that the two terms, on oppositesides of the equal sign, cancel each other), that ẋ x̄ and thatẍ x̄ the EOM can be written as m x̄ (t) c x̄(t) k x̄(t) p(t).The equation of motion expressed with reference to the staticequilibrium position is not affected by static forces.For this reasons, all displacements in further discussions will bereferenced from the equilibrium position and denoted, for simplicity,with x(t).Note that the total displacements, stresses. etc. areinfluenced by the static forces, and must be computedusing the superposition of effects.1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped system
Influence of support motionDynamics ofStructuresGiacomo BoffiFixed reference axisxtot (t)1 DOF Systemmk2Free vibrations ofa SDOF systemFree vibrations ofa damped systemk2cx( t)xg (t)Displacements, deformations and stresses in a structure are induced alsoby a motion of its support.Important examples of support motion are the motion of a buildingfoundation due to earthquake and the motion of the base of a piece ofequipment due to vibrations of the building in which it is housed.Influence of support motion, cont.Fixed reference axisxtot (t)mk2k2Dynamics ofStructuresConsidering a support motion xg (t),defined with respect to a inertial frame ofreference, the total displacement isxtot (t) xg (t) x(t)cGiacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systemand the total acceleration isxg (t)x( t)ẍtot (t) ẍg (t) ẍ(t).While the elastic and damping forces are still proportional to relativedisplacements and velocities, the inertial force is proportional to the totalacceleration,fI (t) mẍtot (t) mẍg (t) mẍ(t).Writing the EOM for a null external load, p(t) 0, is hencem ẍtot (t) c ẋ(t) k x(t) 0,or,m ẍ(t) c ẋ(t) k x(t) m ẍg (t) peff (t).Support motion is sufficient to excite a dynamic system: peff (t) m ẍg (t).Free VibrationsDynamics ofStructuresGiacomo BoffiThe equation of motion,m ẍ(t) c ẋ(t) k x(t) p(t)is a linear differential equation of the second order, with constantcoefficients.Its solution can be expressed in terms of a superposition of aparticular solution, depending on p(t), and a free vibration solution,that is the solution of the so called homogeneous problem, wherep(t) 0.In the following, we will study the solution of the homogeneousproblem, the so-called homogeneous or complementary solution, thatis the free vibrations of the SDOF after a perturbation of theposition of equilibrium.1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped system
Free vibrations of an undamped systemDynamics ofStructuresAn undamped system, where c 0 and no energy dissipation takesplace, is just an ideal notion, as it would be a realization of motusperpetuum. Nevertheless, it is an useful idealization. In this case,the homogeneous equation of motion isGiacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systemm ẍ(t) k x(t) 0which solution is of the form exp st; substituting this solution in theabove equation we have(k s 2 m) exp st 0noting that exp st 6 0, we finally have2(k s m) 0 s r kmAs m and k are positive quantities, s must be purely imaginary.Undamped Free VibrationsDynamics ofStructuresGiacomo BoffiIntroducing the natural circular frequency ωnω2nk ,m1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systemthe solution of the algebraic equation in s isrrq kks 1 i ω2n iωnmm where i 1 and the general integral of the homogeneousequation isx(t) G1 exp(iωn t) G2 exp( iωn t).The solution has an imaginary part?Undamped Free VibrationsThe solution is derived from the general integral imposing the (real)initial conditionsx(0) x0 ,ẋ(0) ẋ0Evaluating x(t) for t 0 and substituting in (39), we have G1 G2 x0iωn G1 iωn G2 ẋ0Solving the linear system we haveix0 ẋ0 /ωnix0 ẋ0 /ωnG1 ,G2 ,2i2isubstituting these values in the general solution and collecting x0and ẋ0 , we finally findexp(iωn t) exp( iωn t)exp(iωn t) exp( iωn t) ẋ0x(t) x0 22iωnDynamics ofStructuresGiacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped system
Undamped Free VibrationsDynamics ofStructuresGiacomo Boffi1 DOF Systemexp(iωn t) exp( iωn t) ẋ0exp(iωn t) exp( iωn t)x0 x(t) 22iωnFree vibrations ofa SDOF systemFree vibrations ofa damped systemUsing the Euler formulas relating the imaginary argumentexponentials and the trigonometric functions, can be rewritten interms of the elementary trigonometric functionsx(t) x0 cos(ωn t) (ẋo /ωn ) sin(ωn t).Considering that for every conceivable initial conditions we can usethe above representation, it is indifferent, and perfectly equivalent,to represent the general integral either in the form of exponentials ofimaginary argument or as a linear combination of sine and cosine ofcircular frequency ωnUndamped Free VibrationsDynamics ofStructuresGiacomo BoffiOtherwise, using the identity exp( iωn t) cos ωn t i sin ωn tx(t) (A iB) (cos ωn t i sin ωn t) (C iD) (cos ωn t i sin ωn t)expanding the product and evidencing the imaginary part of the responsewe have1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systemI(x) i (A sin ωn t B cos ωn t C sin ωn t D cos ωn t) .Imposing that I(x) 0, i.e., that the response is real, we have(A C ) sin ωn t (B D) cos ωn t 0 C A, D B.Substituting in x(t) eventually we havex(t) 2A cos(ωn t) 2B sin(ωn t).Undamped Free VibrationsDynamics ofStructuresGiacomo Boffi1 DOF SystemOur preferred representation of the general integral of undampedfree vibrations isx(t) A cos(ωn t) B sin(ωn t)For the usual initial conditions, we have already seen thatA x0 ,B ẋ0.ωnFree vibrations ofa SDOF systemFree vibrations ofa damped system
Undamped Free VibrationsDynamics ofStructuresGiacomo BoffiSometimes we prefer to write x(t) as a single harmonic, introducinga phase difference φ so that the amplitude of the motion, C , is putin evidence:1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systemx(t) C cos(ωn t ϕ) C (cos ωn t cos ϕ sin ωn t sin ϕ) A cos ωn t B sin ωn tFrom A C cos ϕ and B C sin ϕ we have tan ϕ B/A, from22222A B C (cos ϕ sin ϕ) we have C A2 B 2 andeventually pC A2 B 2x(t) C cos(ωn t ϕ), withϕ arctan(B/A)Undamped Free VibrationsDynamics ofStructuresGiacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemx(t)arctan ẋ0Free vibrations ofa damped systemρx0t ωθT 2πωIt is worth noting that the coefficients A, B and C have thedimension of a length, the coefficient ωn has the dimension of thereciprocal of time and that the coefficient ϕ is an angle, or in otherterms is adimensional.Behavior of Damped SystemsDynamics ofStructuresGiacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systemThe viscous damping modifies the response of a sdof systemintroducing a decay in the amplitude of the response. Depending onthe amount of damping, the response can be oscillatory or not. Theamount of damping that separates the two behaviors is denoted ascritical damping.Under-criticallydamped SDOFCritically dampedSDOFOver-criticallydamped SDOF
The solution of the EOMThe equation of motion for a free vibrating damped system isDynamics ofStructuresGiacomo Boffi1 DOF Systemm ẍ(t) c ẋ(t) k x(t) 0,substituting the solution exp st in the preceding equation andsimplifying, we have that the parameter s must satisfy the equationm s2 c s k 0Free vibrations ofa SDOF systemFree vibrations ofa damped systemUnder-criticallydamped SDOFCritically dampedSDOFOver-criticallydamped SDOFor, after dividing both members by m,cs 2 s ω2n 0mwhose solutions are s r 2 2ccccs ω2n ωn 1 .2m2m2mωn2mωnCritical DampingDynamics ofStructuresGiacomo Boffi1 DOF SystemThe behavior of the solution of the free vibration problem depends of 2ccourse on the sign of the radicand 2mω 1:n 0 the roots s are complex conjugate, 0 the roots are identical, double root,Free vibrations ofa SDOF systemFree vibrations ofa damped systemUnder-criticallydamped SDOFCritically dampedSDOFOver-criticallydamped SDOF 0 the roots are real.The value of c that make the radicand equal to zero is known as thecritical damping, ccr 2mωn 2 mk.Critical DampingDynamics ofStructuresGiacomo Boffi1 DOF SystemFree vibrations ofa SDOF systemA single degree of freedom system is denoted as critically damped,under-critically damped or over-critically damped depending on thevalue of the damping coefficient with respect to the critical damping.Typical building structures are undercritically damped.Free vibrations ofa damped systemUnder-criticallydamped SDOFCritically dampedSDOFOver-criticallydamped SDOF
Damping RatioDynamics ofStructuresGiacomo BoffiIf we introduce the ratio of the damping to the critical damping, orcritical damping ratio ζ,ccζ ,ccr2mωnc ζccr 2ζωn mthe equation of free vibrations can be rewritten as1 DOF SystemFree vibrations ofa SDOF systemFree vibrations ofa damped systemUnder-criticallydamped SDOFCritically dampedSDOFOver-criticallydamped SDOFẍ(t) 2ζωn ẋ(t) ω2n x(t) 0and the roots s1,2 can be rewritten ass ζωn ωnpζ2 1.Free Vibrations of Under-critically Damped SystemsDynamics ofStructuresGiacomo BoffiWe start studying the free vibration response of under-criticallydamped SDOF, as this is
equations, that we are going to write express the dynamic equilibrium of the structural system and are known as the Equations of Motion (EOM). The solution of the EOM gives the requested displacements. The formulation of the EOM is the most important, often the most di cult part of a dynamic analysis. Dynami
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