Our focus will be on deterministjc analysis Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Types of Dynamic Analysis Taking into account linear systems only, we must consider two difgerent defjnitjons of the loading to defjne two types of dynamic analysis Deterministjc Analysis applies when the tjme variatjon of the loading is fully known and we can determine the complete tjme variatjon of all the required response quantjtjes Non‐deterministjc Analysis applies when the tjme variatjon of the loading is essentjally random and is known only in terms of some statjstjcs In a non‐deterministjc or stochastjc analysis the structural response can be known only in terms of statjstjcs of the response quantjtjes
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Types of Dynamic Analysis Taking into account linear systems only, we must consider two difgerent defjnitjons of the loading to defjne two types of dynamic analysis Deterministjc Analysis applies when the tjme variatjon of the loading is fully known and we can determine the complete tjme variatjon of all the required response quantjtjes Non‐deterministjc Analysis applies when the tjme variatjon of the loading is essentjally random and is known only in terms of some statjstjcs In a non‐deterministjc or stochastjc analysis the structural response can be known only in terms of statjstjcs of the response quantjtjes Our focus will be on deterministjc analysis
the loading can be described in terms of an analytjcal functjon, e.g., 𝑞(𝑢) = 𝑞 𝑝 exp (𝛽𝑢) the loading is measured experimentally, hence it is known only in a discrete set of instants; in this case, we say that we know the loading tjme‐history . Periodic Loadings a periodic loading repeats itself with a fjxed period 𝑈 : 𝑞(𝑢) = 𝑞 0 𝑔(𝑢) with 𝑔(𝑢) ≡ 𝑔(𝑢 + 𝑈) Non Periodic Loadings here we see two sub‐cases, Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Types of Dynamic Loading Dealing with deterministjc dynamic loadings we will study, in order of complexity, Harmonic Loadings a force is modulated by a harmonic functjon of tjme, characterized by a frequency 𝜕 and a phase 𝜒 : 𝑞(𝑢) = 𝑞 0 sin (𝜕𝑢 − 𝜒)
the loading can be described in terms of an analytjcal functjon, e.g., 𝑞(𝑢) = 𝑞 𝑝 exp (𝛽𝑢) the loading is measured experimentally, hence it is known only in a discrete set of instants; in this case, we say that we know the loading tjme‐history . Non Periodic Loadings here we see two sub‐cases, Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Types of Dynamic Loading Dealing with deterministjc dynamic loadings we will study, in order of complexity, Harmonic Loadings a force is modulated by a harmonic functjon of tjme, characterized by a frequency 𝜕 and a phase 𝜒 : 𝑞(𝑢) = 𝑞 0 sin (𝜕𝑢 − 𝜒) Periodic Loadings a periodic loading repeats itself with a fjxed period 𝑈 : 𝑞(𝑢) = 𝑞 0 𝑔(𝑢) with 𝑔(𝑢) ≡ 𝑔(𝑢 + 𝑈)
the loading can be described in terms of an analytjcal functjon, e.g., 𝑞(𝑢) = 𝑞 𝑝 exp (𝛽𝑢) the loading is measured experimentally, hence it is known only in a discrete set of instants; in this case, we say that we know the loading tjme‐history . Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Types of Dynamic Loading Dealing with deterministjc dynamic loadings we will study, in order of complexity, Harmonic Loadings a force is modulated by a harmonic functjon of tjme, characterized by a frequency 𝜕 and a phase 𝜒 : 𝑞(𝑢) = 𝑞 0 sin (𝜕𝑢 − 𝜒) Periodic Loadings a periodic loading repeats itself with a fjxed period 𝑈 : 𝑞(𝑢) = 𝑞 0 𝑔(𝑢) with 𝑔(𝑢) ≡ 𝑔(𝑢 + 𝑈) Non Periodic Loadings here we see two sub‐cases,
the loading is measured experimentally, hence it is known only in a discrete set of instants; in this case, we say that we know the loading tjme‐history . Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Types of Dynamic Loading Dealing with deterministjc dynamic loadings we will study, in order of complexity, Harmonic Loadings a force is modulated by a harmonic functjon of tjme, characterized by a frequency 𝜕 and a phase 𝜒 : 𝑞(𝑢) = 𝑞 0 sin (𝜕𝑢 − 𝜒) Periodic Loadings a periodic loading repeats itself with a fjxed period 𝑈 : 𝑞(𝑢) = 𝑞 0 𝑔(𝑢) with 𝑔(𝑢) ≡ 𝑔(𝑢 + 𝑈) Non Periodic Loadings here we see two sub‐cases, the loading can be described in terms of an analytjcal functjon, e.g., 𝑞(𝑢) = 𝑞 𝑝 exp (𝛽𝑢)
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Types of Dynamic Loading Dealing with deterministjc dynamic loadings we will study, in order of complexity, Harmonic Loadings a force is modulated by a harmonic functjon of tjme, characterized by a frequency 𝜕 and a phase 𝜒 : 𝑞(𝑢) = 𝑞 0 sin (𝜕𝑢 − 𝜒) Periodic Loadings a periodic loading repeats itself with a fjxed period 𝑈 : 𝑞(𝑢) = 𝑞 0 𝑔(𝑢) with 𝑔(𝑢) ≡ 𝑔(𝑢 + 𝑈) Non Periodic Loadings here we see two sub‐cases, the loading can be described in terms of an analytjcal functjon, e.g., 𝑞(𝑢) = 𝑞 𝑝 exp (𝛽𝑢) the loading is measured experimentally, hence it is known only in a discrete set of instants; in this case, we say that we know the loading tjme‐history .
A dynamic analysis is required only when the inertjal forces represent a signifjcant portjon of the total load. On the other hand, if the loads and the defmectjons are varying slowly , a statjc analysis will provide an acceptable approximatjon. We will defjne slowly Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Characteristjcs of a Dynamical Problem A dynamical problem is essentjally characterized by the relevance of inertjal forces , arising from the accelerated motjon of structural or serviced masses.
On the other hand, if the loads and the defmectjons are varying slowly , a statjc analysis will provide an acceptable approximatjon. We will defjne slowly Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Characteristjcs of a Dynamical Problem A dynamical problem is essentjally characterized by the relevance of inertjal forces , arising from the accelerated motjon of structural or serviced masses. A dynamic analysis is required only when the inertjal forces represent a signifjcant portjon of the total load.
We will defjne slowly Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Characteristjcs of a Dynamical Problem A dynamical problem is essentjally characterized by the relevance of inertjal forces , arising from the accelerated motjon of structural or serviced masses. A dynamic analysis is required only when the inertjal forces represent a signifjcant portjon of the total load. On the other hand, if the loads and the defmectjons are varying slowly , a statjc analysis will provide an acceptable approximatjon.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Characteristjcs of a Dynamical Problem A dynamical problem is essentjally characterized by the relevance of inertjal forces , arising from the accelerated motjon of structural or serviced masses. A dynamic analysis is required only when the inertjal forces represent a signifjcant portjon of the total load. On the other hand, if the loads and the defmectjons are varying slowly , a statjc analysis will provide an acceptable approximatjon. We will defjne slowly
In many cases it is however possible to simplify the formulatjon of the structural dynamic problem to ordinary difgerentjal equatjons. Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Formulatjon of a Dynamical Problem In a structural system the inertjal forces depend on the tjme derivatjves of displacements while the elastjc forces, equilibratjng the inertjal ones, depend on the spatjal derivatjves of the displacements. ... the natural statement of the problem is hence in terms of partjal difgerentjal equatjons .
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Formulatjon of a Dynamical Problem In a structural system the inertjal forces depend on the tjme derivatjves of displacements while the elastjc forces, equilibratjng the inertjal ones, depend on the spatjal derivatjves of the displacements. ... the natural statement of the problem is hence in terms of partjal difgerentjal equatjons . In many cases it is however possible to simplify the formulatjon of the structural dynamic problem to ordinary difgerentjal equatjons.
Under this assumptjon, the analytjcal problem is greatly simplifjed: 1 the inertjal forces are applied only at the lumped masses, 2 the only defmectjons that infmuence the inertjal forces are the defmectjons of the lumped masses, 3 using methods of statjc analysis we can determine those defmectjons, thus consentjng the formulatjon of the problem in terms of a set of ordinary difgerentjal equatjons, one for each relevant component of the inertjal forces. Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Lumped Masses In many structural problems we can say that the masses are concentrated in a discrete set of lumped masses (e.g., in a multj‐storey building most of the masses is concentrated at the level of the storeys’fmoors) .
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Lumped Masses In many structural problems we can say that the masses are concentrated in a discrete set of lumped masses (e.g., in a multj‐storey building most of the masses is concentrated at the level of the storeys’fmoors) . Under this assumptjon, the analytjcal problem is greatly simplifjed: 1 the inertjal forces are applied only at the lumped masses, 2 the only defmectjons that infmuence the inertjal forces are the defmectjons of the lumped masses, 3 using methods of statjc analysis we can determine those defmectjons, thus consentjng the formulatjon of the problem in terms of a set of ordinary difgerentjal equatjons, one for each relevant component of the inertjal forces.
Of course, a contjnuous system has an infjnite number of degrees of freedom. Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Dynamic Degrees of Freedom The dynamic degrees of freedom (DDOF) in a discretjzed system are the displacements components of the lumped masses associated with the relevant components of the inertjal forces. If a lumped mass can be regarded as a point mass then at most 3 translatjonal DDOFs will suffjce to represent the associated inertjal force. On the contrary, if a lumped mass has a discrete volume its inertjal force depends also on its rotatjons (inertjal couples) and we need at most 6 DDOFs to represent the mass defmectjons and the inertjal force.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Dynamic Degrees of Freedom The dynamic degrees of freedom (DDOF) in a discretjzed system are the displacements components of the lumped masses associated with the relevant components of the inertjal forces. If a lumped mass can be regarded as a point mass then at most 3 translatjonal DDOFs will suffjce to represent the associated inertjal force. On the contrary, if a lumped mass has a discrete volume its inertjal force depends also on its rotatjons (inertjal couples) and we need at most 6 DDOFs to represent the mass defmectjons and the inertjal force. Of course, a contjnuous system has an infjnite number of degrees of freedom.
When the masses are distributed we can simplify our problem expressing the defmectjons in terms of a linear combinatjon of assigned functjons of positjon, the coeffjcients of the linear combinatjon being the generalized coordinates (e..g., the defmectjons of a rectjlinear beam can be expressed in terms of a trigonometric series). Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Generalized Displacements The lumped mass procedure that we have outlined is efgectjve if a large proportjon of the total mass is concentrated in a few points (e.g., in a multj‐storey building one can consider a lumped mass for each storey).
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Generalized Displacements The lumped mass procedure that we have outlined is efgectjve if a large proportjon of the total mass is concentrated in a few points (e.g., in a multj‐storey building one can consider a lumped mass for each storey). When the masses are distributed we can simplify our problem expressing the defmectjons in terms of a linear combinatjon of assigned functjons of positjon, the coeffjcients of the linear combinatjon being the generalized coordinates (e..g., the defmectjons of a rectjlinear beam can be expressed in terms of a trigonometric series).
Even if the method of generalized coordinates has its beauty, we must recognize that for each difgerent problem we have to derive an ad hoc formulatjon, with an evident loss of generality. Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Generalized Displacements, cont. To fully describe a displacement fjeld, we need to combine an infjnity of linearly independent base functjons , but in practjce a good approximatjon can be achieved using just a small number of functjons and degrees of freedom.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Generalized Displacements, cont. To fully describe a displacement fjeld, we need to combine an infjnity of linearly independent base functjons , but in practjce a good approximatjon can be achieved using just a small number of functjons and degrees of freedom. Even if the method of generalized coordinates has its beauty, we must recognize that for each difgerent problem we have to derive an ad hoc formulatjon, with an evident loss of generality.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Finite Element Method The fjnite elements method (FEM) combines aspects of lumped mass and generalized coordinates methods, providing a simple and reliable method of analysis, that can be easily programmed on a digital computer.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Finite Element Method In the FEM, the structure is subdivided in a number of non‐overlapping pieces, called the fjnite elements , delimited by common nodes . The FEM uses piecewise approximatjons (i.e., local to each element) to the fjeld of displacements. In each element the displacement fjeld is derived from the displacements of the nodes that surround each partjcular element, using interpolatjng functjons . The displacement, deformatjon and stress fjelds in each element, as well as the inertjal forces, can thus be expressed in terms of the unknown nodal displacements . The nodal displacements are the dynamical DOFs of the FEM model.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Finite Element Method Some of the most prominent advantages of the FEM method are 1 The desired level of approximatjon can be achieved by further subdividing the structure. 2 The resultjng equatjons are only loosely coupled, leading to an easier computer solutjon. 3 For a partjcular type of fjnite element (e.g., beam, solid, etc) the procedure to derive the displacement fjeld and the element characteristjcs does not depend on the partjcular geometry of the elements, and can easily be implemented in a computer program.
The mathematjcal expressions, ordinary or partjal difgerentjal equatjons, that we are going to write express the dynamic equilibrium of the structural system and are known as the Equatjons of Motjon (EOM). The solutjon of the EOM gives the requested displacements. The formulatjon of the EOM is the most important, ofuen the most diffjcult part of a dynamic analysis. Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Writjng the equatjon of motjon In a deterministjc dynamic analysis, given a prescribed load, we want to evaluate the displacements in each instant of tjme. In many cases a limited number of DDOFs gives a suffjcient accuracy; further, the dynamic problem can be reduced to the determinatjon of the tjme‐histories of some selected component of the response.
The solutjon of the EOM gives the requested displacements. The formulatjon of the EOM is the most important, ofuen the most diffjcult part of a dynamic analysis. Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Writjng the equatjon of motjon In a deterministjc dynamic analysis, given a prescribed load, we want to evaluate the displacements in each instant of tjme. In many cases a limited number of DDOFs gives a suffjcient accuracy; further, the dynamic problem can be reduced to the determinatjon of the tjme‐histories of some selected component of the response. The mathematjcal expressions, ordinary or partjal difgerentjal equatjons, that we are going to write express the dynamic equilibrium of the structural system and are known as the Equatjons of Motjon (EOM).
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Writjng the equatjon of motjon In a deterministjc dynamic analysis, given a prescribed load, we want to evaluate the displacements in each instant of tjme. In many cases a limited number of DDOFs gives a suffjcient accuracy; further, the dynamic problem can be reduced to the determinatjon of the tjme‐histories of some selected component of the response. The mathematjcal expressions, ordinary or partjal difgerentjal equatjons, that we are going to write express the dynamic equilibrium of the structural system and are known as the Equatjons of Motjon (EOM). The solutjon of the EOM gives the requested displacements. The formulatjon of the EOM is the most important, ofuen the most diffjcult part of a dynamic analysis.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Writjng the EOM, cont. We have a choice of techniques to help us in writjng the EOM, namely: the D’Alembert Principle, the Principle of Virtual Displacements, the Variatjonal Approach.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon D’Alembert principle By Newton’s II law of motjon, for any partjcle the rate of change of momentum is equal to the external force, 𝑞(𝑢) = 𝑒 𝑒𝑢(𝑛𝑒 ⃗ 𝑣 ⃗ 𝑒𝑢 ), where ⃗ 𝑣(𝑢) is the partjcle displacement. In structural dynamics, we may regard the mass as a constant, and thus write 𝑞(𝑢) = 𝑛 ̈ ⃗ 𝑣, ⃗ where each operatjon of difgerentjatjon with respect to tjme is denoted with a dot. If we write 𝑞(𝑢) − 𝑛 ̈ ⃗ 𝑣 = 0 ⃗ and interpret the term −𝑛 ̈ 𝑣 as an Inertjal Force that contrasts the ⃗ acceleratjon of the partjcle, we have an equatjon of equilibrium for the partjcle.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon D’Alembert principle, cont. The concept that a mass develops an inertjal force opposing its acceleratjon is known as the D’Alembert principle, and using this principle we can write the EOM as a simple equatjon of equilibrium. The term ⃗ 𝑞(𝑢) must comprise each difgerent force actjng on the partjcle, including the reactjons of kinematjc or elastjc constraints, internal forces and external, autonomous forces. In many simple problems, D’Alembert principle is the most direct and convenient method for the formulatjon of the EOM .
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Principle of virtual displacements In a reasonably complex dynamic system, with e.g. artjculated rigid bodies and external/internal constraints, the direct formulatjon of the EOM using D’Alembert principle may result diffjcult. In these cases, applicatjon of the Principle of Virtual Displacements is very convenient, because the reactjve forces do not enter the equatjons of motjon, that are directly writuen in terms of the motjons compatjble with the restraints/constraints of the system.
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Principle of Virtual Displacements, cont. For example, considering an assemblage of rigid bodies, the pvd states that necessary and suffjcient conditjon for equilibrium is that, for every virtual displacement (i.e., any infjnitesimal displacement compatjble with the restraints) the total work done by all the external forces is zero. For an assemblage of rigid bodies, writjng the EOM requires 1 to identjfy all the external forces, comprising the inertjal forces, and to express their values in terms of the ddof ; 2 to compute the work done by these forces for difgerent virtual displacements, one for each ddof ; 3 to equate to zero all these work expressions. The pvd is partjcularly convenient also because we have only scalar equatjons, even if the forces and displacements are of vectorial nature.
The method to be used in a partjcular problem is mainly a matuer of convenience (and, to some extent, of personal taste). Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Variatjonal approach Variatjonal approaches do not consider directly the forces actjng on the dynamic system, but are concerned with the variatjons of kinetjc and potentjal energy and lead, as well as the pvd , to a set of scalar equatjons. For example, the equatjon of motjon of a generical system can be derived in terms of the Lagrangian functjon , ℒ = 𝑈 − 𝑊 where 𝑈 and 𝑊 are, respectjvely, the kinetjc and the potentjal energy of the system expressed in terms of a vector ⃗ 𝑟 of indipendent coordinates 𝑒𝑢 � 𝜖𝑀 𝑒 � = 𝜖𝑀 , 𝑗 = 1, … , N . 𝜖 ̇ 𝑟 𝑗 𝜖𝑟 𝑗
Introductjon Characteristjcs of a Dynamical Problem Formulatjon of a Dynamical Problem Formulatjon of the equatjons of motjon Variatjonal approach Variatjonal approaches do not consider directly the forces actjng on the dynamic system, but are concerned with the variatjons of kinetjc and potentjal energy and lead, as well as the pvd , to a set of scalar equatjons. For example, the equatjon of motjon of a generical system can be derived in terms of the Lagrangian functjon , ℒ = 𝑈 − 𝑊 where 𝑈 and 𝑊 are, respectjvely, the kinetjc and the potentjal energy of the system expressed in terms of a vector ⃗ 𝑟 of indipendent coordinates 𝑒𝑢 � 𝜖𝑀 𝑒 � = 𝜖𝑀 , 𝑗 = 1, … , N . 𝜖 ̇ 𝑟 𝑗 𝜖𝑟 𝑗 The method to be used in a partjcular problem is mainly a matuer of convenience (and, to some extent, of personal taste).
It is diffjcult to integrate the above equatjon in the general case, but it’s easy when the motjon occurs in a small neighborhood of the equilibrium positjon. ̈ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system 1 DOF System Structural dynamics is all about the motjon of a system in the neighborhood of a point of equilibrium. We’ll start by studying the most simple of systems, a single degree of freedom system, without external forces, subjected to a perturbatjon of the equilibrium. If our system has a constant mass 𝑛 and it’s subjected to a generical, non‐linear, internal force 𝐺 = 𝐺(𝑧, ̇ 𝑧) , where 𝑧 is the displacement and ̇ 𝑧 the velocity of the partjcle, the equatjon of motjon is 𝑧 = 1 𝑛𝐺(𝑧, ̇ 𝑧) = 𝑔(𝑧, ̇ 𝑧).
̈ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system 1 DOF System Structural dynamics is all about the motjon of a system in the neighborhood of a point of equilibrium. We’ll start by studying the most simple of systems, a single degree of freedom system, without external forces, subjected to a perturbatjon of the equilibrium. If our system has a constant mass 𝑛 and it’s subjected to a generical, non‐linear, internal force 𝐺 = 𝐺(𝑧, ̇ 𝑧) , where 𝑧 is the displacement and ̇ 𝑧 the velocity of the partjcle, the equatjon of motjon is 𝑧 = 1 𝑛𝐺(𝑧, ̇ 𝑧) = 𝑔(𝑧, ̇ 𝑧). It is diffjcult to integrate the above equatjon in the general case, but it’s easy when the motjon occurs in a small neighborhood of the equilibrium positjon.
In an infjnitesimal neighborhood of 𝑧 eq. , the equatjon of motjon can be studied in terms of a linear, constant coeffjcients difgerentjal equatjon of second order. ̇ ̈ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system 1 DOF System, cont. In a positjon of equilibrium, 𝑧 eq. , the velocity and the acceleratjon are zero, and hence 𝑔(𝑧 eq. , 0) = 0 . The force can be linearized in a neighborhood of 𝑧 eq. , 0 : 𝑧) = 𝑔(𝑧 eq. , 0) + 𝜖𝑔 𝜖𝑧(𝑧 − 𝑧 eq. ) + 𝜖𝑔 𝑔(𝑧, ̇ 𝑧( ̇ 𝑧 − 0) + 𝑃(𝑧, ̇ 𝑧). 𝜖 ̇ Assuming that 𝑃(𝑧, ̇ 𝑧) is small in a neighborhood of 𝑧 eq. , we can write the equatjon of motjon 𝑦 + 𝑏 ̇ 𝑦 + 𝑐𝑦 = 0 𝜖𝑔 𝜖𝑔 where 𝑦 = 𝑧 − 𝑧 eq. , 𝑏 = − 𝑧 � and 𝑐 = − 𝜖𝑧 � . 𝜖 ̇ 𝑧=0 𝑧=𝑧 eq
̈ ̇ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system 1 DOF System, cont. In a positjon of equilibrium, 𝑧 eq. , the velocity and the acceleratjon are zero, and hence 𝑔(𝑧 eq. , 0) = 0 . The force can be linearized in a neighborhood of 𝑧 eq. , 0 : 𝑧) = 𝑔(𝑧 eq. , 0) + 𝜖𝑔 𝜖𝑧(𝑧 − 𝑧 eq. ) + 𝜖𝑔 𝑔(𝑧, ̇ 𝑧( ̇ 𝑧 − 0) + 𝑃(𝑧, ̇ 𝑧). 𝜖 ̇ Assuming that 𝑃(𝑧, ̇ 𝑧) is small in a neighborhood of 𝑧 eq. , we can write the equatjon of motjon 𝑦 + 𝑏 ̇ 𝑦 + 𝑐𝑦 = 0 𝜖𝑔 𝜖𝑔 where 𝑦 = 𝑧 − 𝑧 eq. , 𝑏 = − 𝑧 � and 𝑐 = − 𝜖𝑧 � . 𝜖 ̇ 𝑧=0 𝑧=𝑧 eq In an infjnitesimal neighborhood of 𝑧 eq. , the equatjon of motjon can be studied in terms of a linear, constant coeffjcients difgerentjal equatjon of second order.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system 1 DOF System, cont. A linear constant coeffjcient difgerentjal equatjon has the homogeneous integral 𝑦 = 𝐵 exp (𝑡𝑢) , that substjtuted in the equatjon of motjon gives 𝑡 2 + 𝑏𝑡 + 𝑐 = 0 whose solutjons are 2 ∓ �𝑏 2 𝑡 1,2 = −𝑏 4 − 𝑐. The general integral is hence 𝑦(𝑢) = 𝐵 1 exp (𝑡 1 𝑢) + 𝐵 2 exp (𝑡 2 𝑢).
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system 1 DOF System, cont. Given that for a free vibratjon problem 𝐵 1 , 𝐵 2 are given by the initjal conditjons, the nature of the solutjon depends on the sign of the real part of 𝑡 1 , 𝑡 2 , because 𝑡 𝑗 = 𝑠 𝑗 + 𝚥𝑟 𝑗 and exp (𝑡 𝑗 𝑢) = exp (𝚥𝑟 𝑗 𝑢) exp (𝑠 𝑗 𝑢). If one of the 𝑠 𝑗 > 0 , the response grows infjnitely over tjme, even for an infjnitesimal perturbatjon of the equilibrium, so that in this case we have an unstable equilibrium . If both 𝑠 𝑗 < 0 , the response decrease over tjme, so we have a stable equilibrium . Finally, if both 𝑠 𝑗 = 0 the roots 𝑡 𝑗 are purely imaginary and the response is harmonic with constant amplitude.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system 1 DOF System, cont. The roots being 2 ∓ �𝑏 2 𝑡 1,2 = −𝑏 4 − 𝑐, if 𝑏 > 0 and 𝑐 > 0 both roots are negatjve or complex conjugate with negatjve real part, the system is asymptotjcally stable, if 𝑏 = 0 and 𝑐 > 0 , the roots are purely imaginary, the equilibrium is indifgerent, the oscillatjons are harmonic, if 𝑏 < 0 or 𝑐 < 0 at least one of the roots has a positjve real part, and the system is unstable.
The entjre mass, 𝑛 , is concentrated in a rigid block, its positjon completely described by the coordinate 𝑦(𝑢) . The energy‐loss ( 𝑏 ̇ 𝑦 ) is given by a massless damper, its damping constant being 𝑑 . The elastjc resistance ( 𝑐 𝑦 ) is provided by a massless spring of stjfgness 𝑙 For completeness we consider also an external loading, the tjme‐varying force 𝑞(𝑢) . 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The famous box car In a single degree of freedom ( sdof ) system each property, 𝑛 , 𝑏 and 𝑐 , can be conveniently represented in a single physical element
The energy‐loss ( 𝑏 ̇ 𝑦 ) is given by a massless damper, its damping constant being 𝑑 . The elastjc resistance ( 𝑐 𝑦 ) is provided by a massless spring of stjfgness 𝑙 For completeness we consider also an external loading, the tjme‐varying force 𝑞(𝑢) . 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The famous box car In a single degree of freedom ( sdof ) system each property, 𝑛 , 𝑏 and 𝑐 , can be conveniently represented in a single physical element The entjre mass, 𝑛 , is concentrated in a rigid block, its positjon completely described by the coordinate 𝑦(𝑢) .
The elastjc resistance ( 𝑐 𝑦 ) is provided by a massless spring of stjfgness 𝑙 For completeness we consider also an external loading, the tjme‐varying force 𝑞(𝑢) . 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The famous box car In a single degree of freedom ( sdof ) system each property, 𝑛 , 𝑏 and 𝑐 , can be conveniently represented in a single physical element The entjre mass, 𝑛 , is concentrated in a rigid block, its positjon completely described by the coordinate 𝑦(𝑢) . The energy‐loss ( 𝑏 ̇ 𝑦 ) is given by a massless damper, its damping constant being 𝑑 .
For completeness we consider also an external loading, the tjme‐varying force 𝑞(𝑢) . 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The famous box car In a single degree of freedom ( sdof ) system each property, 𝑛 , 𝑏 and 𝑐 , can be conveniently represented in a single physical element The entjre mass, 𝑛 , is concentrated in a rigid block, its positjon completely described by the coordinate 𝑦(𝑢) . The energy‐loss ( 𝑏 ̇ 𝑦 ) is given by a massless damper, its damping constant being 𝑑 . The elastjc resistance ( 𝑐 𝑦 ) is provided by a massless spring of stjfgness 𝑙
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The famous box car In a single degree of freedom ( sdof ) system each property, 𝑛 , 𝑏 and 𝑐 , can be conveniently represented in a single physical element The entjre mass, 𝑛 , is concentrated in a rigid block, its positjon completely described by the coordinate 𝑦(𝑢) . The energy‐loss ( 𝑏 ̇ 𝑦 ) is given by a massless damper, its damping constant being 𝑑 . The elastjc resistance ( 𝑐 𝑦 ) is provided by a massless spring of stjfgness 𝑙 For completeness we consider also an external loading, the tjme‐varying force 𝑞(𝑢) .
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The famous box car In a single degree of freedom ( sdof ) system each property, 𝑛 , 𝑏 and 𝑐 , can be conveniently represented in a single physical element The entjre mass, 𝑛 , is concentrated in a rigid block, its positjon completely described by the coordinate 𝑦(𝑢) . The energy‐loss ( 𝑏 ̇ 𝑦 ) is given by a massless damper, its damping constant being 𝑑 . The elastjc resistance ( 𝑐 𝑦 ) is provided by a massless spring of stjfgness 𝑙 For completeness we consider also an external loading, the tjme‐varying force 𝑞(𝑢) .
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The famous box car In a single degree of freedom ( sdof ) system each property, 𝑛 , 𝑏 and 𝑐 , can be conveniently represented in a single physical element The entjre mass, 𝑛 , is concentrated in a rigid block, its positjon completely described by the coordinate 𝑦(𝑢) . The energy‐loss ( 𝑏 ̇ 𝑦 ) is given by a massless damper, its damping constant being 𝑑 . The elastjc resistance ( 𝑐 𝑦 ) is provided by a massless spring of stjfgness 𝑙 For completeness we consider also an external loading, the tjme‐varying force 𝑞(𝑢) . x ( t ) x k f S ( t ) m p ( t ) f D ( t ) f I ( t ) p ( t ) c ( a ) ( b )
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Equatjon of motjon of the basic dynamic system x ( t ) x k f S ( t ) m p ( t ) f D ( t ) f I ( t ) p ( t ) c ( a ) ( b ) The equatjon of motjon can be writuen using the D’Alembert Principle, expressing the equilibrium of all the forces actjng on the mass including the inertjal force . The forces are the external force, 𝑞(𝑢) , positjve in the directjon of motjon and the resistjng forces, i.e., the inertjal force 𝑔 I (𝑢) , the damping force 𝑔 D (𝑢) and the elastjc force, 𝑔 S (𝑢) , that are opposite to the directjon of the acceleratjon, velocity and displacement.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Equatjon of motjon of the basic dynamic system, cont. The equatjon of motjon can be writuen using the D’Alembert Principle, expressing the equilibrium of all the forces actjng on the mass including the inertjal force . x ( t ) x k f S ( t ) m p ( t ) f D ( t ) f I ( t ) p ( t ) c ( a ) ( b ) The equatjon of motjon, merely expressing the equilibrium of these forces, writjng the resistjng forces and the external force across the equal sign 𝑔 I (𝑢) + 𝑔 D (𝑢) + 𝑔 S (𝑢) = 𝑞(𝑢)
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system EOM of the basic dynamic system, cont. According to D’Alembert principle, the inertjal force is the product of the mass and acceleratjon 𝑔 I (𝑢) = 𝑛 ̈ 𝑦(𝑢). Assuming a viscous damping mechanism, the damping force is the product of the damping constant 𝑑 and the velocity, 𝑔 D (𝑢) = 𝑑 ̇ 𝑦(𝑢). Finally, the elastjc force is the product of the elastjc stjfgness 𝑙 and the displacement, 𝑔 S (𝑢) = 𝑙 𝑦(𝑢).
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system EOM of the basic dynamic system, cont. Let’s write again the difgerentjal equatjon of dynamic equilibrium 𝑔 I (𝑢) + 𝑔 D (𝑢) + 𝑔 S (𝑢) = 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 𝑞(𝑢). The resistjng forces , i.e., the forces in the lefu member of the EoM are proportjonal to the defmectjon 𝑦(𝑢) or one of its tjme derivatjves, ̇ 𝑦(𝑢) , ̈ 𝑦(𝑢) . The equatjon of motjon is a linear difgerentjal equatjon of the second order, with constant coeffjcients. The resistjng forces are, by conventjon, positjve when opposite to the directjon of motjon, i.e., resistjng the motjon.
Expressing the displacement as the sum of a constant, statjc displacement and a dynamic displacement, 𝑦(𝑢) = Δ s 𝑢 + ̄ 𝑦(𝑢), and substjtutjng in the EOM we have 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 Δ s 𝑢 + 𝑙 ̄ 𝑦(𝑢) = 𝑞(𝑢) + 𝑋. 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of statjc forces ∆ st ¯ x ( t ) x x ( t ) k k ∆ st W m p ( t ) f S ( t ) p ( t ) f D ( t ) c f I ( t ) ( a ) ( b ) Considering the presence of a constant force 𝑋 , the EOM is 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 𝑞(𝑢) + 𝑋.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of statjc forces ∆ st ¯ x ( t ) x x ( t ) k k ∆ st W m p ( t ) f S ( t ) p ( t ) f D ( t ) c f I ( t ) ( a ) ( b ) Considering the presence of a constant force 𝑋 , the EOM is 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 𝑞(𝑢) + 𝑋. Expressing the displacement as the sum of a constant, statjc displacement and a dynamic displacement, 𝑦(𝑢) = Δ s 𝑢 + ̄ 𝑦(𝑢), and substjtutjng in the EOM we have 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 Δ s 𝑢 + 𝑙 ̄ 𝑦(𝑢) = 𝑞(𝑢) + 𝑋.
The equatjon of motjon expressed with reference to the statjc equilibrium positjon is not afgected by statjc forces. For this reasons, all displacements in further discussions will be referenced from the equilibrium positjon and denoted, for simplicity, with 𝑦(𝑢) . Note that the total displacements, stresses. etc. are infmuenced by the statjc forces, and must be computed using the superpositjon of efgects. 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of statjc forces, cont. Recognizing that 𝑙 Δ s 𝑢 = 𝑋 (so that the two terms, on opposite sides of the 𝑦 ≡ ̇̄ 𝑦 ≡ ̈̄ equal sign, cancel each other), that ̇ 𝑦 and that ̈ 𝑦 the EOM can be writuen as 𝑛 ̈̄ 𝑦(𝑢) + 𝑑 ̇̄ 𝑦(𝑢) + 𝑙 ̄ 𝑦(𝑢) = 𝑞(𝑢).
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of statjc forces, cont. Recognizing that 𝑙 Δ s 𝑢 = 𝑋 (so that the two terms, on opposite sides of the 𝑦 ≡ ̇̄ 𝑦 ≡ ̈̄ equal sign, cancel each other), that ̇ 𝑦 and that ̈ 𝑦 the EOM can be writuen as 𝑛 ̈̄ 𝑦(𝑢) + 𝑑 ̇̄ 𝑦(𝑢) + 𝑙 ̄ 𝑦(𝑢) = 𝑞(𝑢). The equatjon of motjon expressed with reference to the statjc equilibrium positjon is not afgected by statjc forces. For this reasons, all displacements in further discussions will be referenced from the equilibrium positjon and denoted, for simplicity, with 𝑦(𝑢) . Note that the total displacements, stresses. etc. are infmuenced by the statjc forces, and must be computed using the superpositjon of efgects.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of support motjon x tot ( t ) m Fixed reference axis k k 2 2 c x g ( t ) x ( t ) Displacements, deformatjons and stresses in a structure are induced also by a motjon of its support. Important examples of support motjon are the motjon of a building foundatjon due to earthquake and the motjon of the base of a piece of equipment due to vibratjons of the building in which it is housed.
While the elastjc and damping forces are stjll proportjonal to relatjve displacements and velocitjes, the inertjal force is proportjonal to the total acceleratjon, 𝑔 I (𝑢) = −𝑛 ̈ 𝑦 tot (𝑢) = 𝑛 ̈ 𝑦 g (𝑢) + 𝑛 ̈ 𝑦(𝑢). Writjng the EOM for a null external load, 𝑞(𝑢) = 0 , is hence 𝑛 ̈ 𝑦 tot (𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 0, or, 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = −𝑛 ̈ 𝑦 g (𝑢) ≡ 𝑞 eff (𝑢). Support motjon is suffjcient to excite a dynamic system: 𝑞 eff (𝑢) = −𝑛 ̈ 𝑦 g (𝑢) . ̈ ̈ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of support motjon, cont. x tot ( t ) Considering a support motjon 𝑦 g (𝑢) , defjned with m respect to a inertjal frame of reference, the total Fixed reference axis displacement is k k 2 2 𝑦 tot (𝑢) = 𝑦 g (𝑢) + 𝑦(𝑢) c and the total acceleratjon is 𝑦 tot (𝑢) = 𝑦 g (𝑢) + ̈ 𝑦(𝑢). x g ( t ) x ( t )
Writjng the EOM for a null external load, 𝑞(𝑢) = 0 , is hence 𝑛 ̈ 𝑦 tot (𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 0, or, 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = −𝑛 ̈ 𝑦 g (𝑢) ≡ 𝑞 eff (𝑢). Support motjon is suffjcient to excite a dynamic system: 𝑞 eff (𝑢) = −𝑛 ̈ 𝑦 g (𝑢) . ̈ ̈ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of support motjon, cont. x tot ( t ) Considering a support motjon 𝑦 g (𝑢) , defjned with m respect to a inertjal frame of reference, the total Fixed reference axis displacement is k k 2 2 𝑦 tot (𝑢) = 𝑦 g (𝑢) + 𝑦(𝑢) c and the total acceleratjon is 𝑦 tot (𝑢) = 𝑦 g (𝑢) + ̈ 𝑦(𝑢). x g ( t ) x ( t ) While the elastjc and damping forces are stjll proportjonal to relatjve displacements and velocitjes, the inertjal force is proportjonal to the total acceleratjon, 𝑔 I (𝑢) = −𝑛 ̈ 𝑦 tot (𝑢) = 𝑛 ̈ 𝑦 g (𝑢) + 𝑛 ̈ 𝑦(𝑢).
Support motjon is suffjcient to excite a dynamic system: 𝑞 eff (𝑢) = −𝑛 ̈ 𝑦 g (𝑢) . ̈ ̈ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of support motjon, cont. x tot ( t ) Considering a support motjon 𝑦 g (𝑢) , defjned with m respect to a inertjal frame of reference, the total Fixed reference axis displacement is k k 2 2 𝑦 tot (𝑢) = 𝑦 g (𝑢) + 𝑦(𝑢) c and the total acceleratjon is 𝑦 tot (𝑢) = 𝑦 g (𝑢) + ̈ 𝑦(𝑢). x g ( t ) x ( t ) While the elastjc and damping forces are stjll proportjonal to relatjve displacements and velocitjes, the inertjal force is proportjonal to the total acceleratjon, 𝑔 I (𝑢) = −𝑛 ̈ 𝑦 tot (𝑢) = 𝑛 ̈ 𝑦 g (𝑢) + 𝑛 ̈ 𝑦(𝑢). Writjng the EOM for a null external load, 𝑞(𝑢) = 0 , is hence 𝑛 ̈ 𝑦 tot (𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 0, or, 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = −𝑛 ̈ 𝑦 g (𝑢) ≡ 𝑞 eff (𝑢).
̈ ̈ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Infmuence of support motjon, cont. x tot ( t ) Considering a support motjon 𝑦 g (𝑢) , defjned with m respect to a inertjal frame of reference, the total Fixed reference axis displacement is k k 2 2 𝑦 tot (𝑢) = 𝑦 g (𝑢) + 𝑦(𝑢) c and the total acceleratjon is 𝑦 tot (𝑢) = 𝑦 g (𝑢) + ̈ 𝑦(𝑢). x g ( t ) x ( t ) While the elastjc and damping forces are stjll proportjonal to relatjve displacements and velocitjes, the inertjal force is proportjonal to the total acceleratjon, 𝑔 I (𝑢) = −𝑛 ̈ 𝑦 tot (𝑢) = 𝑛 ̈ 𝑦 g (𝑢) + 𝑛 ̈ 𝑦(𝑢). Writjng the EOM for a null external load, 𝑞(𝑢) = 0 , is hence 𝑛 ̈ 𝑦 tot (𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 0, or, 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = −𝑛 ̈ 𝑦 g (𝑢) ≡ 𝑞 eff (𝑢). Support motjon is suffjcient to excite a dynamic system: 𝑞 eff (𝑢) = −𝑛 ̈ 𝑦 g (𝑢) .
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Free Vibratjons The equatjon of motjon, 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 𝑞(𝑢) is a linear difgerentjal equatjon of the second order, with constant coeffjcients. Its solutjon can be expressed in terms of a superpositjon of a partjcular solutjon , depending on 𝑞(𝑢) , and a free vibratjon solutjon, that is the solutjon of the so called homogeneous problem , where 𝑞(𝑢) = 0 . In the following, we will study the solutjon of the homogeneous problem, the so‐called homogeneous or complementary solutjon, that is the free vibratjons of the SDOF afuer a perturbatjon of the positjon of equilibrium.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Free vibratjons of an undamped system An undamped system, where 𝑑 = 0 and no energy dissipatjon takes place, is just an ideal notjon, as it would be a realizatjon of motus perpetuum . Nevertheless, it is an useful idealizatjon. In this case, the homogeneous equatjon of motjon is 𝑛 ̈ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 0 which solutjon is of the form 𝑦(𝑢) = 𝐵 exp 𝑡𝑢 ; substjtutjng this solutjon in the above equatjon we have 𝐵 (𝑙 + 𝑡 2 𝑛) exp 𝑡𝑢 = 0 notjng that 𝐵 exp 𝑡𝑢 ≠ 0 , we fjnally have (𝑙 + 𝑡 2 𝑛) = 0 ⇒ 𝑡 = ±�− 𝑙 𝑛.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons Substjtutjng our two difgerent roots in the general solutjons we have 𝑦(𝑢) = 𝐵 exp (+�− 𝑙 𝑛 𝑢) + 𝐶 exp (−�− 𝑙 𝑛 𝑢). Please note that, because both 𝑛 and 𝑙 are positjve quantjtjes, the radicand is negatjve and the roots in the expression above are purely imaginary.
But exp (𝑗𝜕 n 𝑢) = cos (𝑗𝜕 n 𝑢) + 𝑗 sin (𝑗𝜕 n 𝑢) ... The solutjon has an imaginary part? 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons 𝑙 Introducing the natural circular frequency 𝜕 n , defjned by 𝜕 2 n = 𝑛 , we can rewrite the solutjon of the algebraic equatjon in 𝑡 as 𝑡 = ±�− 𝑙 𝑛 = ±√−1� 𝑙 𝑛 = ±𝑗�𝜕 2 𝑜 = ±𝑗𝜕 n where 𝑗 = √−1 and the general integral of the homogeneous equatjon is hence 𝑦(𝑢) = 𝐻 1 exp (𝑗𝜕 n 𝑢) + 𝐻 2 exp (−𝑗𝜕 n 𝑢), where we have writuen 𝐻 1 and 𝐻 2 to emphasize that both are possibly complex numbers.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons 𝑙 Introducing the natural circular frequency 𝜕 n , defjned by 𝜕 2 n = 𝑛 , we can rewrite the solutjon of the algebraic equatjon in 𝑡 as 𝑡 = ±�− 𝑙 𝑛 = ±√−1� 𝑙 𝑛 = ±𝑗�𝜕 2 𝑜 = ±𝑗𝜕 n where 𝑗 = √−1 and the general integral of the homogeneous equatjon is hence 𝑦(𝑢) = 𝐻 1 exp (𝑗𝜕 n 𝑢) + 𝐻 2 exp (−𝑗𝜕 n 𝑢), where we have writuen 𝐻 1 and 𝐻 2 to emphasize that both are possibly complex numbers. But exp (𝑗𝜕 n 𝑢) = cos (𝑗𝜕 n 𝑢) + 𝑗 sin (𝑗𝜕 n 𝑢) ... The solutjon has an imaginary part?
̇ ̇ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons The solutjon is derived from the general integral imposing the (real) initjal conditjons 𝑦(0) = 𝑦 0 , 𝑦(0) = ̇ 𝑦 0 Evaluatjng 𝑦(𝑢) and ̇ 𝑦(𝑢) for 𝑢 = 0 and imposing the initjal conditjons leads to 𝐻 1 + 𝐻 2 = 𝑦 0 � 𝑗𝜕 n 𝐻 1 − 𝑗𝜕 n 𝐻 2 = ̇ 𝑦 0 Solving the linear system we have 𝐻 1 = 𝑗𝑦 0 + ̇ 𝑦 0 /𝜕 n 𝐻 2 = 𝑗𝑦 0 − ̇ 𝑦 0 /𝜕 n , , 2𝑗 2𝑗 substjtutjng these values in the general solutjon and collectjng 𝑦 0 and ̇ 𝑦 0 , we fjnally fjnd 𝑦(𝑢) = exp (𝑗𝜕 n 𝑢) + exp (−𝑗𝜕 n 𝑢) 𝑦 0 + exp (𝑗𝜕 n 𝑢) − exp (−𝑗𝜕 n 𝑢) 𝑦 0 2 2𝑗 𝜕 n
Considering that for every conceivable initjal conditjons we can use the above representatjon, it is indifgerent, and perfectly equivalent, to represent the general integral either in the form of exponentjals of imaginary argument or as a linear combinatjon of sine and cosine of circular frequency 𝜕 n ̇ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons 𝑦(𝑢) = exp (𝑗𝜕 n 𝑢) + exp (−𝑗𝜕 n 𝑢) 𝑦 0 + exp (𝑗𝜕 n 𝑢) − exp (−𝑗𝜕 n 𝑢) 𝑦 0 2 2𝑗 𝜕 n Using the Euler formulas relatjng the imaginary argument exponentjals and the trigonometric functjons, can be rewrituen in terms of the elementary trigonometric functjons 𝑦(𝑢) = 𝑦 0 cos (𝜕 n 𝑢) + ( ̇ 𝑦 𝑝 /𝜕 n ) sin (𝜕 n 𝑢).
̇ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons 𝑦(𝑢) = exp (𝑗𝜕 n 𝑢) + exp (−𝑗𝜕 n 𝑢) 𝑦 0 + exp (𝑗𝜕 n 𝑢) − exp (−𝑗𝜕 n 𝑢) 𝑦 0 2 2𝑗 𝜕 n Using the Euler formulas relatjng the imaginary argument exponentjals and the trigonometric functjons, can be rewrituen in terms of the elementary trigonometric functjons 𝑦(𝑢) = 𝑦 0 cos (𝜕 n 𝑢) + ( ̇ 𝑦 𝑝 /𝜕 n ) sin (𝜕 n 𝑢). Considering that for every conceivable initjal conditjons we can use the above representatjon, it is indifgerent, and perfectly equivalent, to represent the general integral either in the form of exponentjals of imaginary argument or as a linear combinatjon of sine and cosine of circular frequency 𝜕 n
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons Otherwise, using the identjty exp (±𝑗𝜕 n 𝑢) = cos 𝜕 n 𝑢 ± 𝑗 sin 𝜕 n 𝑢 𝑦(𝑢) = (𝐵 + 𝑗𝐶) ( cos 𝜕 n 𝑢 + 𝑗 sin 𝜕 n 𝑢) + (𝐷 − 𝑗𝐸) × ( cos 𝜕 n 𝑢 − 𝑗 sin 𝜕 n 𝑢) expanding the product and evidencing the imaginary part of the response we have 𝐽(𝑦) = 𝑗 (𝐵 sin 𝜕 n 𝑢 + 𝐶 cos 𝜕 n 𝑢 − 𝐷 sin 𝜕 n 𝑢 − 𝐸 cos 𝜕 n 𝑢) . Imposing that 𝐽(𝑦) = 0 , i.e., that the response is real, we have (𝐵 − 𝐷) sin 𝜕 n 𝑢 + (𝐶 − 𝐸) cos 𝜕 n 𝑢 = 0 → 𝐷 = 𝐵, 𝐸 = 𝐶. Substjtutjng in 𝑦(𝑢) eventually we have 𝑦(𝑢) = 2𝐵 cos (𝜕 n 𝑢) − 2𝐶 sin (𝜕 n 𝑢).
̇ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons Our preferred representatjon of the general integral of undamped free vibratjons is 𝑦(𝑢) = 𝐵 cos (𝜕 n 𝑢) + 𝐶 sin (𝜕 n 𝑢) For the usual initjal conditjons, we have already seen that 𝑦 0 𝐵 = 𝑦 0 , 𝐶 = . 𝜕 n
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons Sometjmes we prefer to write 𝑦(𝑢) as a single harmonic, introducing a phase difgerence 𝜚 so that the amplitude of the motjon, 𝐷 , is put in evidence: 𝑦(𝑢) = 𝐷 cos (𝜕 n 𝑢 − 𝜒) = 𝐷 ( cos 𝜕 n 𝑢 cos 𝜒 + sin 𝜕 n 𝑢 sin 𝜒) = 𝐵 cos 𝜕 n 𝑢 + 𝐶 sin 𝜕 n 𝑢 From 𝐵 = 𝐷 cos 𝜒 and 𝐶 = 𝐷 sin 𝜒 we have tan 𝜒 = 𝐶/𝐵 , from 𝐵 2 + 𝐶 2 = 𝐷 2 ( cos 2 𝜒 + sin 2 𝜒) we have 𝐷 = √𝐵 2 + 𝐶 2 and eventually with �𝐷 = �𝐵 2 + 𝐶 2 𝑦(𝑢) = 𝐷 cos (𝜕 n 𝑢 − 𝜒), 𝜒 = arctan (𝐶/𝐵)
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Undamped Free Vibratjons x ( t ) arctan ˙ x 0 ρ x 0 t − θ ω T = 2 π ω It is worth notjng that the coeffjcients 𝐵 , 𝐶 and 𝐷 have the dimension of a length, the coeffjcient 𝜕 n has the dimension of the reciprocal of tjme and that the coeffjcient 𝜒 is an angle, or in other terms is adimensional.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Behavior of Damped Systems The viscous damping modifjes the response of a sdof system introducing a decay in the amplitude of the response. Depending on the amount of damping, the response can be oscillatory or not. The amount of damping that separates the two behaviors is denoted as critjcal damping .
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The solutjon of the EOM The equatjon of motjon for a free vibratjng damped system is 𝑛 ̈ 𝑦(𝑢) + 𝑑 ̇ 𝑦(𝑢) + 𝑙 𝑦(𝑢) = 0, substjtutjng the solutjon exp 𝑡𝑢 in the preceding equatjon and simplifying, we have that the parameter 𝑡 must satjsfy the equatjon 𝑛 𝑡 2 + 𝑑 𝑡 + 𝑙 = 0 or, afuer dividing both members by 𝑛 , 𝑡 2 + 𝑑 𝑛 𝑡 + 𝜕 2 n = 0 whose solutjons are 2 2 𝑡 = − 𝑑 2𝑛 ∓ �� 𝑑 𝑑 𝑑 ∓ �� − 𝜕 2 2𝑛� n = 𝜕 n �− � − 1� . 2𝑛𝜕 n 2𝑛𝜕 n
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Critjcal Damping The behavior of the solutjon of the free vibratjon problem depends of course 2 𝑑 on the sign of the radicand Δ = � 2𝑛𝜕 n � − 1 : Δ < 0 the roots 𝑡 are complex conjugate, Δ = 0 the roots are identjcal, double root, Δ > 0 the roots are real. The value of 𝑑 that make the radicand equal to zero is known as the critjcal damping , 𝑑 cr = 2𝑛𝜕 n = 2√𝑛𝑙.
Typical building structures are undercritjcally damped. 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Critjcal Damping A single degree of freedom system is denoted as critjcally damped , under‐critjcally damped or over‐critjcally damped depending on the value of the damping coeffjcient with respect to the critjcal damping.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Critjcal Damping A single degree of freedom system is denoted as critjcally damped , under‐critjcally damped or over‐critjcally damped depending on the value of the damping coeffjcient with respect to the critjcal damping. Typical building structures are undercritjcally damped.
̈ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Damping Ratjo If we introduce the ratjo of the damping to the critjcal damping, or critjcal damping ratjo 𝜂 , 𝜂 = 𝑑 𝑑 = , 𝑑 = 𝜂𝑑 cr = 2𝜂𝜕 n 𝑛 𝑑 cr 2𝑛𝜕 n the equatjon of free vibratjons can be rewrituen as 𝑦(𝑢) + 𝜕 2 𝑦(𝑢) + 2𝜂𝜕 n ̇ n 𝑦(𝑢) = 0 and the roots 𝑡 1,2 can be rewrituen as 𝑡 = −𝜂𝜕 n ∓ 𝜕 n �𝜂 2 − 1.
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Free Vibratjons of Under‐critjcally Damped Systems We start studying the free vibratjon response of under‐critjcally damped SDOF, as this is the most important case in structural dynamics. The determinant being negatjve, the roots 𝑡 1,2 are 𝑡 = −𝜂𝜕 n ∓ 𝜕 n √−1�1 − 𝜂 2 = −𝜂𝜕 n ∓ 𝑗𝜕 D with the positjon that 𝜕 D = 𝜕 n �1 − 𝜂 2 . is the damped frequency ; the general integral of the equatjon of motjon is, collectjng the terms in exp (−𝜂𝜕 n 𝑢) 𝑦(𝑢) = exp (−𝜂𝜕 n 𝑢) [𝐻 1 exp (−𝑗𝜕 D 𝑢) + 𝐻 2 exp (+𝑗𝜕 D 𝑢)]
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Initjal Conditjons By imposing the initjal conditjons, 𝑣(0) = 𝑣 0 , ̇ 𝑣(0) = 𝑤 0 , afuer a bit of algebra we can write the equatjon of motjon for the given initjal conditjons, namely 𝑦(𝑢) = exp (−𝜂𝜕 n 𝑢) � exp (𝑗𝜕 D 𝑢) + exp (−𝑗𝜕 D 𝑢) 𝑣 0 + 2 exp (𝑗𝜕 D 𝑢) − exp (−𝑗𝜕 D 𝑢) 𝑤 0 + 𝜂𝜕 n 𝑣 0 � . 2𝑗 𝜕 D Using the Euler formulas, we fjnally have the preferred format of the general integral: 𝑦(𝑢) = exp (−𝜂𝜕 n 𝑢) [𝐵 cos (𝜕 D 𝑢) + 𝐶 sin (𝜕 D 𝑢)] with 𝐶 = 𝑤 0 + 𝜂𝜕 n 𝑣 0 𝐵 = 𝑣 0 , . 𝜕 D
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system The Damped Free Response x ( t ) ρ x 0 t � � 2 � x 0 + ζωx 0 ˙ ρ = x 2 0 + ω D T = 2 π ω
1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Critjcally damped SDOF In this case, 𝜂 = 1 and 𝑡 1,2 = −𝜕 n , so that the general integral must be writuen in the form 𝑦(𝑢) = exp (−𝜕 n 𝑢)(𝐵 + 𝐶𝑢). The solutjon for given initjal conditjon is 𝑦(𝑢) = exp (−𝜕 n 𝑢)(𝑣 0 + (𝑤 0 + 𝜕 n 𝑣 0 )𝑢), note that, if 𝑤 0 = 0 , the solutjon asymptotjcally approaches zero without crossing the zero axis.
as 𝜂𝜕 n > 𝜕 , for increasing 𝑢 the general integral goes to zero, and that as for increasing 𝜂 we have that ̂ 𝜕 → 𝜂𝜕 n , the velocity with which the response approaches zero slows down for increasing 𝜂 . Note that: ̂ ̂ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Over‐critjcally damped SDOF In this case, 𝜂 > 1 and 𝑡 = −𝜂𝜕 n ∓ 𝜕 n �𝜂 2 − 1 = −𝜂𝜕 n ∓ ̂ 𝜕 where 𝜕 = 𝜕 n �𝜂 2 − 1 and, afuer some rearrangement, the general integral for the over‐damped SDOF can be writuen 𝑦(𝑢) = exp (−𝜂𝜕 n 𝑢) (𝐵 cosh ( ̂ 𝜕𝑢) + 𝐶 sinh ( ̂ 𝜕𝑢))
as for increasing 𝜂 we have that ̂ 𝜕 → 𝜂𝜕 n , the velocity with which the response approaches zero slows down for increasing 𝜂 . ̂ ̂ 1 DOF System Free vibratjons of a SDOF system Free vibratjons of a damped system Over‐critjcally damped SDOF In this case, 𝜂 > 1 and 𝑡 = −𝜂𝜕 n ∓ 𝜕 n �𝜂 2 − 1 = −𝜂𝜕 n ∓ ̂ 𝜕 where 𝜕 = 𝜕 n �𝜂 2 − 1 and, afuer some rearrangement, the general integral for the over‐damped SDOF can be writuen 𝑦(𝑢) = exp (−𝜂𝜕 n 𝑢) (𝐵 cosh ( ̂ 𝜕𝑢) + 𝐶 sinh ( ̂ 𝜕𝑢)) Note that: as 𝜂𝜕 n > 𝜕 , for increasing 𝑢 the general integral goes to zero, and that
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