-
An Introduction to Dynamics of Structures
Giacomo Boffi
http://intranet.dica.polimi.it/people/boffiβgiacomo
Dipartimento di Ingegneria Civile Ambientale e TerritorialePolitecnico di Milano
March 10, 2020
-
Outline
Part 1Introduction
Characteristics of a Dynamical Problem
Formulation of a Dynamical Problem
Formulation of the equations of motion
Part 2
1 DOF System
Free vibrations of a SDOF system
Free vibrations of a damped system
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Definitions
Letβs start with some definitions from the Oxford Dictionary of English:
Dynamic adj., constantly changingDynamics noun, the branch of Mechanics concerned with the
motion of bodies under the action of forcesDynamic Loading a Loading that varies over time
Dynamic Response the Response (deflections and stresses) of a system to adynamic loading; Dynamic Responses vary over time
Dynamics of Structures all of the above, applied to a structural system,i.e., a system designed to stay in equilibrium.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Definitions
Letβs start with some definitions from the Oxford Dictionary of English:Dynamic adj., constantly changing
Dynamics noun, the branch of Mechanics concerned with themotion of bodies under the action of forces
Dynamic Loading a Loading that varies over timeDynamic Response the Response (deflections and stresses) of a system to a
dynamic loading; Dynamic Responses vary over timeDynamics of Structures all of the above, applied to a structural system,
i.e., a system designed to stay in equilibrium.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Definitions
Letβs start with some definitions from the Oxford Dictionary of English:Dynamic adj., constantly changingDynamics noun, the branch of Mechanics concerned with the
motion of bodies under the action of forces
Dynamic Loading a Loading that varies over timeDynamic Response the Response (deflections and stresses) of a system to a
dynamic loading; Dynamic Responses vary over timeDynamics of Structures all of the above, applied to a structural system,
i.e., a system designed to stay in equilibrium.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Definitions
Letβs start with some definitions from the Oxford Dictionary of English:Dynamic adj., constantly changingDynamics noun, the branch of Mechanics concerned with the
motion of bodies under the action of forcesDynamic Loading a Loading that varies over time
Dynamic Response the Response (deflections and stresses) of a system to adynamic loading; Dynamic Responses vary over time
Dynamics of Structures all of the above, applied to a structural system,
i.e., a system designed to stay in equilibrium.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Definitions
Letβs start with some definitions from the Oxford Dictionary of English:Dynamic adj., constantly changingDynamics noun, the branch of Mechanics concerned with the
motion of bodies under the action of forcesDynamic Loading a Loading that varies over time
Dynamic Response the Response (deflections and stresses) of a system to adynamic loading; Dynamic Responses vary over time
Dynamics of Structures all of the above, applied to a structural system,
i.e., a system designed to stay in equilibrium.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Definitions
Letβs start with some definitions from the Oxford Dictionary of English:Dynamic adj., constantly changingDynamics noun, the branch of Mechanics concerned with the
motion of bodies under the action of forcesDynamic Loading a Loading that varies over time
Dynamic Response the Response (deflections and stresses) of a system to adynamic loading; Dynamic Responses vary over time
Dynamics of Structures all of the above, applied to a structural system,
i.e., a system designed to stay in equilibrium.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Definitions
Letβs start with some definitions from the Oxford Dictionary of English:Dynamic adj., constantly changingDynamics noun, the branch of Mechanics concerned with the
motion of bodies under the action of forcesDynamic Loading a Loading that varies over time
Dynamic Response the Response (deflections and stresses) of a system to adynamic loading; Dynamic Responses vary over time
Dynamics of Structures all of the above, applied to a structural system,
i.e., a system designed to stay in equilibrium.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Definitions
Letβs start with some definitions from the Oxford Dictionary of English:Dynamic adj., constantly changingDynamics noun, the branch of Mechanics concerned with the
motion of bodies under the action of forcesDynamic Loading a Loading that varies over time
Dynamic Response the Response (deflections and stresses) of a system to adynamic loading; Dynamic Responses vary over time
Dynamics of Structures all of the above, applied to a structural system,i.e., a system designed to stay in equilibrium.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Dynamics of Structures
Our aim is to determine the stresses and deflections that a dynamic loadinginduces in a structure that remains in the neighborhood of a point ofequilibrium.
Methods of dynamic analysis are extensions of the methods of standardstatic analysis, or to say it better, static analysis is indeed a special case ofdynamic analysis.
If we restrict ourselves to analysis of linear systems, it is so convenientto use the principle of superposition to study the combined effects of staticand dynamic loadings that different methods, of different character, areapplied to these different loadings.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Dynamics of Structures
Our aim is to determine the stresses and deflections that a dynamic loadinginduces in a structure that remains in the neighborhood of a point ofequilibrium.
Methods of dynamic analysis are extensions of the methods of standardstatic analysis, or to say it better, static analysis is indeed a special case ofdynamic analysis.
If we restrict ourselves to analysis of linear systems, it is so convenientto use the principle of superposition to study the combined effects of staticand dynamic loadings that different methods, of different character, areapplied to these different loadings.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Analysis
Taking into account linear systems only, we must consider two differentdefinitions of the loading to define two types of dynamic analysis
Deterministic Analysis applies when the time variation of the loading is fullyknown and we can determine the complete time variation ofall the required response quantities
Nonβdeterministic Analysis applies when the time variation of the loading isessentially random and is known only in terms of somestatisticsIn a nonβdeterministic or stochastic analysis the structuralresponse can be known only in terms of statistics of theresponse quantities
Our focus will be on deterministic analysis
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Analysis
Taking into account linear systems only, we must consider two differentdefinitions of the loading to define two types of dynamic analysisDeterministic Analysis applies when the time variation of the loading is fully
known and we can determine the complete time variation ofall the required response quantities
Nonβdeterministic Analysis applies when the time variation of the loading isessentially random and is known only in terms of somestatisticsIn a nonβdeterministic or stochastic analysis the structuralresponse can be known only in terms of statistics of theresponse quantities
Our focus will be on deterministic analysis
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Analysis
Taking into account linear systems only, we must consider two differentdefinitions of the loading to define two types of dynamic analysisDeterministic Analysis applies when the time variation of the loading is fully
known and we can determine the complete time variation ofall the required response quantities
Nonβdeterministic Analysis applies when the time variation of the loading isessentially random and is known only in terms of somestatisticsIn a nonβdeterministic or stochastic analysis the structuralresponse can be known only in terms of statistics of theresponse quantities
Our focus will be on deterministic analysis
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Analysis
Taking into account linear systems only, we must consider two differentdefinitions of the loading to define two types of dynamic analysisDeterministic Analysis applies when the time variation of the loading is fully
known and we can determine the complete time variation ofall the required response quantities
Nonβdeterministic Analysis applies when the time variation of the loading isessentially random and is known only in terms of somestatisticsIn a nonβdeterministic or stochastic analysis the structuralresponse can be known only in terms of statistics of theresponse quantities
Our focus will be on deterministic analysis
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Analysis
Taking into account linear systems only, we must consider two differentdefinitions of the loading to define two types of dynamic analysisDeterministic Analysis applies when the time variation of the loading is fully
known and we can determine the complete time variation ofall the required response quantities
Nonβdeterministic Analysis applies when the time variation of the loading isessentially random and is known only in terms of somestatisticsIn a nonβdeterministic or stochastic analysis the structuralresponse can be known only in terms of statistics of theresponse quantities
Our focus will be on deterministic analysis
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Loading
Dealing with deterministic dynamic loadings we will study, in order of complexity,Harmonic Loadings a force is modulated by a harmonic function of time,
characterized by a frequency π and a phase π:π(π‘) = π0 sin(ππ‘ β π)
Periodic Loadings a periodic loading repeats itself with a fixed period π:π(π‘) = π0 π(π‘) with π(π‘) β‘ π(π‘ + π)
Non Periodic Loadings here we see two subβcases,
the loading can be described in terms of an analytical function,e.g.,
π(π‘) = ππ exp(πΌπ‘)the loading is measured experimentally, hence it is known onlyin a discrete set of instants; in this case, we say that we knowthe loading timeβhistory.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Loading
Dealing with deterministic dynamic loadings we will study, in order of complexity,Harmonic Loadings a force is modulated by a harmonic function of time,
characterized by a frequency π and a phase π:π(π‘) = π0 sin(ππ‘ β π)
Periodic Loadings a periodic loading repeats itself with a fixed period π:π(π‘) = π0 π(π‘) with π(π‘) β‘ π(π‘ + π)
Non Periodic Loadings here we see two subβcases,
the loading can be described in terms of an analytical function,e.g.,
π(π‘) = ππ exp(πΌπ‘)the loading is measured experimentally, hence it is known onlyin a discrete set of instants; in this case, we say that we knowthe loading timeβhistory.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Loading
Dealing with deterministic dynamic loadings we will study, in order of complexity,Harmonic Loadings a force is modulated by a harmonic function of time,
characterized by a frequency π and a phase π:π(π‘) = π0 sin(ππ‘ β π)
Periodic Loadings a periodic loading repeats itself with a fixed period π:π(π‘) = π0 π(π‘) with π(π‘) β‘ π(π‘ + π)
Non Periodic Loadings here we see two subβcases,
the loading can be described in terms of an analytical function,e.g.,
π(π‘) = ππ exp(πΌπ‘)the loading is measured experimentally, hence it is known onlyin a discrete set of instants; in this case, we say that we knowthe loading timeβhistory.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Loading
Dealing with deterministic dynamic loadings we will study, in order of complexity,Harmonic Loadings a force is modulated by a harmonic function of time,
characterized by a frequency π and a phase π:π(π‘) = π0 sin(ππ‘ β π)
Periodic Loadings a periodic loading repeats itself with a fixed period π:π(π‘) = π0 π(π‘) with π(π‘) β‘ π(π‘ + π)
Non Periodic Loadings here we see two subβcases,the loading can be described in terms of an analytical function,e.g.,
π(π‘) = ππ exp(πΌπ‘)
the loading is measured experimentally, hence it is known onlyin a discrete set of instants; in this case, we say that we knowthe loading timeβhistory.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Types of Dynamic Loading
Dealing with deterministic dynamic loadings we will study, in order of complexity,Harmonic Loadings a force is modulated by a harmonic function of time,
characterized by a frequency π and a phase π:π(π‘) = π0 sin(ππ‘ β π)
Periodic Loadings a periodic loading repeats itself with a fixed period π:π(π‘) = π0 π(π‘) with π(π‘) β‘ π(π‘ + π)
Non Periodic Loadings here we see two subβcases,the loading can be described in terms of an analytical function,e.g.,
π(π‘) = ππ exp(πΌπ‘)the loading is measured experimentally, hence it is known onlyin a discrete set of instants; in this case, we say that we knowthe loading timeβhistory.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Characteristics of a Dynamical Problem
A dynamical problem is essentially characterized by the relevance of inertialforces, arising from the accelerated motion of structural or serviced masses.
A dynamic analysis is required only when the inertial forces represent asignificant portion of the total load.
On the other hand, if the loads and the deflections are varying slowly, a staticanalysis will provide an acceptable approximation.
We will define slowly
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Characteristics of a Dynamical Problem
A dynamical problem is essentially characterized by the relevance of inertialforces, arising from the accelerated motion of structural or serviced masses.
A dynamic analysis is required only when the inertial forces represent asignificant portion of the total load.
On the other hand, if the loads and the deflections are varying slowly, a staticanalysis will provide an acceptable approximation.
We will define slowly
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Characteristics of a Dynamical Problem
A dynamical problem is essentially characterized by the relevance of inertialforces, arising from the accelerated motion of structural or serviced masses.
A dynamic analysis is required only when the inertial forces represent asignificant portion of the total load.
On the other hand, if the loads and the deflections are varying slowly, a staticanalysis will provide an acceptable approximation.
We will define slowly
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Characteristics of a Dynamical Problem
A dynamical problem is essentially characterized by the relevance of inertialforces, arising from the accelerated motion of structural or serviced masses.
A dynamic analysis is required only when the inertial forces represent asignificant portion of the total load.
On the other hand, if the loads and the deflections are varying slowly, a staticanalysis will provide an acceptable approximation.
We will define slowly
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Formulation of a Dynamical Problem
In a structural system the inertial forces depend on the time derivatives ofdisplacements while the elastic forces, equilibrating the inertial ones,depend on the spatial derivatives of the displacements. ... the naturalstatement of the problem is hence in terms of partial differential equations.
In many cases it is however possible to simplify the formulation of thestructural dynamic problem to ordinary differential equations.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Formulation of a Dynamical Problem
In a structural system the inertial forces depend on the time derivatives ofdisplacements while the elastic forces, equilibrating the inertial ones,depend on the spatial derivatives of the displacements. ... the naturalstatement of the problem is hence in terms of partial differential equations.
In many cases it is however possible to simplify the formulation of thestructural dynamic problem to ordinary differential equations.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Lumped Masses
In many structural problems we can say that the masses are concentrated ina discrete set of lumped masses (e.g., in a multiβstorey building most of themasses is concentrated at the level of the storeysβfloors).
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 the deflections
of the lumped masses,3 using methods of static analysis we can determine those deflections,
thus consenting the formulation of the problem in terms of a set of ordinarydifferential equations, one for each relevant component of the inertial forces.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Lumped Masses
In many structural problems we can say that the masses are concentrated ina discrete set of lumped masses (e.g., in a multiβstorey building most of themasses is concentrated at the level of the storeysβfloors).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 the deflections
of the lumped masses,3 using methods of static analysis we can determine those deflections,
thus consenting the formulation of the problem in terms of a set of ordinarydifferential equations, one for each relevant component of the inertial forces.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Dynamic Degrees of Freedom
The dynamic degrees of freedom (DDOF) in a discretized system are thedisplacements components of the lumped masses associated with therelevant components of the inertial forces.If a lumped mass can be regarded as a pointmass then at most 3translational DDOFs will suffice to represent the associated inertial force.On the contrary, if a lumped mass has a discrete volume its inertial forcedepends also on its rotations (inertial couples) and we need at most 6 DDOFsto represent the mass deflections and the inertial force.
Of course, a continuous system has an infinite number of degrees offreedom.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Dynamic Degrees of Freedom
The dynamic degrees of freedom (DDOF) in a discretized system are thedisplacements components of the lumped masses associated with therelevant components of the inertial forces.If a lumped mass can be regarded as a pointmass then at most 3translational DDOFs will suffice to represent the associated inertial force.On the contrary, if a lumped mass has a discrete volume its inertial forcedepends also on its rotations (inertial couples) and we need at most 6 DDOFsto represent the mass deflections and the inertial force.
Of course, a continuous system has an infinite number of degrees offreedom.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Generalized Displacements
The lumped mass procedure that we have outlined is effective if a largeproportion of the total mass is concentrated in a few points (e.g., in amultiβstorey building one can consider a lumped mass for each storey).
When the masses are distributed we can simplify our problem expressing thedeflections in terms of a linear combination of assigned functions of position,the coefficients of the linear combination being the generalized coordinates(e..g., the deflections of a rectilinear beam can be expressed in terms of atrigonometric series).
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Generalized Displacements
The lumped mass procedure that we have outlined is effective if a largeproportion of the total mass is concentrated in a few points (e.g., in amultiβstorey building one can consider a lumped mass for each storey).When the masses are distributed we can simplify our problem expressing thedeflections in terms of a linear combination of assigned functions of position,the coefficients of the linear combination being the generalized coordinates(e..g., the deflections of a rectilinear beam can be expressed in terms of atrigonometric series).
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Generalized Displacements, cont.
To fully describe a displacement field, we need to combine an infinity oflinearly independent base functions, but in practice a good approximationcan be achieved using just a small number of functions and degrees offreedom.
Even if the method of generalized coordinates has its beauty, wemust recognize that for each different problem we have to derive anad hoc formulation, with an evident loss of generality.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Generalized Displacements, cont.
To fully describe a displacement field, we need to combine an infinity oflinearly independent base functions, but in practice a good approximationcan be achieved using just a small number of functions and degrees offreedom.
Even if the method of generalized coordinates has its beauty, wemust recognize that for each different problem we have to derive anad hoc formulation, with an evident loss of generality.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Finite Element Method
The finite elements method (FEM) combines aspects of lumped mass andgeneralized coordinates methods, providing a simple and reliable method ofanalysis, that can be easily programmed on a digital computer.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Finite Element Method
In the FEM, the structure is subdivided in a number of nonβoverlappingpieces, called the finite elements, delimited by common nodes.The FEM uses piecewise approximations (i.e., local to each element) tothe field of displacements.In each element the displacement field is derived from thedisplacements of the nodes that surround each particular element,using interpolating functions.The displacement, deformation and stress fields in each element, aswell as the inertial forces, can thus be expressed in terms of theunknown nodal displacements.The nodal displacements are the dynamical DOFs of the FEM model.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Finite Element Method
Some of the most prominent advantages of the FEM method are1 The desired level of approximation can be achieved by further
subdividing the structure.2 The resulting equations are only loosely coupled, leading to an easier
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 of theelements, and can easily be implemented in a computer program.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Writing the equation of motion
In a deterministic dynamic analysis, given a prescribed load, we want toevaluate the displacements in each instant of time.In many cases a limited number of DDOFs gives a sufficient accuracy; further,the dynamic problem can be reduced to the determination of thetimeβhistories of some selected component of the response.
The mathematical expressions, ordinary or partial differential equations, thatwe are going to write express the dynamic equilibrium of the structuralsystem 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 difficultpart of a dynamic analysis.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Writing the equation of motion
In a deterministic dynamic analysis, given a prescribed load, we want toevaluate the displacements in each instant of time.In many cases a limited number of DDOFs gives a sufficient accuracy; further,the dynamic problem can be reduced to the determination of thetimeβhistories of some selected component of the response.The mathematical expressions, ordinary or partial differential equations, thatwe are going to write express the dynamic equilibrium of the structuralsystem 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 difficultpart of a dynamic analysis.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Writing the equation of motion
In a deterministic dynamic analysis, given a prescribed load, we want toevaluate the displacements in each instant of time.In many cases a limited number of DDOFs gives a sufficient accuracy; further,the dynamic problem can be reduced to the determination of thetimeβhistories of some selected component of the response.The mathematical expressions, ordinary or partial differential equations, thatwe are going to write express the dynamic equilibrium of the structuralsystem 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 difficultpart of a dynamic analysis.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Writing the EOM, cont.
We have a choice of techniques to help us in writing the EOM, namely:the DβAlembert Principle,the Principle of Virtual Displacements,the Variational Approach.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
DβAlembert principle
By Newtonβs II law of motion, for any particle the rate of change ofmomentum is equal to the external force,
οΏ½βοΏ½(π‘) = πππ‘(πποΏ½βοΏ½ππ‘ ),
where οΏ½βοΏ½(π‘) is the particle displacement.In structural dynamics, we may regard the mass as a constant, and thus write
οΏ½βοΏ½(π‘) = π Μβπ’,where each operation of differentiation with respect to time is denoted witha dot.If we write
οΏ½βοΏ½(π‘) β π Μβπ’ = 0and interpret the term βπ Μβπ’ as an Inertial Force that contrasts theacceleration of the particle, we have an equation of equilibrium for theparticle.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
DβAlembert principle, cont.
The concept that a mass develops an inertial force opposing its accelerationis known as the DβAlembert principle, and using this principle we can writethe EOM as a simple equation of equilibrium.The term οΏ½βοΏ½(π‘)must comprise each different force acting on the particle,including the reactions of kinematic or elastic constraints, internal forces andexternal, autonomous forces.In many simple problems, DβAlembert principle is the most direct andconvenient method for the formulation of the EOM.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Principle of virtual displacements
In a reasonably complex dynamic system, with e.g. articulated rigid bodiesand external/internal constraints, the direct formulation of the EOM usingDβAlembert principle may result difficult.In these cases, application of the Principle of Virtual Displacements is veryconvenient, because the reactive forces do not enter the equations ofmotion, that are directly written in terms of the motions compatible with therestraints/constraints of the system.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Principle of Virtual Displacements, cont.
For example, considering an assemblage of rigid bodies, the pvd states thatnecessary and sufficient condition for equilibrium is that, for every virtualdisplacement (i.e., any infinitesimal displacement compatible with therestraints) the total work done by all the external forces is zero.For an assemblage of rigid bodies, writing the EOM requires
1 to identify all the external forces, comprising the inertial forces, and toexpress 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 vectorial nature.
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Variational approach
Variational approaches do not consider directly the forces acting on thedynamic system, but are concerned with the variations of kinetic andpotential energy and lead, as well as the pvd, to a set of scalar equations.
For example, the equation of motion of a generical system can be derived interms of the Lagrangian function, β = π β π where π and π are,respectively, the kinetic and the potential energy of the system expressed interms of a vector οΏ½βοΏ½ of indipendent coordinates
πππ‘
ππΏποΏ½ΜοΏ½π
= ππΏπππ, π = 1,β¦ ,N.
The method to be used in a particular problem is mainly a matter ofconvenience (and, to some extent, of personal taste).
-
Introduction Characteristics of a Dynamical Problem Formulation of a Dynamical Problem Formulation of the equations of motion
Variational approach
Variational approaches do not consider directly the forces acting on thedynamic system, but are concerned with the variations of kinetic andpotential energy and lead, as well as the pvd, to a set of scalar equations.
For example, the equation of motion of a generical system can be derived interms of the Lagrangian function, β = π β π where π and π are,respectively, the kinetic and the potential energy of the system expressed interms of a vector οΏ½βοΏ½ of indipendent coordinates
πππ‘
ππΏποΏ½ΜοΏ½π
= ππΏπππ, π = 1,β¦ ,N.
The method to be used in a particular problem is mainly a matter ofconvenience (and, to some extent, of personal taste).
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
1 DOF System
Structural dynamics is all about the motion of a system in the neighborhoodof a point of equilibrium.Weβll start by studying the most simple of systems, a single degree offreedom system, without external forces, subjected to a perturbation of theequilibrium.If our system has a constantmassπ and itβs subjected to a generical,nonβlinear, internal force πΉ = πΉ(π¦, οΏ½ΜοΏ½), where π¦ is the displacement and οΏ½ΜοΏ½ thevelocity of the particle, the equation of motion is
οΏ½ΜοΏ½ = 1ππΉ(π¦, οΏ½ΜοΏ½) = π(π¦, οΏ½ΜοΏ½).
It is difficult to integrate the above equation in the general case, but itβs easywhen the motion occurs in a small neighborhood of the equilibrium position.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
1 DOF System
Structural dynamics is all about the motion of a system in the neighborhoodof a point of equilibrium.Weβll start by studying the most simple of systems, a single degree offreedom system, without external forces, subjected to a perturbation of theequilibrium.If our system has a constantmassπ and itβs subjected to a generical,nonβlinear, internal force πΉ = πΉ(π¦, οΏ½ΜοΏ½), where π¦ is the displacement and οΏ½ΜοΏ½ thevelocity of the particle, the equation of motion is
οΏ½ΜοΏ½ = 1ππΉ(π¦, οΏ½ΜοΏ½) = π(π¦, οΏ½ΜοΏ½).
It is difficult to integrate the above equation in the general case, but itβs easywhen the motion occurs in a small neighborhood of the equilibrium position.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
1 DOF System, cont.
In a position of equilibrium, π¦eq., the velocity and the acceleration are zero,and hence π(π¦eq., 0) = 0.The force can be linearized in a neighborhood of π¦eq., 0:
π(π¦, οΏ½ΜοΏ½) = π(π¦eq., 0) +ππππ¦(π¦ β π¦eq.) +
ππποΏ½ΜοΏ½ (οΏ½ΜοΏ½ β 0) + π(π¦, οΏ½ΜοΏ½).
Assuming that π(π¦, οΏ½ΜοΏ½) is small in a neighborhood of π¦eq., we can write theequation of motion
οΏ½ΜοΏ½ + ποΏ½ΜοΏ½ + ππ₯ = 0
where π₯ = π¦ β π¦eq., π = βππποΏ½ΜοΏ½ οΏ½ΜοΏ½=0
and π = β ππππ¦ π¦=π¦eq.
In an infinitesimal neighborhood of π¦eq., the equation of motion can bestudied in terms of a linear, constant coefficients differential equation ofsecond order.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
1 DOF System, cont.
In a position of equilibrium, π¦eq., the velocity and the acceleration are zero,and hence π(π¦eq., 0) = 0.The force can be linearized in a neighborhood of π¦eq., 0:
π(π¦, οΏ½ΜοΏ½) = π(π¦eq., 0) +ππππ¦(π¦ β π¦eq.) +
ππποΏ½ΜοΏ½ (οΏ½ΜοΏ½ β 0) + π(π¦, οΏ½ΜοΏ½).
Assuming that π(π¦, οΏ½ΜοΏ½) is small in a neighborhood of π¦eq., we can write theequation of motion
οΏ½ΜοΏ½ + ποΏ½ΜοΏ½ + ππ₯ = 0
where π₯ = π¦ β π¦eq., π = βππποΏ½ΜοΏ½ οΏ½ΜοΏ½=0
and π = β ππππ¦ π¦=π¦eq.
In an infinitesimal neighborhood of π¦eq., the equation of motion can bestudied in terms of a linear, constant coefficients differential equation ofsecond order.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
1 DOF System, cont.
A linear constant coefficient differential equation has the homogeneousintegral π₯ = π΄ exp(π π‘), that substituted in the equation of motion gives
π 2 + ππ + π = 0
whose solutions are
π 1,2 = βπ2 β
π24 β π.
The general integral is hence
π₯(π‘) = π΄1 exp(π 1π‘) + π΄2 exp(π 2π‘).
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
1 DOF System, cont.
Given that for a free vibration problem π΄1, π΄2 are given by the initialconditions, the nature of the solution depends on the sign of the real part ofπ 1, π 2, because π π = ππ + π€ππ and
exp(π ππ‘) = exp(π€πππ‘) exp(πππ‘).
If one of the ππ > 0, the response grows infinitely over time, even for aninfinitesimal perturbation of the equilibrium, so that in this case we have anunstable equilibrium.If both ππ < 0, the response decrease over time, so we have a stableequilibrium.Finally, if both ππ = 0 the roots π π are purely imaginary and the response isharmonic with constant amplitude.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
1 DOF System, cont.
The roots being
π 1,2 = βπ2 β
π24 β π,
if π > 0 and π > 0 both roots are negative or complex conjugate withnegative real part, the system is asymptotically stable,if π = 0 and π > 0, the roots are purely imaginary, the equilibrium isindifferent, the oscillations are harmonic,if π < 0 or π < 0 at least one of the roots has a positive real part, andthe system is unstable.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The famous box car
In a single degree of freedom (sdof) system each property,π, π and π, canbe conveniently represented in a single physical element
The entire mass,π, is concentrated in a rigid block, its position completely describedby the coordinate π₯(π‘).The energyβloss (π οΏ½ΜοΏ½) is given by a massless damper, its damping constant being π.The elastic resistance (π π₯) is provided by a massless spring of stiffness πFor completeness we consider also an external loading, the timeβvarying force π(π‘).
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The famous box car
In a single degree of freedom (sdof) system each property,π, π and π, canbe conveniently represented in a single physical element
The entire mass,π, is concentrated in a rigid block, its position completely describedby the coordinate π₯(π‘).
The energyβloss (π οΏ½ΜοΏ½) is given by a massless damper, its damping constant being π.The elastic resistance (π π₯) is provided by a massless spring of stiffness πFor completeness we consider also an external loading, the timeβvarying force π(π‘).
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The famous box car
In a single degree of freedom (sdof) system each property,π, π and π, canbe conveniently represented in a single physical element
The entire mass,π, is concentrated in a rigid block, its position completely describedby the coordinate π₯(π‘).The energyβloss (π οΏ½ΜοΏ½) is given by a massless damper, its damping constant being π.
The elastic resistance (π π₯) is provided by a massless spring of stiffness πFor completeness we consider also an external loading, the timeβvarying force π(π‘).
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The famous box car
In a single degree of freedom (sdof) system each property,π, π and π, canbe conveniently represented in a single physical element
The entire mass,π, is concentrated in a rigid block, its position completely describedby the coordinate π₯(π‘).The energyβloss (π οΏ½ΜοΏ½) is given by a massless damper, its damping constant being π.The elastic resistance (π π₯) is provided by a massless spring of stiffness π
For completeness we consider also an external loading, the timeβvarying force π(π‘).
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The famous box car
In a single degree of freedom (sdof) system each property,π, π and π, canbe conveniently represented in a single physical element
The entire mass,π, is concentrated in a rigid block, its position completely describedby the coordinate π₯(π‘).The energyβloss (π οΏ½ΜοΏ½) is given by a massless damper, its damping constant being π.The elastic resistance (π π₯) is provided by a massless spring of stiffness πFor completeness we consider also an external loading, the timeβvarying force π(π‘).
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The famous box car
In a single degree of freedom (sdof) system each property,π, π and π, canbe conveniently represented in a single physical element
The entire mass,π, is concentrated in a rigid block, its position completely describedby the coordinate π₯(π‘).The energyβloss (π οΏ½ΜοΏ½) is given by a massless damper, its damping constant being π.The elastic resistance (π π₯) is provided by a massless spring of stiffness πFor completeness we consider also an external loading, the timeβvarying force π(π‘).
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The famous box car
In a single degree of freedom (sdof) system each property,π, π and π, canbe conveniently represented in a single physical element
The entire mass,π, is concentrated in a rigid block, its position completely describedby the coordinate π₯(π‘).The energyβloss (π οΏ½ΜοΏ½) is given by a massless damper, its damping constant being π.The elastic resistance (π π₯) is provided by a massless spring of stiffness πFor completeness we consider also an external loading, the timeβvarying force π(π‘).
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Equation of motion of the basic dynamic system
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
The equation of motion can be written using the DβAlembert Principle,expressing the equilibrium of all the forces acting on the mass including theinertial force.The forces are the external force, π(π‘), positive in the direction of motionand the resisting forces, i.e., the inertial force πI(π‘), the damping force πD(π‘)and the elastic force, πS(π‘), that are opposite to the direction of theacceleration, velocity and displacement.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Equation of motion of the basic dynamic system, cont.
The equation of motion can be written using the DβAlembert Principle,expressing the equilibrium of all the forces acting on the mass including theinertial force.
(a)
m
c
k
p(t)
xx(t)
(b)
p(t)fD(t)fS(t)
fI(t)
The equation of motion, merely expressing the equilibrium of these forces,writing the resisting forces and the external force across the equal sign
πI(π‘) + πD(π‘) + πS(π‘) = π(π‘)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
EOM of the basic dynamic system, cont.
According to DβAlembert principle, the inertial force is the product of themass and acceleration
πI(π‘) = π οΏ½ΜοΏ½(π‘).
Assuming a viscous damping mechanism, the damping force is the product ofthe damping constant π and the velocity,
πD(π‘) = π οΏ½ΜοΏ½(π‘).
Finally, the elastic force is the product of the elastic stiffness π and thedisplacement,
πS(π‘) = π π₯(π‘).
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
EOM of the basic dynamic system, cont.
Letβs write again the differential equation of dynamic equilibrium
πI(π‘) + πD(π‘) + πS(π‘) = π οΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = π(π‘).
The resisting forces, i.e., the forces in the left member of the EoM areproportional to the deflection π₯(π‘) or one of its time derivatives, οΏ½ΜοΏ½(π‘), οΏ½ΜοΏ½(π‘).The equation of motion is a linear differential equation of the second order,with constant coefficients.The resisting forces are, by convention, positive when opposite to the direction ofmotion, i.e., resisting the motion.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of static forces
(a)
m
c
k
p(t)
x(t) x
βst xΜ(t)
(b)
p(t)
kβstfS(t)fD(t)
W
fI(t)
Considering the presence of a constant forceπ, the EOM is
π οΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = π(π‘) +π.
Expressing the displacement as the sum of a constant, static displacement and adynamic displacement,
π₯(π‘) = Ξsπ‘ + οΏ½ΜοΏ½(π‘),and substituting in the EOM we have
π οΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π Ξsπ‘ + π οΏ½ΜοΏ½(π‘) = π(π‘) +π.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of static forces
(a)
m
c
k
p(t)
x(t) x
βst xΜ(t)
(b)
p(t)
kβstfS(t)fD(t)
W
fI(t)
Considering the presence of a constant forceπ, the EOM is
π οΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = π(π‘) +π.
Expressing the displacement as the sum of a constant, static displacement and adynamic displacement,
π₯(π‘) = Ξsπ‘ + οΏ½ΜοΏ½(π‘),and substituting in the EOM we have
π οΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π Ξsπ‘ + π οΏ½ΜοΏ½(π‘) = π(π‘) +π.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of static forces, cont.
Recognizing that π Ξsπ‘ = π (so that the two terms, on opposite sides of theequal sign, cancel each other), that οΏ½ΜοΏ½ β‘ ΜοΏ½ΜοΏ½ and that οΏ½ΜοΏ½ β‘ ΜοΏ½ΜοΏ½ the EOM can bewritten as
π ΜοΏ½ΜοΏ½(π‘) + π ΜοΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) = π(π‘).
The equation of motion expressed with reference to the static equilibriumposition is not affected by static forces.For this reasons, all displacements in further discussions will be referencedfrom the equilibrium position and denoted, for simplicity, with π₯(π‘).
Note that the total displacements, stresses. etc. are influenced bythe static forces, and must be computed using the superposition ofeffects.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of static forces, cont.
Recognizing that π Ξsπ‘ = π (so that the two terms, on opposite sides of theequal sign, cancel each other), that οΏ½ΜοΏ½ β‘ ΜοΏ½ΜοΏ½ and that οΏ½ΜοΏ½ β‘ ΜοΏ½ΜοΏ½ the EOM can bewritten as
π ΜοΏ½ΜοΏ½(π‘) + π ΜοΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) = π(π‘).The equation of motion expressed with reference to the static equilibriumposition is not affected by static forces.For this reasons, all displacements in further discussions will be referencedfrom the equilibrium position and denoted, for simplicity, with π₯(π‘).
Note that the total displacements, stresses. etc. are influenced bythe static forces, and must be computed using the superposition ofeffects.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of support motion
Fix
edre
fere
nce
axis
xtot(t)
xg(t) x(t)
k
2k
2
m
c
Displacements, deformations and stresses in a structure are induced also by amotion of its support.Important examples of support motion are the motion of a building foundation dueto earthquake and the motion of the base of a piece of equipment due to vibrationsof the building in which it is housed.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of support motion, cont.Fix
edre
fere
nce
axis
xtot(t)
xg(t) x(t)
k
2k
2
m
c
Considering a support motion π₯g(π‘), defined withrespect to a inertial frame of reference, the totaldisplacement is
π₯tot(π‘) = π₯g(π‘) + π₯(π‘)
and the total acceleration is
οΏ½ΜοΏ½tot(π‘) = οΏ½ΜοΏ½g(π‘) + οΏ½ΜοΏ½(π‘).
While the elastic and damping forces are still proportional to relative displacements andvelocities, the inertial force is proportional to the total acceleration,
πI(π‘) = βποΏ½ΜοΏ½tot(π‘) = ποΏ½ΜοΏ½g(π‘) +ποΏ½ΜοΏ½(π‘).
Writing the EOM for a null external load, π(π‘) = 0, is henceποΏ½ΜοΏ½tot(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = 0, or,ποΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = βπ οΏ½ΜοΏ½g(π‘) β‘ πeff(π‘).
Support motion is sufficient to excite a dynamic system: πeff(π‘) = βπ οΏ½ΜοΏ½g(π‘).
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of support motion, cont.Fix
edre
fere
nce
axis
xtot(t)
xg(t) x(t)
k
2k
2
m
c
Considering a support motion π₯g(π‘), defined withrespect to a inertial frame of reference, the totaldisplacement is
π₯tot(π‘) = π₯g(π‘) + π₯(π‘)
and the total acceleration is
οΏ½ΜοΏ½tot(π‘) = οΏ½ΜοΏ½g(π‘) + οΏ½ΜοΏ½(π‘).
While the elastic and damping forces are still proportional to relative displacements andvelocities, the inertial force is proportional to the total acceleration,
πI(π‘) = βποΏ½ΜοΏ½tot(π‘) = ποΏ½ΜοΏ½g(π‘) +ποΏ½ΜοΏ½(π‘).
Writing the EOM for a null external load, π(π‘) = 0, is henceποΏ½ΜοΏ½tot(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = 0, or,ποΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = βπ οΏ½ΜοΏ½g(π‘) β‘ πeff(π‘).
Support motion is sufficient to excite a dynamic system: πeff(π‘) = βπ οΏ½ΜοΏ½g(π‘).
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of support motion, cont.Fix
edre
fere
nce
axis
xtot(t)
xg(t) x(t)
k
2k
2
m
c
Considering a support motion π₯g(π‘), defined withrespect to a inertial frame of reference, the totaldisplacement is
π₯tot(π‘) = π₯g(π‘) + π₯(π‘)
and the total acceleration is
οΏ½ΜοΏ½tot(π‘) = οΏ½ΜοΏ½g(π‘) + οΏ½ΜοΏ½(π‘).
While the elastic and damping forces are still proportional to relative displacements andvelocities, the inertial force is proportional to the total acceleration,
πI(π‘) = βποΏ½ΜοΏ½tot(π‘) = ποΏ½ΜοΏ½g(π‘) +ποΏ½ΜοΏ½(π‘).
Writing the EOM for a null external load, π(π‘) = 0, is henceποΏ½ΜοΏ½tot(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = 0, or,ποΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = βπ οΏ½ΜοΏ½g(π‘) β‘ πeff(π‘).
Support motion is sufficient to excite a dynamic system: πeff(π‘) = βπ οΏ½ΜοΏ½g(π‘).
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Influence of support motion, cont.Fix
edre
fere
nce
axis
xtot(t)
xg(t) x(t)
k
2k
2
m
c
Considering a support motion π₯g(π‘), defined withrespect to a inertial frame of reference, the totaldisplacement is
π₯tot(π‘) = π₯g(π‘) + π₯(π‘)
and the total acceleration is
οΏ½ΜοΏ½tot(π‘) = οΏ½ΜοΏ½g(π‘) + οΏ½ΜοΏ½(π‘).
While the elastic and damping forces are still proportional to relative displacements andvelocities, the inertial force is proportional to the total acceleration,
πI(π‘) = βποΏ½ΜοΏ½tot(π‘) = ποΏ½ΜοΏ½g(π‘) +ποΏ½ΜοΏ½(π‘).
Writing the EOM for a null external load, π(π‘) = 0, is henceποΏ½ΜοΏ½tot(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = 0, or,ποΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = βπ οΏ½ΜοΏ½g(π‘) β‘ πeff(π‘).
Support motion is sufficient to excite a dynamic system: πeff(π‘) = βπ οΏ½ΜοΏ½g(π‘).
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Free Vibrations
The equation of motion,
π οΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = π(π‘)
is a linear differential equation of the second order, with constantcoefficients.Its solution can be expressed in terms of a superposition of a particularsolution, depending on π(π‘), and a free vibration solution, that is thesolution of the so called homogeneous problem, where π(π‘) = 0.In the following, we will study the solution of the homogeneous problem,the soβcalled homogeneous or complementary solution, that is the freevibrations of the SDOF after a perturbation of the position of equilibrium.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Free vibrations of an undamped system
An undamped system, where π = 0 and no energy dissipation takes place, isjust an ideal notion, as it would be a realization ofmotus perpetuum.Nevertheless, it is an useful idealization. In this case, the homogeneousequation of motion is
π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = 0which solution is of the form π₯(π‘) = π΄ exp π π‘; substituting this solution inthe above equation we have
π΄ (π + π 2π) exp π π‘ = 0
noting that π΄ exp π π‘ β 0, we finally have
(π + π 2π) = 0 β π = Β± β ππ.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
Substituting our two different roots in the general solutions we have
π₯(π‘) = π΄ exp(+ β ππ π‘) + π΅ exp(β βππ π‘).
Please note that, because bothπ and π are positive quantities, the radicandis negative and the roots in the expression above are purely imaginary.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
Introducing the natural circular frequency πn, defined by π2n =ππ , we can
rewrite the solution of the algebraic equation in π as
π = Β± β ππ = Β±ββ1ππ = Β±π π
2π = Β±ππn
where π = ββ1 and the general integral of the homogeneous equation ishence
π₯(π‘) = πΊ1 exp(ππnπ‘) + πΊ2 exp(βππnπ‘),where we have written πΊ1 and πΊ2 to emphasize that both are possiblycomplex numbers.
But exp(ππnπ‘) = cos(ππnπ‘) + π sin(ππnπ‘) ... The solution has an imaginarypart?
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
Introducing the natural circular frequency πn, defined by π2n =ππ , we can
rewrite the solution of the algebraic equation in π as
π = Β± β ππ = Β±ββ1ππ = Β±π π
2π = Β±ππn
where π = ββ1 and the general integral of the homogeneous equation ishence
π₯(π‘) = πΊ1 exp(ππnπ‘) + πΊ2 exp(βππnπ‘),where we have written πΊ1 and πΊ2 to emphasize that both are possiblycomplex numbers.But exp(ππnπ‘) = cos(ππnπ‘) + π sin(ππnπ‘) ... The solution has an imaginarypart?
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
The solution is derived from the general integral imposing the (real) initialconditions
π₯(0) = π₯0, οΏ½ΜοΏ½(0) = οΏ½ΜοΏ½0Evaluating π₯(π‘) and οΏ½ΜοΏ½(π‘) for π‘ = 0 and imposing the initial conditions leads to
πΊ1 + πΊ2 = π₯0ππnπΊ1 β ππnπΊ2 = οΏ½ΜοΏ½0
Solving the linear system we have
πΊ1 =ππ₯0 + οΏ½ΜοΏ½0/πn
2π , πΊ2 =ππ₯0 β οΏ½ΜοΏ½0/πn
2π ,substituting these values in the general solution and collecting π₯0 and οΏ½ΜοΏ½0, wefinally find
π₯(π‘) = exp(ππnπ‘) + exp(βππnπ‘)2 π₯0 +exp(ππnπ‘) β exp(βππnπ‘)
2ποΏ½ΜοΏ½0πn
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
π₯(π‘) = exp(ππnπ‘) + exp(βππnπ‘)2 π₯0 +exp(ππnπ‘) β exp(βππnπ‘)
2ποΏ½ΜοΏ½0πn
Using the Euler formulas relating the imaginary argument exponentials andthe trigonometric functions, can be rewritten in terms of the elementarytrigonometric functions
π₯(π‘) = π₯0 cos(πnπ‘) + (οΏ½ΜοΏ½π/πn) sin(πnπ‘).
Considering that for every conceivable initial conditions we can use theabove representation, it is indifferent, and perfectly equivalent, to representthe general integral either in the form of exponentials of imaginary argumentor as a linear combination of sine and cosine of circular frequency πn
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
π₯(π‘) = exp(ππnπ‘) + exp(βππnπ‘)2 π₯0 +exp(ππnπ‘) β exp(βππnπ‘)
2ποΏ½ΜοΏ½0πn
Using the Euler formulas relating the imaginary argument exponentials andthe trigonometric functions, can be rewritten in terms of the elementarytrigonometric functions
π₯(π‘) = π₯0 cos(πnπ‘) + (οΏ½ΜοΏ½π/πn) sin(πnπ‘).
Considering that for every conceivable initial conditions we can use theabove representation, it is indifferent, and perfectly equivalent, to representthe general integral either in the form of exponentials of imaginary argumentor as a linear combination of sine and cosine of circular frequency πn
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
Otherwise, using the identity exp(Β±ππnπ‘) = cosπnπ‘ Β± π sinπnπ‘
π₯(π‘) = (π΄ + ππ΅) (cosπnπ‘ + π sinπnπ‘) + (πΆ β ππ·) Γ (cosπnπ‘ β π sinπnπ‘)
expanding the product and evidencing the imaginary part of the response we have
πΌ(π₯) = π (π΄ sinπnπ‘ + π΅ cosπnπ‘ β πΆ sinπnπ‘ β π· cosπnπ‘) .
Imposing that πΌ(π₯) = 0, i.e., that the response is real, we have
(π΄ β πΆ) sinπnπ‘ + (π΅ β π·) cosπnπ‘ = 0 β πΆ = π΄, π· = π΅.
Substituting in π₯(π‘) eventually we have
π₯(π‘) = 2π΄ cos(πnπ‘) β 2π΅ sin(πnπ‘).
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
Our preferred representation of the general integral of undamped freevibrations is
π₯(π‘) = π΄ cos(πnπ‘) + π΅ sin(πnπ‘)For the usual initial conditions, we have already seen that
π΄ = π₯0, π΅ =οΏ½ΜοΏ½0πn
.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
Sometimes we prefer to write π₯(π‘) as a single harmonic, introducing a phasedifference π so that the amplitude of the motion, πΆ, is put in evidence:
π₯(π‘) = πΆ cos(πnπ‘ β π) = πΆ (cosπnπ‘ cosπ + sinπnπ‘ sinπ)= π΄ cosπnπ‘ + π΅ sinπnπ‘
From π΄ = πΆ cosπ and π΅ = πΆ sinπ we have tanπ = π΅/π΄, fromπ΄2 + π΅2 = πΆ2(cos2 π + sin2 π) we have πΆ = βπ΄2 + π΅2 and eventually
π₯(π‘) = πΆ cos(πnπ‘ β π), withπΆ = π΄2 + π΅2π = arctan(π΅/π΄)
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Undamped Free Vibrations
x(t)
t
x0
arctan xΜ0
β ΞΈΟT = 2ΟΟ
Ο
It is worth noting that the coefficients π΄, π΅ and πΆ have the dimension of alength, the coefficient πn has the dimension of the reciprocal of time andthat the coefficient π is an angle, or in other terms is adimensional.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Behavior of Damped Systems
The viscous damping modifies the response of a sdof system introducing adecay in the amplitude of the response. Depending on the amount ofdamping, the response can be oscillatory or not. The amount of dampingthat separates the two behaviors is denoted as critical damping.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The solution of the EOM
The equation of motion for a free vibrating damped system is
π οΏ½ΜοΏ½(π‘) + π οΏ½ΜοΏ½(π‘) + π π₯(π‘) = 0,substituting the solution exp π π‘ in the preceding equation and simplifying,we have that the parameter π must satisfy the equation
ππ 2 + π π + π = 0or, after dividing both members byπ,
π 2 + ππ π + π2n = 0
whose solutions are
π = β π2π βπ2π
2β π2n = πn β
π2ππn
β π2ππn
2β 1 .
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Critical Damping
The behavior of the solution of the free vibration problem depends of course
on the sign of the radicand Ξ = π2ππn2β 1:
Ξ < 0 the roots π are complex conjugate,Ξ = 0 the roots are identical, double root,Ξ > 0 the roots are real.
The value of π that make the radicand equal to zero is known as the criticaldamping,
πcr = 2ππn = 2βππ.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Critical Damping
A single degree of freedom system is denoted as critically damped,underβcritically damped or overβcritically damped depending on the value ofthe damping coefficient with respect to the critical damping.
Typical building structures are undercritically damped.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Critical Damping
A single degree of freedom system is denoted as critically damped,underβcritically damped or overβcritically damped depending on the value ofthe damping coefficient with respect to the critical damping.
Typical building structures are undercritically damped.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Damping Ratio
If we introduce the ratio of the damping to the critical damping, or criticaldamping ratio π,
π = ππcr= π2ππn
, π = ππcr = 2ππnπ
the equation of free vibrations can be rewritten as
οΏ½ΜοΏ½(π‘) + 2ππnοΏ½ΜοΏ½(π‘) + π2nπ₯(π‘) = 0
and the roots π 1,2 can be rewritten as
π = βππn β πn π2 β 1.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Free Vibrations of Underβcritically Damped Systems
We start studying the free vibration response of underβcritically dampedSDOF, as this is the most important case in structural dynamics.The determinant being negative, the roots π 1,2 are
π = βππn β πnββ1 1 β π2 = βππn β ππD
with the position thatπD = πn 1 β π2.
is the damped frequency; the general integral of the equation of motion is,collecting the terms in exp(βππnπ‘)
π₯(π‘) = exp(βππnπ‘) [πΊ1 exp(βππDπ‘) + πΊ2 exp(+ππDπ‘)]
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Initial Conditions
By imposing the initial conditions, π’(0) = π’0, οΏ½ΜοΏ½(0) = π£0, after a bit ofalgebra we can write the equation of motion for the given initial conditions,namely
π₯(π‘) = exp(βππnπ‘)exp(ππDπ‘) + exp(βππDπ‘)
2 π’0+
exp(ππDπ‘) β exp(βππDπ‘)2π
π£0 + ππn π’0πD
.
Using the Euler formulas, we finally have the preferred format of the generalintegral:
π₯(π‘) = exp(βππnπ‘) [π΄ cos(πDπ‘) + π΅ sin(πDπ‘)]with
π΄ = π’0, π΅ =π£0 + ππn π’0
πD.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
The Damped Free Response
x(t)
t
x0
Ο
T = 2ΟΟ
Ο =
βx20 +
(xΜ0+ΞΆΟx0
ΟD
)2
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Critically damped SDOF
In this case, π = 1 and π 1,2 = βπn, so that the general integral must bewritten in the form
π₯(π‘) = exp(βπnπ‘)(π΄ + π΅π‘).The solution for given initial condition is
π₯(π‘) = exp(βπnπ‘)(π’0 + (π£0 + πn π’0)π‘),
note that, if π£0 = 0, the solution asymptotically approaches zero withoutcrossing the zero axis.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Overβcritically damped SDOF
In this case, π > 1 and
π = βππn β πn π2 β 1 = βππn β οΏ½ΜοΏ½
whereοΏ½ΜοΏ½ = πn π2 β 1
and, after some rearrangement, the general integral for the overβdampedSDOF can be written
π₯(π‘) = exp(βππnπ‘) (π΄ cosh(οΏ½ΜοΏ½π‘) + π΅ sinh(οΏ½ΜοΏ½π‘))
Note that:
as ππn > οΏ½ΜοΏ½, for increasing π‘ the general integral goes to zero, and thatas for increasing π we have that οΏ½ΜοΏ½ β ππn, the velocity with which theresponse approaches zero slows down for increasing π.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Overβcritically damped SDOF
In this case, π > 1 and
π = βππn β πn π2 β 1 = βππn β οΏ½ΜοΏ½
whereοΏ½ΜοΏ½ = πn π2 β 1
and, after some rearrangement, the general integral for the overβdampedSDOF can be written
π₯(π‘) = exp(βππnπ‘) (π΄ cosh(οΏ½ΜοΏ½π‘) + π΅ sinh(οΏ½ΜοΏ½π‘))
Note that:as ππn > οΏ½ΜοΏ½, for increasing π‘ the general integral goes to zero, and that
as for increasing π we have that οΏ½ΜοΏ½ β ππn, the velocity with which theresponse approaches zero slows down for increasing π.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Overβcritically damped SDOF
In this case, π > 1 and
π = βππn β πn π2 β 1 = βππn β οΏ½ΜοΏ½
whereοΏ½ΜοΏ½ = πn π2 β 1
and, after some rearrangement, the general integral for the overβdampedSDOF can be written
π₯(π‘) = exp(βππnπ‘) (π΄ cosh(οΏ½ΜοΏ½π‘) + π΅ sinh(οΏ½ΜοΏ½π‘))
Note that:as ππn > οΏ½ΜοΏ½, for increasing π‘ the general integral goes to zero, and thatas for increasing π we have that οΏ½ΜοΏ½ β ππn, the velocity with which theresponse approaches zero slows down for increasing π.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Measuring damping
The real dissipative behavior of a structural system is complex and verydifficult to assess.For convenience, it is customary to express the real dissipative behavior interms of an equivalent viscous damping.In practice, wemeasure the response of a SDOF structural system undercontrolled testing conditions and find the value of the viscous damping (ordamping ratio) for which our simplified model best matches themeasurements.For example, we could require that, under free vibrations, the real structureand the simplified model exhibit the same decay of the vibration amplitude.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Logarithmic Decrement
Consider a SDOF system in free vibration and two positive peaks, π’π andπ’π+π, occurring at times π‘π = π (2π/πD) and π‘π+π = (π +π) (2π/πD).The ratio of these peaks is
π’ππ’π+π
= exp(βππnπ2π/πD)exp(βππn(π + π)2π/πD)= exp(2ππππn/πD)
Substituting πD = πn 1 β π2 and taking the logarithm of both members weobtain
ln( π’ππ’π+π) = πΏ = 2ππ π
1 β π2
where we have introduced πΏ, the logarithmic decrement; solving for π, wefinally get
π = πΏ (2ππ)2 + πΏ2 β12 .
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Recursive Formula for π
The equation of the logarithmic decrement, that we write as
πΏπ2π = π =
π1 β π2
can be formally solved for π:π = π 1 β π2
obtaining an equation that can be interpreted as generating a sequence
ππ+1 = π 1 β π2π .
Starting our sequence of successive approximations with π0 = 0 we obtain π1 = πΏ 2ππ = πand usually the following iterate π2 = π 1 β π21 = π 1 β π2 has converged to the truevalue with a number of digits that exceeds the experimental accuracy.
While the recursive formula is useful in itself, it is also useful as a first example of findingbetter approximations of a systemβs parameter using an iterative procedure.
-
1 DOF System Free vibrations of a SDOF system Free vibrations of a damped system
Recursive Formula for π
The equation of the logarithmic decrement, that we write as
πΏπ2π = π =
π1 β π2
can be formally solved for π:π = π 1 β π2
obtaining an equation that can be interpreted as generating a sequence
ππ+1 = π 1 β π2π .
Starting our sequence of successive approximations with π0 = 0 we obtain π1 = πΏ 2ππ = πand usually the following iterate π2 = π 1 β π21 = π 1 β π2 has converged to the truevalue with a number of digits that exceeds the experimental accuracy.While the recursive formula is useful in itself, it is also useful as a first example of findingbetter approximations of a systemβs parameter using an iterative procedure.
IntroductionIntroductionCharacteristics of a Dynamical ProblemFormulation of a Dynamical ProblemFormulation of the equations of motion
Single Degree of Freedom System1 DOF SystemFree vibrations of a SDOF systemFree vibrations of a damped systemUnder-critically damped SDOFCritically damped SDOFOver-critically damped SDOF