AHMADIAN Lectures 1 9

7/29/2019 AHMADIAN Lectures 1 9

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School of Mechanical EngineeringIran University of Science and Technology

Advanced Vibrations

Lecture One

By: H. [email protected]


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Preliminaries:Multi-Degree-of-Freedom Systems

1. THE EQUATION OF MOTION

2. FREE VIBRATIONS

3. EIGEN PROBLEM

4. MODE SHAPES

5. RESPONSE TO INITIAL EXCITATIONS

6. COORDINATE TRANSFORMATION

7. ORTHOGONALITY OF MODES, NATURALCOORDINATES

8. BEAT PHENOMENON


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THE EQUATION OF MOTION OF MULTIDEGREE OF FREEIDOM SYSTEMS


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FREE VIBRATIONS OF UNDAMPEDSYSTEMS, NATURAL MODES

Synchronous motion


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EIGEN PROBLEM


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MODE SHAPES


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EXAMPLE


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EXAMPLE


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RESPONSE TO INITIAL EXCITATIONS


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EXAMPLE

X1(0)= 1.2 cm. The other initial conditions are zero.


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COORDINATE TRANSFORMATION,COUPLING


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ORTHOGONALITY OF MODES,NATURAL COORDINATES


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EXAMPLE

Introducing the linear transformation:


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BEAT PHENOMENON


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BEAT PHENOMENONThen, considering the initial conditions:


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WE COVERED:Multi-Degree-of-Freedom Systems

1. THE EQUATION OF MOTION

2. FREE VIBRATIONS

3. EIGEN PROBLEM

4. MODE SHAPES5. RESPONSE TO INITIAL EXCITATIONS

6. COORDINATE TRANSFORMATION

7. ORTHOGONALITY OF MODES, NATURALCOORDINATES

8. BEAT PHENOMENON


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Advanced Vibrations

Lecture Two



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Preliminaries:Multi-Degree-of-Freedom Systems

1. RESPONSE TO HARMONICEXCITATIONS

2. UNDAMPED VIBRATION ABSORBERS3. RESPONSE TO NONPERODIC

EXCITATIONS


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RESPONSE TO HARMONIC EXCITATIONS


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EXAMPLE:

Plot the frequency responsecurves when F2=0.


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EXAMPLE:


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UNDAMPED VIBRATION ABSORBERS


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UNDAMPED VIBRATION ABSORBERS

The shaded area indicates theregion which the performance ofthe absorber can be regardedas satisfactory.

One disadvantage of thevibration absorber is that twonew resonance frequencies are

created.


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RESPONSE TO NON-PERIODICEXCITATIONS

Modal

Forces

ModalCoordinates


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RESPONSE TO NON-PERIODICEXCITATIONS


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EXAMPLE:


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EXAMPLE:


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Advanced Vibrations

Lecture Three

Elements of Analytical Dynamics



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Elements of Analytical DynamicsNewton's laws were formulated for a single particle

Can be extended to systems of particles.

The equations of motion are expressed in terms ofphysical coordinates vector and force vector. For this reason, Newtonian mechanicsis often referred to

as vectorial mechanics.

The drawback is that it requires one free-body diagram foreach of the masses,

Necessitating the inclusion of reaction forces andinteracting forces.

These reaction and constraint forces play the role ofunknowns, which makes it necessary to work with asurplus of equations of motion, one additional equationfor every unknown force.


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Elements of Analytical DynamicsAnalytical mechanics, oranalytical dynamics,

considers the system as a whole: Not separate individual components,

This excludes the reaction and constraintforces automatically.

This approach, due to Lagrange, permits theformulation of problems of dynamics in terms of:

two scalar functions the kinetic energy and the

potential energy, andan infinitesimal expression, the virtual work

performed by the nonconservative forces.


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Elements of Analytical DynamicsIn analytical mechanics the equations of motion areformulated in terms of generalized coordinates andgeneralized forces:

Not necessarily physical coordinates and forces.

The formulation is independent of any special system

of coordinates.

The development of analytical mechanics required theintroduction of the concept of virtual displacements,

Ied to the development of the calculus of variations.

For this reason, analytical mechanics is often referredto as the variational approach to mechanics.


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6 Elements of Analytical Dynamics

6.1 DOF and Generalized Coordinates

6.2 The Principle of Virtual Work

6.3 The Principle of D'Alembert

6.4 The Extended Hamilton's Principle

6.5 Lagrange's Equations


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6.1 DEGREES OF FREEDOM ANDGENERALIZED COORDINATES

A source of possible difficulties in using Newton'sequations is use of physical coordinates, which may notalways be independent.

Independent coordinates

The generalized coordinates are not unique


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6.2 THE PRINCIPLE OF VIRTUAL WORK

The principle of virtual work, due to JohannBernoul l i, is basically a statement of the staticequilibrium of a mechanical system.We consider a system of N particles and definethe virtual displacements, as infinitesimal changes

in the coordinates.

The virtual displacements must be consistent withthe system constraints, but are otherwise

arbitrary.The virtual displacements, being infinitesimal,obey the rules of differential calculus.


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THE PRINCIPLE OF VIRTUAL WORK

applied force constraint forceresultant force on each particle

The virtual work performed by

the constraint forces is zero


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THE PRINCIPLE OF VIRTUAL WORKWhen r i are independent,

If not to switch to a set of generalized

coordinates:

Generalized forces


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THE PRINCIPLE OF D'ALEMBERT

The virtual work principle can be extended todynamics, in which form it is known as

d'Alembert's principle.

Lagrange version of d'Alembertls principle


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THE EXTENDED HAMILTON'SPRINCIPLE

The virtual work of all the applied forces,

The kinetic energy of particle mi


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It is convenient to choose

Extended Hamilton's principle


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where V is the potential energy

Or in terms of the independent generalized coordinates


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A Note:

The work performed by the force F in moving

the particle m from positionr1 to positionr2 isresponsible for a change in the kinetic energy

from TI to T2.


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A Note (continued):

The work performed by conservative forces in movinga particle from r1 to r2 is equal to the negative of thechange in the potential energy from V1 to V2

( )


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A Note (continued):

6 El f A l i l D i


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6 Elements of Analytical Dynamics

6.1 DOF and Generalized Coordinates

6.2 The Principle of Virtual Work

6.3 The Principle of D'Alembert6.4 The Extended Hamilton's Principle

6.5 Lagrange's Equations


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School of Mechanical Engineering

Iran University of Science and Technology

Advanced Vibrations

Lecture Four:LAGRANGE'S EQUATIONS


THE EXTENDED HAMILTON'S


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Lagrangianam il ton 's pr inc ip le

THE EXTENDED HAMILTON'S PRINCIPLE


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THE EXTENDED HAMILTON'S PRINCIPLE

Use the extended Hamilton's principle to derive the equations of motion for

the two-degree-of-freedom system.


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Only the virtual displacements are arbitrary.



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LAGRANGE'S EQUATIONS


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For many problems the extended Hamilton'sprinciple is not the most efficient method forderiving equations of motion:

Involves routine operations that must becarried out every time the principle is applied,

The integrations by parts.

The extended Hamilton's principle is used togenerate a more expeditious method for derivingequations of motion, Lagrange's equations.



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Derive Lagrange's equations of motion for thesystem



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LAGRANGE S EQUATIONS

Final word:


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Final word:

Lagrange's equations are more efficient, theextended Hamilton principle is more versatile.

In fact, it can produce results in cases in which

Lagrange's equations cannot, most notably in thecase of distributed-parameter systems.


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Advanced Vibrations

Lecture Five:MULTI-DEGREE-OF-FREEDOM

SYSTEMS


7 Multi Degree of Freedom Systems


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7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for LinearSystems

7.2 Flexibility and Stiffness InfluenceCoefficients

7.3 Properties of the Stiffness and MassCoefficients

7.4 Lagrange's Equations Linearized aboutEquilibrium

7.5 Linear Transformations : Coupling

7.6 Undamped Free Vibration :TheEigenvalue Problem

7.7 Orthogonality of Modal Vectors

7.8 Systems Admitting Rigid-Body Motions

7.9 Decomposition of the Response in

Terms of Modal Vectors7.10 Response to Initial Excitations byModal Analysis

7.11 Eigenvalue Problem in Terms of aSingle Symmetric Matrix

7.12 Geometric Interpretation of theEigenvalue Problem

7.13 Rayleigh's Quotient and Its Properties

7.14 Response to Harmonic ExternalExcitations

7.15 Response to External Excitations byModal Analysis

7.15.1 Undamped systems

7.15.2 Systems with proportionaldamping

7.16 Systems with Arbitrary ViscousDamping

7.17 Discrete-Time Systems

7 1 EQUATIONS OF MOTION FOR


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7.1 EQUATIONS OF MOTION FORLINEAR SYSTEMS

7 2 FLEXIBILITY AND STIFFNESS


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7.2 FLEXIBILITY AND STIFFNESSINFLUENCE COEFFICIENTS

The stiffness coefficients can be obtained byother means, not necessarily involving theequations of motion.

The stiffness coefficients are more properlyknown as stiffness influence coefficients, andcan be derived by using its definition.

There is one more type of influence coefficients,

namely, flexibility influence coefficients.They are intimately related to the stiffness

influence coefficients.



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We define the flexibility influence coefficienta i jas the displacement of pointx i, due to a unitforce, F j = 1.



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The stiffness influence coefficientk i j is the forcerequired atx i to produce a unit displacement atpointx j, and displacements at all other points arezero.To obtain zero displacements at all points the

forces must simply hold these points fixed.



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Example:


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Example:

7 3 PROPERTIES OF THE STIFFNESS


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7.3 PROPERTIES OF THE STIFFNESSAND MASS COEFFICIENTS

The potential energy of a single linear spring:

By analogy the elastic potential energy for a

system is:


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Maxwell's reciprocity theorem:


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Maxwell s reciprocity theorem:

7 4 LAGRANGE'S EQUATIONS


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7.4 LAGRANGE S EQUATIONSLINEARIZED ABOUT EQUILIBRIUM

Rayleigh's dissipation function



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7.5 LINEAR TRANSFORMATIONS.


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7.5 LINEAR TRANSFORMATIONS.COUPLING

Derivation of the matrices M' and K' in a


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Derivation of the matrices M and K in amore natural manner

7.Multi-Degree-of-Freedom Systems


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7.Multi Degree of Freedom Systems7.1 Equations of Motion for LinearSystems




















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Advanced Vibrations

Lecture 6Multi-Degree-of-Freedom Systems (Ch7)




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7.Multi Degree of Freedom Systems7.1 Equations of Motion for LinearSystems



















7.6 UNDAMPED FREE VIBRATION.


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7.6 UNDAMPED FREE VIBRATION.THE EIGENVALUE PROBLEM

synchronous motion



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In general, all frequencies are distinct, except:

In degeneratecases,

They cannot occur in one-dimensional structures;

They can occur in two-dimensional symmetricstructures.

characteristic polynomial



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THE EIGENVALUE PROBLEM

The shape of the natural modes is unique but

the amplitude is not.

A very convenient normalization scheme

consists of setting:



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THE EIGENVALUE PROBLEM

Free vibration for the initial excitations


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7.7 ORTHOGONALITY OF MODAL


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VECTORS

7.7 ORTHOGONALITY OF MODAL


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VECTORS

7.8 SYSTEMS ADMITTING RIGID-BODY


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MOTIONS


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MOTIONS



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MOTIONS



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MOTIONS



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7.1 Equations of Motion for LinearSystems




















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Advanced Vibrations

Lecture 7




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Terms of Modal Vectors



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Terms of Modal Vectors

The modal vectors are orthonormal with respectto the mass matrix M,

7.10 Response to Initial Excitations by


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Modal Analysis



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Modal Analysis

Modal Coordinates



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Modal Analysis

7.10 Response to Initial Excitations byM d l A l i


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Modal Analysis

We wish to demonstrate that each of the naturalmodes can be excited independently of the other;

7.11 Eigenvalue Problem in Terms of aSi l S t i M t i


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Single Symmetric Matrix



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Eigenvalue Problem



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Eigenvalue Problem



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Eigenvalue Problem

Solving the eigenvalue problem by finding the

principle axes of the ellipse.



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Eigenvalue Problem

Transforming to canonical form implies

elimination of cross products:

7.12 Geometric Interpretation of theEi l P bl


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Eigenvalue Problem



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Eigenvalue Problem

Obtaining the angle, one may calculate theeigenvalues and eigenvectors:



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Eigenvalue Problem

Example: Solving the eigenvalue problem byfinding the principal axes of the correspondingellipse.



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Eigenvalue Problem



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Eigenvalue Problem



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7.2 Flexibility and Stiffness Influence

Coefficients7.3 Properties of the Stiffness and MassCoefficients






7.9 Decomposition of the Response inTerms of Modal Vectors

7.10 Response to Initial Excitations byModal Analysis


7.12 Geometric Interpretation of the

Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties



7.15.1 Undamped systems 7.15.2 Systems with proportional

damping




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Advanced Vibrations

Lecture 8




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damping



7.13 RAYLEIGH'S QUOTIENT AND ITSPROPERTIES


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PROPERTIES

Rayleigh's quotient



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PROPERTIES



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PROPERTIES



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PROPERTIES



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PROPERTIES

Of special interest in vibrations is the fundamentalfrequency.

Rayleigh's quotient is an upper bound for thelowest eigenvalue.



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PROPERTIESExample:

Simulates gravity loading



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PROPERTIES

Exact solution



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PROPERTIES

7.14 RESPONSE TO HARMONICEXTERNAL EXCITATIONS


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EXTERNAL EXCITATIONS

7.14 RESPONSE TO HARMONICEXTERNAL EXCITATIONS


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EXTERNAL EXCITATIONS

This approach is feasible only for systems with a smallnumber of degrees of freedom.

For large systems, it becomes necessary to adopt anapproach based on the idea of decoupling the equationsof motion.

7.15 RESPONSE TO EXTERNALEXCITATIONS BY MODAL ANALYSIS:


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EXCITATIONS BY MODAL ANALYSIS:Undamped systems

7.15 RESPONSE TO EXTERNALEXCITATIONS BY MODAL ANALYSIS


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EXCITATIONS BY MODAL ANALYSIS

7.15 RESPONSE TO EXTERNALEXCITATIONS BY MODAL ANALYSIS


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Harmonic excitation



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Transient Vibration



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Systems admitting rigid-body modes



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Systems with proportional damping



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Harmonic excitation



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Transient Vibration

7.Multi-Degree-of-Freedom Systems7 1 Equations of Motion for Linear 7 11 Eigenvalue Problem in Terms of a


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damping




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Advanced Vibrations

Lecture 9


7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear 7.11 Eigenvalue Problem in Terms of a


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damping



7.16 SYSTEMS WITH ARBITRARYVISCOUS DAMPING


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VISCOUS DAMPING



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VISCOUS DAMPINGNonsymmetric

The eigenvalues/vectors are in general complex.

7.16 SYSTEMS WITH ARBITRARYVISCOUS DAMPING: Orthogonality


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VISCOUS DAMPING: Orthogonality

Left eigenvectors Right eigenvectors



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The right eigenvectors xi arebior thogonal to the lefteigenvectors yj.



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The bi-orthogonality property forms the basis for amodal analysis for the response of systems witharbitrary viscous damping.

Biorthonormality Relations



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Assume an arbitrary 2n-dimensional state vector:

The expansion theorem forms the basis for a

state space modal analysis:



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7.16 SYSTEMS WITH ARBITRARY VISCOUSDAMPING: Harmonic Excitations


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7.16 SYSTEMS WITH ARBITRARY VISCOUSDAMPING: Harmonic Excitations


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7.16 SYSTEMS WITH ARBITRARY VISCOUSDAMPING:Arbitrary Excitations


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The state transition matrix



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Example 7.12. Determine the response of the system to the

excitation:



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7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for LinearSystems



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Systems










Single Symmetric Matrix






damping



AHMADIAN Lectures 1 9

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