Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and...

School of Mechanical EngineeringIran University of Science and Technology

Advanced Vibrations Lecture One

Elements of Analytical Dynamics

By: H. [email protected]


Elements of Analytical DynamicsNewton's laws were formulated for a single particleCan be extended to systems of particles.The equations of motion are expressed in terms of

physical coordinates vector and force vector. For this reason, Newtonian mechanics is often referred to

as vectorial mechanics.The drawback is that it requires one free-body diagram for each of the masses, Necessitating the inclusion of reaction forces and

interacting forces. These reaction and constraint forces play the role of

unknowns, which makes it necessary to work with a surplus of equations of motion, one additional equation for every unknown force.


Elements of Analytical DynamicsAnalytical mechanics, or analytical dynamics, considers the system as a whole: Not separate individual components,This excludes the reaction and constraint

forces automatically.This approach, due to Lagrange, permits the formulation of problems of dynamics in terms of: two scalar functions the kinetic energy and the

potential energy, and an infinitesimal expression, the virtual work

performed by the nonconservative forces.


Elements of Analytical DynamicsIn analytical mechanics the equations of motion are formulated in terms of generalized coordinates and generalized forces: Not necessarily physical coordinates and forces. The formulation is independent of any special system

of coordinates.

The development of analytical mechanics required the introduction of the concept of virtual displacements, Ied to the development of the calculus of variations. For this reason, analytical mechanics is often referred

to as the variational approach to mechanics.


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


6.1 DEGREES OF FREEDOM AND GENERALIZED COORDINATESA source of possible difficulties in using Newton's equations is use of physical coordinates, which may not always be independent.

Independent coordinates

The generalized coordinates are not unique


6.2 THE PRINCIPLE OF VIRTUAL WORKThe principle of virtual work, due to Johann Bernoulli, is basically a statement of the static equilibrium of a mechanical system.We consider a system of N particles and define the virtual displacements, as infinitesimal changes in the coordinates.The virtual displacements must be consistent with the system constraints, but are otherwise arbitrary.The virtual displacements, being infinitesimal, obey the rules of differential calculus.


THE PRINCIPLE OF VIRTUAL WORK

applied force constraint forceresultant force on each particle

The virtual work performed by the constraint forces is zero


THE PRINCIPLE OF VIRTUAL WORKWhen ri are independent,

If not to switch to a set of generalized coordinates:

Generalized forces


THE PRINCIPLE OF D'ALEMBERT

The virtual work principle can be extended to dynamics, in which form it is known as d'Alembert's principle.

Lagrange version of d'Alembertls principle


Example1:,

,


Example2:Derive the equation of motion for the systems of Example 1 by means of the generalizedd'Alembert's principle.


THE EXTENDED HAMILTON'S PRINCIPLE

The virtual work of all the applied forces,

The kinetic energy of particle mi



It is convenient to choose

Extended Hamilton's principle



where V is the potential energy

Or in terms of the independent generalized coordinates


A Note:

The work performed by the force F in moving the particle m from position r1 to position r2 is responsible for a change in the kinetic energy from TI to T2.


A Note (continued):

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


A Note (continued):


Example3:


Example3: cont.


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


Advanced Vibrations Lecture Two:

LAGRANGE'S EQUATIONS




LagrangianHamilton's principle



Use the extended Hamilton's principle to derive the equations of motion forthe two-degree-of-freedom system.



Only the virtual displacements are arbitrary.



For many problems the extended Hamilton's principle is not the most efficient method for deriving equations of motion: Involves routine operations that must be

carried out every time the principle is applied, The integrations by parts.

The extended Hamilton's principle is used to generate a more expeditious method for deriving equations of motion, Lagrange's equations.



Derive Lagrange's equations of motion for the system


Example of a non-natural systemThe system consists of a mass m connected to a rigid ring through a viscous damper and two nonlinear springs. The mass m is subjected to external damping forces proportional to the absolute velocities X and Y, where the proportionality constant is h.

The Rayleigh dissipation function:

Two nonlinear springs


The Kinetic Energy:

When T2 = T, T1 = T0 = 0, the system is said to be natural


The potential energy/The Lagrangian:

U=V-T0


The Rayleigh dissipation function:


Lagrange's equations of motion

The gyroscopic matrix

The damping matrixThe circulatory matrix


Final word:

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

In fact, it can produce results in cases in which Lagrange's equations cannot, most notably in the case of distributed-parameter systems.


Advanced Vibrations Lecture Three:

MULTI-DEGREE-OF-FREEDOM SYSTEMS



7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis

7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional

damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems


7.1 EQUATIONS OF MOTION FOR LINEAR SYSTEMS


7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTSThe stiffness coefficients can be obtained by other means, not necessarily involving the equations of motion. The stiffness coefficients are more properly

known as stiffness influence coefficients, and can 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.


7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTS

We define the flexibility influence coefficient aijas the displacement of point xi, due to a unit force, Fj = 1.



The stiffness influence coefficient kij is the force required at xi to produce a unit displacement at point xj, and displacements at all other points are zero. To obtain zero displacements at all points the

forces must simply hold these points fixed.


Example:


7.3 PROPERTIES OF THE STIFFNESS AND MASS COEFFICIENTSThe potential energy of a single linear spring:

By analogy the elastic potential energy for a system is:


Symmetry Property:


Maxwell's reciprocity theorem:


7.4 LAGRANGE'S EQUATIONS LINEARIZED ABOUT EQUILIBRIUM

Rayleigh's dissipation function


7.4 LAGRANGE'S EQUATIONS LINEARIZED ABOUT EQUILIBRIUM


7.5 LINEAR TRANSFORMATIONS. COUPLING


Derivation of the matrices M' and K' in a more natural manner


Advanced Vibrations Lecture Four:

Multi-Degree-of-Freedom Systems (Ch7)



7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM

synchronous motion



In general, all frequencies are distinct, except:In degenerate cases, They cannot occur in one-dimensional structures; They can occur in two-dimensional symmetric

structures.

characteristic polynomial



The shape of the natural modes is unique but the amplitude is not.

A very convenient normalization scheme consists of setting:


THE SYMMETRIC EIGENVALUE PROBLEM: LINEAR CONSERVATIVE NATURAL SYSTEMS

Standard eigenvalue problem


THE SYMMETRIC EIGENVALUE PROBLEM: LINEAR CONSERVATIVE NATURAL SYSTEMS

Eigenvalues and eigenvectors associated with real symmetric matrices are real.

To demonstrate these properties, we consider these eigenvalue, eigenvector are complex:

Norm of a vector is a positive number , therefore:

The eigenvalues of a real symmetric matrix are real.

As a corollary, the eigenvectors of a real symmetric matrix are real.


Free vibration for the initial excitations


7.7 ORTHOGONALITY OF MODAL VECTORS


7.8 SYSTEMS ADMITTING RIGID-BODY MOTIONS



The orthogonality of the rigid-body mode to the elastic modes is equivalent to the preservation of zero angular momentum in pure elastic motion.


Advanced Vibrations

Lecture Five



7.9 Decomposition of the Response in Terms of Modal Vectors


7.9 Decomposition of the Response in Terms of Modal Vectors

The modal vectors are orthonormal with respect to the mass matrix M,


7.10 Response to Initial Excitations by Modal Analysis



Modal Coordinates



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


7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix


7.12 Geometric Interpretation of the Eigenvalue Problem


7.12 Geometric Interpretation of the Eigenvalue ProblemSolving the eigenvalue problem by finding the principle axes of the ellipse.



Transforming to canonical form implies elimination of cross products:



Obtaining the angle, one may calculate the eigenvalues and eigenvectors:



Example: Solving the eigenvalue problem by finding the principal axes of the corresponding ellipse.


Advanced Vibrations

Lecture Six



7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES

Rayleigh's quotient



Of special interest in vibrations is the fundamental frequency.

Rayleigh's quotient is an upper bound for the lowest eigenvalue.


7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIESExample:

Simulates gravity loading



Exact solution


7.14 RESPONSE TO HARMONIC EXTERNAL EXCITATIONS


7.14 RESPONSE TO HARMONIC EXTERNAL EXCITATIONS

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

For large systems, it becomes necessary to adopt an approach based on the idea of decoupling the equations of motion.


7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Undamped systems


7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS


7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS Harmonic excitation


7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Transient Vibration


7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Systems admitting rigid-body modes


7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Systems with proportional damping


7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Harmonic excitation


7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Transient Vibration


Advanced Vibrations

Lecture Seven





damping7.16 Systems with Arbitrary Viscous Damping7.17 Discrete-Time Systems


7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING



Nonsymmetric

The eigenvalues/vectors are in general complex.


7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Orthogonality

Left eigenvectors Right eigenvectors



The right eigenvectors xi are biorthogonal to the left eigenvectors yj.



The bi-orthogonality property forms the basis for a modal analysis for the response of systems with arbitrary viscous damping.

Biorthonormality Relations



Assume an arbitrary 2n-dimensional state vector:

The expansion theorem forms the basis for a state space modal analysis:


7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Harmonic Excitations


7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Arbitrary Excitations



The state transition matrix


7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPINGExample 7.12. Determine the response of the system to the excitation:




damping7.16 Systems with Arbitrary Viscous Damping7.17 Discrete-Time Systems


Advanced Vibrations

Discrete-Time Systems

Lecture Eight



NON CONSERVATIVE SYSTEMSExample 1:Solve the eigenvalue problem for the linearized model of shown system about the trivial equilibrium(X0 = Y0 = 0), for the parameter values:

m= 1 kg, Ω = 2 rad/s,

c =0.1 N.s/m, h = 0.2 N.s/m

kx = 5N/m, ky = 10 N/m,


Example 1:THE NONSYMMETRIC EIGENVALUE PROBLEM


Example 1:THE NONSYMMETRIC EIGENVALUE PROBLEM

The response may beevaluated in the modal domain:


Response Evaluation in Time Domain

An nth power approximation

The transition matrix

Semigroup property


Convergence Criteria:

The rate of convergence depends on t×max IλiI, in which max IλiI denotes the magnitude of the eigenvalue of A of largest modulus (the semigroup property, can be used to expedite the convergence of the series).


DISCRETE-TIME SYSTEMS

The discrete-time equivalent of above equation is given by the sequence:


DISCRETE-TIME SYSTEMS,Example 2:Compute the discrete-time response sequence of the system represented by matrix A to an initial impulse applied in the 1st

DOF.


Example 2 (cont.) :The largest eigenvalue modulus is 4.7673 and if

we choose t=0.05s and an accuracy of 10-4, then the transition matrix can be computed with five terms.


Example 2 (cont.) :


Defective SystemsExample 3:

1 2 3 1 2 1 2, , 0.m m m m k k k c c= = = = = = =1 0 0 1 1 0 0 0 00 1 0 , 1 2 1 , 0 0 00 0 1 0 1 1 0 0

M m K k Cc

− = = − − = −


State Space Form

Note: A is defective as it fails to have a linearly independent set of 6 eigenvectors (has repeated eigenvalues).

( )

1 1

5 4 3 2 2 2

0 0 0 1 0 00 0 0 0 1 0

0 0 0 0 0 0 1,

0 0 0 02 0 0 0

0 0 0

4 3 3 0.

kI mA

M K M C cm

α

α α βα α α

α α β

λ λ βλ αλ αβλ α λ α β

− −

=

= = − − − = −

− −

+ + + + + =


Undamped Systems with Rigid-Body Modes

There is only one rigid body mode.Zero eigen-values must occur with

multiplicity of 2The generalized eigen-problem is

defective.For each regular state rigid-body

mode, there will be a corresponding generalized state rigid-body mode


Generalized Eigenvectors: Jordan Form

Using a linearly independent set of generalized eigenvectors A is transformed into block-diagonal Jordan form:

,A X X J=


Undamped Systems with Rigid-Body Modes

[ ] [ ] ( )1 2 1 2

2 2

1, 0

0

1 0 1 01 0 1 01 0 1 00 0 0 10 0 0 10 0 0 1

rr

r

A X X X X

A X X

λλ

λ

= =

= ⇒ =


Undamped Eigenvalues c=0.( )5 4 3 2 2 2

1,2 3,4 5,6

5 63 4

5 6

5 63 4

5 63 4

5 6

5 63 4

4 3 3 0.

0 0.,0., , 3

1 11 0 1 1 1 02 2 2 2

1 0 0 0 1 1 1 0 0 01 11 0 1 1 1 02 2 2 2,

0 12 2

0 1 0 0

0 12 2

i i

X Y


β λ λ α λ α

λ λλ λ

λ λλ λλ λ

λ λλ λ

λ λλ λλ λ

+ + + + + =

= ⇒ = = ± = ±

− − − − − − − −

− − − − = =

− − − − − −

.1 10 1 1 12 2

0 1 0 0 1 11 10 1 1 12 2

− − − − − −

Right Eigenvectors Left Eigenvectors


Damped Eigenvalues c≠0.( )5 4 3 2 2 2

1 2

3,4

5,6

1,22

2

2

4 3 3 0.

0., - 0.0672,11, -0.0500 0.9962 ,5

0.0164 1.7299 .

1 11 1.00451 1.013600 1.00450 1.0136

ii

X


λ λα β λ

λ

λλλ

+ + + + + =

= == = ⇒ = ± = ±

=

-


Response of Damped System to Initial Excitation

( ) (0), (0) [0,0,1,0,0,0] 'Atx t e x x= =The state transition matrix


Response of Damped System to Initial Excitation


Advanced Vibrations

Distributed-Parameter Systems: Exact Solutions

(Lecture Nine)



Distributed-Parameter Systems: Exact Solutions Relation between Discrete and

Distributed Systems . Transverse Vibration of Strings Derivation of the String Vibration

Problem by the Extended Hamilton Principle

Bending Vibration of Beams Free Vibration: The Differential

Eigenvalue Problem Orthogonality of Modes

Expansion Theorem Systems with Lumped Masses at

the Boundaries

Eigenvalue Problem and Expansion Theorem for Problems with Lumped Masses at the Boundaries

Rayleigh's Quotient . The Variational Approach to the Differential Eigenvalue Problem

Response to Initial Excitations Response to External Excitations Systems with External Forces at

Boundaries The Wave Equation Traveling Waves in Rods of

Finite Length


Introduction

The motion of distributed-parameter systems is governed by partial differential equations: to be satisfied over the domain of the system,

and is subject to boundary conditions at the end

points of the domain. Such problems are known as boundary-value

problems.


RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS


RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS

Ignoring 2nd order term


DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE


DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE

EOM

BC’s


BENDING VIBRATION OF BEAMS


BENDING VIBRATION OF BEAMS:EHP


Distributed-Parameter Systems: Exact Solutions Relation between Discrete and

Distributed Systems . Transverse Vibration of Strings Derivation of the String Vibration

Problem by the Extended Hamilton Principle

Bending Vibration of Beams Free Vibration: The Differential

Eigenvalue Problem Orthogonality of Modes

Expansion Theorem Systems with Lumped Masses at

the Boundaries

Eigenvalue Problem and Expansion Theorem for Problems with Lumped Masses at the Boundaries

Rayleigh's Quotient . The Variational Approach to the Differential Eigenvalue Problem

Response to Initial Excitations Response to External Excitations Systems with External Forces at

Boundaries The Wave Equation Traveling Waves in Rods of

Finite Length

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