lecture 20

School of Mechanical EngineeringIran University of Science and Technology

Advanced Vibrations

Distributed-Parameter Systems: Approximate Methods

Lecture 20

By: H. Ahmadianahmadian@iust.ac.ir

Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method

RAYLEIGH'S PRINCIPLE

The lowest eigenvalue is the minimum value that Rayleigh's quotient can take by letting the trial function Y(x) vary at will.

The minimum value is achieved when Y(x) coincides with the lowest eigenfunction Y1(x).

RAYLEIGH'S PRINCIPLEConsider the differential eigenvalue problem for a string in transverse vibration fixed at x=0 and supported by a spring of stiffness k at x=L.

Exact solutions are possible only in relatively few cases, Most of them characterized by constant tension and uniform mass

density. In seeking an approximate solution, sacrifices must be made, in the

sense that something must be violated. Almost always, one forgoes the exact solution of the differential

equation, which will be satisfied only approximately, But insists on satisfying both boundary conditions exactly.

RAYLEIGH'S PRINCIPLERayleigh's principle, suggests a way of approximating the lowest eigenvalue, without solving the differential eigenvalue problem directly.

Minimizing Rayleigh's quotient is equivalent to solving the differential equation in a weighted average sense, where the weighting function is Y(x).

RAYLEIGH'S PRINCIPLEBoundary conditions do not appear explicitly in the weighted average form of Rayleigh's quotient.

To taken into account the characteristics of the system as much as possible, the trial functions used in conjunction with the weighted average form of Rayleigh's quotient must satisfy all the boundary conditions of the problem.

Comparison functions: trial functions that are as many times differentiable as the order of the system and satisfy all the boundary conditions.

The trial functions must be from the class of comparison functions.The differentiability of the trial functions is

seldom an issue. But the satisfaction of all the boundary

conditions, particularly the satisfaction of the natural boundary conditions can be.

In view of this, we wish to examine the implications of violating the natural boundary conditions.

Rayligh’s quotient involves Vmax and Tref, which are defined for trial functions that are half as many times differentiable as the order of the system and need satisfy only the geometric boundary conditions, as the natural boundary conditions are accounted for in some

fashion.

RAYLEIGH'S PRINCIPLETrial functions that are half as many times

differentiable as the order of the system and satisfy the geometric boundary conditions alone as admissible functions. In using admissible functions in conjunction with the

energy form of Rayleigh's quotient, the natural boundary conditions are still violated.

But, the deleterious effect of this violation is somewhat mitigated by the fact that the energy form of Rayleigh's quotient, includes contributions to Vmax from springs at boundaries and to Tref from masses at boundaries.

But if comparison functions are available, then their use is preferable over the use of admissible functions, because the results are likely to be more accurate.

Example: Lowest natural frequency of the fixed-free tapered rod in axial vibration

The 1st mode of a uniform clamped-free rod as a trial function:

A comparison function

THE RAYLEIGH-RITZ METHODThe method was developed by Ritz as an extension of Rayleigh's energy method. Although Rayleigh claimed that the method

originated with him, the form in which the method is generally used is due to Ritz.

The first step in the Rayleigh-Ritz method is to construct the minimizing sequence:

independent trial functionsundetermined coefficients

THE RAYLEIGH-RITZ METHOD

The independence of the trial functions implies the independence of the coefficients, which in turn implies the independence of the variations

Solving the equations amounts to determining the coefficients, as wellas to determining

THE RAYLEIGH-RITZ METHODTo illustrate the Rayleigh-Ritz process, we consider the differential eigenvalue problem for the string in transverse vibration:

Example : Solve the eigenvalue problem for the fixed-free tapered rod in axial vibration

The comparison functions

Example :

Example : n = 2

Example : n = 3

Example : The Ritz eigenvalues for the two approximations are:

The improvement in the first two Ritz natural frequencies is very small, indicates the chosen comparison functions

resemble very closely the actual natural modes.Convergence to the lowest eigenvalue with six

decimal places accuracy is obtained with 11 terms:

Truncation

Approximation of a system with an infinite number of DOFs by a discrete system with n degrees of freedom implies truncation:

Constraints tend to increase the stiffness of a system:

The nature of the Ritz eigenvalues requires further elaboration.

TruncationA question of particular interest is how the eigenvalues of the (n +1)-DOF approximation relate to the eigenvalues of the n-DOF approximation.

We observe that the extra term in series does not affect the mass and stiffness coefficients computed on the basis of an n-term series (embedding property):

TruncationFor matrices with embedding property the eigenvalues satisfy the separation theorem:

Advanced Vibrations

Separation Theorem for Natural Systems Lecture 21

Max-Min Characterization of the EigenvaluesThe lowest eigenvalue of a vibrating system is the

minimum value Rayleigh's quotient:

The question is whether statements similar to those made for 1st eigenvalue can be made for the intermediate eigenvalues.

Modify v by omitting the first eigenvector v1,

Max-Min Characterization of the EigenvaluesRayleigh's quotient has the minimum value of for

all trial vectors v orthogonal to the first eigenvector v1, where the minimum is reached at v = v2 :

The approach can be extended to higher eigenvalues by constraining the trial vector v to be orthogonal to a suitable number of lower eigenvectors:

Max-Min Characterization of the EigenvaluesConsider a given n-vector w and constrain the

trial vector v to be from the (n-1)-dimensional Euclidean space of constraint orthogonal to w,

n-dimensional ellipsoid associatedwith the real symmetric positive definite matrix A

(n-1) x (n-1) real symmetric positive definitematrix Ã corresponding to the (n-1)-dimensional ellipsoid of constraint resulting

How the eigenvalues of the constrained system Ãrelate to the eigenvalues of the original unconstrained system A ?

Concentrating first on and introduce the definition:

Max-Min Characterization of the Eigenvalues

The longest principal axis of the (n-1)-dimensional ellipsoid of constraint associated with Ã is generally shorter than that corresponding to A, so

Iff w coincides with one of the higher eigenvectors, then

This is consistent with the fact that constraints tend to increase the system stiffness.

Max-Min Characterization of the EigenvaluesThe question remains as to the highest value

can reach:Consider the trial vector:

The choice of V in the given form was motivated by the desire to define therange of as sharply as possible. Indeed, any other choice of V would replace in the

right inequality by a higher value.

Max-Min Characterization of the EigenvaluesRayleigh’s theorem for systems with one constraint:The 1st eigenvalue of a system with one constraint lies between

the 1st and the 2nd eigenvalue of the original unconstrained system,

The right side of the inequality, can be reinterpreted as:

The 2nd eigenvalue of a real symmetric positive definite matrix A is the maximum value that call he given to min(vTAv/vTv) subject to vTw =0.

The preceding theorem can be extended to any number r of constraints, providing a characterization of

Max-Min Characterization of the EigenvaluesWe consider r independent n-vectors w1, w2, ... , wr

and introduce the definition:

In the special case in which the arbitrary constraint vectors wicoincide with the eigenvectors vi of A (i = 1,2,... , r),

Assuming:

Max-Min Characterization of the EigenvaluesThe eigenvalue of a real symmetric positive

definite matrix A is the maximum value that can be given to min(vTAv/vTv)by the imposition of the r constraints vTwi =0 (i = 1,2, ... ,r),

SEPARATION THEOREM FOR NATURAL SYSTEMSThe Courant-Fischer maximin theorem characterizes

the eigenvalues of a real symmetric positive definite matrix subjected to given constraints (a reduction in the number of DOFs):

SEPARATION THEOREM FOR NATURAL SYSTEMS

Next, we assume that the trial vector v is subjected to r -1constraints:

Moreover, we assume an additional constraint:

The constraints are equivalent to:

To complete the picture, we must have a relation

between

Hence, combining inequalities:

Known as the separation theorem,

Advanced Vibrations

Lecture 22

Rayleigh-Ritz method (contd.)How to choose suitable comparison functions, or

admissible functions: the requirement that all boundary conditions, or

merely the geometric boundary conditions be satisfied is too broad to serve as a guideline.

There may be several sets of functions that could be used and the rate of convergence tends to vary from set to set.It is imperative that the functions be from a complete

set, because otherwise convergence may not be possible:

power series, trigonometric functions, Bessel functions, Legendre polynomials, etc.

Rayleigh-Ritz method Extreme care must be exercised when the end involves a discrete

component, such as a spring or a lumped mass, As an illustration, we consider a rod in axial vibration fixed at x=0

and restrained by a spring of stiffness k at x=L:

If we choose as admissible functions the eigenfunctions of a uniform fixed-free rod, then the rate of convergence will be very poor:

The rate of convergence can be vastly improved by using comparison functions:

Rayleigh-Ritz methodExample : Consider the case in which the end x = L of

the rod of previous example is restrained by a spring of stiffness k = EA/L and obtain the solution of the eigenvalue problem derived by the Rayleigh-Ritz method:

1) Using admissible functions2) Using the comparison functions

Example: Using Admissible Functions

Example: Using Admissible Functions, Setting n=2

Example: Using Admissible Functions, Setting n=3

Example: Using Admissible Functions, The convergence using admissible functions is

extremely slow. Using n = 30, none of the natural frequencies

has reached convergence with six decimal places accuracy:

Example: Using Comparison Function

Convergence to six decimal places is reached by the three lowest natural frequencies as follows:

AN ENHANCED RAYLEIGH-RITZ METHOD

Improving accuracy, and hence convergence rate, by combining admissible functions from several families, each family possessing different dynamic

characteristics of the system under consideration

Free end Fixed end

AN ENHANCED RAYLEIGH-RITZ METHODThe linear combination can be made to satisfy the boundary condition for a spring-supported end

Example: Use the given comparison function given in conjunction with Rayleigh's energy method to estimate the lowest natural frequency of the rod of previous example.

AN ENHANCED RAYLEIGH-RITZ METHODIt is better to regard a1 and a2 as independent

undetermined coefficients, and let the Rayleigh- Ritz process determine these coefficients.This motivates us to create a new class of functions

referred to as quasi-comparison functions defined as linear combinations of admissible

functions capable of satisfying all the boundary conditions of the problem

AN ENHANCED RAYLEIGH-RITZ METHODOne word of caution is in order: Each of the two sets of admissible functions is completeAs a result, a given function in one set can be

expanded in terms of the functions in the other set. • The implication is that, as the number of terms n

increases, the two sets tend to become dependent.

• When this happens, the mass and stiffness matrices tend to become singular and the eigensolutions meaningless.

But, because convergence to the lower modes tends to be so fast, in general the singularity problem does not have the chance to materialize.

Solve the problem of privious example using the quasi-comparison functions

Example: n=2

Example: n=3

Advanced Vibrations

Lecture 23

Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz MethodThe Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method

The Assumed-Modes Method: System Response

known trial functions

Example: Use the assumed-modes method in conjunction with a three-term series

to obtain the response of the tapered rod of previous Example to the uniformly distributed force

Damping Effects:

Procedure for the Assumed-Modes Method

1. Select a set of N admissible functions.

2. Compute the coefficients kij of the stiffness matrix.

3. Compute the coefficients mij of the mass matrix

4. Determine expressions for the generalized forces Pi(t) corresponding to the applied force p(x,t).

5. Form the equations of motion.

A 2-DOF model for axial vibration of a uniform cantilever bar

Assumed-Modes Method: Bending of Bernoulli-Euler Beams

A missile on launch pad

Advanced Vibrations

Lecture 24

THE GALERKIN METHOD

The approximate solution is assumed in the form

known independent comparisonfunctions from a complete set

residual

Galerkin's method is more general in scope and can be used for both conservative and non-conservative systems.

THE GALERKIN METHOD

The residual is orthogonal to every trial function.As n increases without bounds, the residual

can remain orthogonal to an infinite set of independent functions only if it tends itself to zero, or

Demonstrates the convergence of Galerkin's method.

THE GALERKIN METHODFind the natural frequencies of vibration of a fixed-

fixed beam of length L, bending stiffness EI, and mass per unit length m using the Galerkin method with the following trial (comparison) functions:

The approximate solution

The governingequation

THE GALERKIN METHODThe residual:

The Galerkin method gives:

THE GALERKIN METHOD

THE GALERKIN METHODConsider a viscously damped beam in transverse vibration.

THE GALERKIN METHOD

THE COLLOCATION METHODThe main difference between the collocation

method and Galerkin's method lies in the weighting functions, the collocation method represent spatial Dirac

delta functions.

THE COLLOCATION METHOD: A beam in transverse vibration

THE COLLOCATION METHOD: The tapered rod

Consider the tapered rod fixed at x=0 and spring-supported at x=L. Solve the problem by the collocation method in two different ways: 1)using the locations xi = iL/n (i = 1,2, . . . , n) 2)using the locations xi =(2i-1)L/2n (i=1,2,. . ., n)

Give results for n = 2 and n = 3.List the three lowest natural frequencies for n =

2,3, . . . ,30 and discuss the nature of the convergence for both cases.

For xi = iL/n the natural frequencies increase as n increases: The specified locations tend to make the rod

longer than it actually is. Because an increased length, while everything

else remains the same, tends to reduce the stiffness, The approximate natural frequencies are lower

than the actual natural frequencies.

On the other hand, the locations xi=(2i-1)L/2n tend to make the rod shorter than it actually is, So that the stiffness of the model is larger than

the stiffness of the actual system. As a result, the approximate natural frequencies

are larger than the actual natural frequencies.

This points to the arbitrariness and lack of predictability inherent in the collocation method, with the nature of the results depending on the choice of locations.

Advanced Vibrations

THE FINITE ELEMENT METHODLecture 25

INTRODUCTION TO THE FINITE ELEMENT METHOD

Finite element method is the most importantdevelopment in the static and dynamic analysisof structures in the second half of the twentiethcentury.Although the finite element method was

developed independently, it was soonrecognized as the most important variant of theRayleigh-Ritz method.

As with the classical Rayleigh-Ritz method, the finite element method also envisions approximate solutions to problems of vibrating distributed systems in the form of linear combinations of known trial functions. Moreover, the expressions for the stiffness and

mass matrices defining the eigenvalue problem are the same as for the classical Rayleigh-Ritz method.

The basic difference between the two approaches lies in the nature of the trial functions. in the classical Rayleigh-Ritz method the trial

functions are global functions, in the finite element method they are local

functions extending over small sub-domains of the system, namely, over finite elements.

In finite element modeling deflection shapes are limited to a portion (finite element) of the structure, with the elements being assembled to for the structural system.The elements are joined together at nodes, or

joints, and displacement compatibility is enforced at these joints.

ELEMENT STIFFNESS AND MASS MATRICES AND FORCE VECTOR

Uniform bar element undergoing axial deformation:

The shape functions must satisfy the BCs:

Considering axial deformation of the uniform element under static loads:

Transverse Motion: Bernoulli-Euler Beam Theory

Transverse Motion: Bernoulli-Euler Beam TheoryFor a beam loaded only at its ends, the equilibrium equation is:

Transverse Motion: Bernoulli-Euler Beam Theory

ExampleDetermine the generalized load vector for a beam element subjected to a uniform transverse load.

Torsion

ASSEMBLY OF SYSTEM MATRICES:• Scheme for the assembly of global matrices

from element matrices for second-order systems using linear interpolation functions

ASSEMBLY OF SYSTEM MATRICES:

• Scheme for the assembly of global matrices from element matrices for fourth-order systems

BOUNDARY CONDITIONS

• The Finite Element formulations inherently satisfy Free boundary conditions.

• Fixed BC’s:

BOUNDARY CONDITIONS

• Supported Spring :

• Lumped Mass:

Example 10.1:The eigenvalue problem for the tapered rod in axial vibration

1. Use the element stiffness and mass matrices with variable cross sections given by:

2. Approximate the stiffness and mass distributions over the finite elements (piece wise constant ) as:

Example 10.1:Variable cross section rod element

Example 10.1:Assembelled Stiffness Matrix

Example 10.1:Assembelled Mass Matrix

Example 10.1:Exact Parameter Distributions

Example 10.1: Convergence Study

Advanced Vibrations Superaccurate finite element eigenvalue computation

Lecture 26

Superaccurate finite element eigenvalue computation

The consistent finite element formulation:It is theoretically sound and also, provides an assured upper bound on the lowest

eigenvalue.

Mass lumping producing a diagonal mass matrix An attractive option for the engineer confronted

with large complex systems.

A Fixed-Free Rod Finite Element Code

Consistent Mass Model

Lumped Mass Model

Convergence Study of the 1st Mode

Consistent Mass Model

Lumped Mass Model

Assembly of the linear finite elements over this mesh using the lumped mass matrix leads to:

Provided:

If:The consistent finite element formulation leads

to an overestimation of eigenvalues and The lumped finite element formulation leads to

an underestimation of eigenvalue;

then it stands to reason thatan intermediate formulation should exist that is

accurately superior to both formulations.

Superaccurate finite element eigenvalue computationLinear combinations of the lumped and the consistent mass matrices give various forms of nonconsistent mass matrices:

where the constraint is imposed for mass conservation.

1=+ βα

Optimal element mass distributionWrite the general finite difference approximation:

Three-nodes string element

More details in:

PARAMETRIC MODELS AND ERROR ANALYSIS FOR RODS

Rod parametric model:

PARAMETRIC MODELS AND ERROR ANALYSIS FOR RODS

The equation of the ith node in the assembled finite element model

0 /2 .4 1/12

k EA dxNo new reqθ

PARAMETRIC MODELS AND ERROR ANALYSIS FOR BEAMS

PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

lecture 20 - iust

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