lecture 20 - iust
TRANSCRIPT
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Approximate Methods
Lecture 20
By: H. [email protected]
School of Mechanical EngineeringIran University of Science and Technology
Distributed-Parameter Systems: Approximate Methods
Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method
School of Mechanical EngineeringIran University of Science and Technology
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).
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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.
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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).
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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.
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RAYLEIGH'S PRINCIPLE
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.
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RAYLEIGH'S PRINCIPLE
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.
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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.
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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
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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
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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
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THE RAYLEIGH-RITZ METHOD
Solving the equations amounts to determining the coefficients, as wellas to determining
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THE RAYLEIGH-RITZ METHODTo illustrate the Rayleigh-Ritz process, we consider the differential eigenvalue problem for the string in transverse vibration:
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THE RAYLEIGH-RITZ METHOD
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Example : Solve the eigenvalue problem for the fixed-free tapered rod in axial vibration
The comparison functions
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Example :
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Example : n = 2
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Example : n = 2
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Example : n = 3
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Example : n = 3
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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:
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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.
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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):
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TruncationFor matrices with embedding property the eigenvalues satisfy the separation theorem:
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Distributed-Parameter Systems: Approximate Methods
Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Separation Theorem for Natural Systems Lecture 21
By: H. [email protected]
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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,
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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:
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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 à corresponding to the (n-1)-dimensional ellipsoid of constraint resulting
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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 à 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.
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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.
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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
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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:
Then
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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),
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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):
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SEPARATION THEOREM FOR NATURAL SYSTEMS
Next, we assume that the trial vector v is subjected to r -1constraints:
Moreover, we assume an additional constraint:
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SEPARATION THEOREM FOR NATURAL SYSTEMS
The constraints are equivalent to:
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SEPARATION THEOREM FOR NATURAL SYSTEMS
To complete the picture, we must have a relation
between
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SEPARATION THEOREM FOR NATURAL SYSTEMS
Hence, combining inequalities:
Known as the separation theorem,
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Advanced Vibrations
Distributed-Parameter Systems: Approximate Methods
Lecture 22
By: H. [email protected]
School of Mechanical EngineeringIran University of Science and Technology
Distributed-Parameter Systems: Approximate Methods
Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method
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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.
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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:
.
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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
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Example: Using Admissible Functions
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Example: Using Admissible Functions, Setting n=2
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Example: Using Admissible Functions, Setting n=3
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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:
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Example: Using Comparison Function
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Example: Using Comparison Function
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Example: Using Comparison Function
Convergence to six decimal places is reached by the three lowest natural frequencies as follows:
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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
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AN ENHANCED RAYLEIGH-RITZ METHODThe linear combination can be made to satisfy the boundary condition for a spring-supported end
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AN ENHANCED RAYLEIGH-RITZ METHOD
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.
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AN ENHANCED RAYLEIGH-RITZ METHOD
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AN ENHANCED RAYLEIGH-RITZ METHOD
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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
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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.
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AN ENHANCED RAYLEIGH-RITZ METHOD
Solve the problem of privious example using the quasi-comparison functions
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Example: n=2
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Example: n=3
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AN ENHANCED RAYLEIGH-RITZ METHOD
School of Mechanical EngineeringIran University of Science and Technology
Distributed-Parameter Systems: Approximate Methods
Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Approximate Methods
Lecture 23
By: H. [email protected]
School of Mechanical EngineeringIran University of Science and Technology
Distributed-Parameter Systems: Approximate Methods
Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz MethodThe Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method
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The Assumed-Modes Method: System Response
known trial functions
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The Assumed-Modes Method: System Response
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The Assumed-Modes Method: System Response
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The Assumed-Modes Method: System Response
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
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The Assumed-Modes Method: System Response
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The Assumed-Modes Method: System Response
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The Assumed-Modes Method: System Response
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Damping Effects:
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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.
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A 2-DOF model for axial vibration of a uniform cantilever bar
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A 2-DOF model for axial vibration of a uniform cantilever bar
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Assumed-Modes Method: Bending of Bernoulli-Euler Beams
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A missile on launch pad
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A missile on launch pad
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Distributed-Parameter Systems: Approximate Methods
Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz MethodThe Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Approximate Methods
Lecture 24
By: H. [email protected]
School of Mechanical EngineeringIran University of Science and Technology
Distributed-Parameter Systems: Approximate Methods
Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz MethodThe Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method
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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.
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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.
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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
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THE GALERKIN METHODThe residual:
The Galerkin method gives:
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THE GALERKIN METHOD
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THE GALERKIN METHOD
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THE GALERKIN METHOD
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THE GALERKIN METHODConsider a viscously damped beam in transverse vibration.
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THE GALERKIN METHOD
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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.
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THE COLLOCATION METHOD: A beam in transverse vibration
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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.
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THE COLLOCATION METHOD: The tapered rod
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THE COLLOCATION METHOD: The tapered rod
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THE COLLOCATION METHOD: The tapered rod
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THE COLLOCATION METHOD: The tapered rod
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THE COLLOCATION METHOD: The tapered rod
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.
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THE COLLOCATION METHOD: The tapered rod
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.
School of Mechanical EngineeringIran University of Science and Technology
Distributed-Parameter Systems: Approximate Methods
Rayleigh's PrincipleThe Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz MethodThe Assumed-Modes Method: System Response The Galerkin MethodThe Collocation Method
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Advanced Vibrations
THE FINITE ELEMENT METHODLecture 25
By: H. [email protected]
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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.
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INTRODUCTION TO THE FINITE ELEMENT 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.
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INTRODUCTION TO THE FINITE ELEMENT 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.
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INTRODUCTION TO THE FINITE ELEMENT METHOD
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.
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ELEMENT STIFFNESS AND MASS MATRICES AND FORCE VECTOR
Uniform bar element undergoing axial deformation:
The shape functions must satisfy the BCs:
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ELEMENT STIFFNESS AND MASS MATRICES AND FORCE VECTOR
Considering axial deformation of the uniform element under static loads:
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ELEMENT STIFFNESS AND MASS MATRICES AND FORCE VECTOR
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Transverse Motion: Bernoulli-Euler Beam Theory
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Transverse Motion: Bernoulli-Euler Beam TheoryFor a beam loaded only at its ends, the equilibrium equation is:
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Transverse Motion: Bernoulli-Euler Beam Theory
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ExampleDetermine the generalized load vector for a beam element subjected to a uniform transverse load.
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Torsion
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ASSEMBLY OF SYSTEM MATRICES:• Scheme for the assembly of global matrices
from element matrices for second-order systems using linear interpolation functions
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ASSEMBLY OF SYSTEM MATRICES:
• Scheme for the assembly of global matrices from element matrices for fourth-order systems
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BOUNDARY CONDITIONS
• The Finite Element formulations inherently satisfy Free boundary conditions.
• Fixed BC’s:
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BOUNDARY CONDITIONS
• Supported Spring :
• Lumped Mass:
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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:
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Example 10.1:Variable cross section rod element
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Example 10.1:Variable cross section rod element
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Example 10.1:Variable cross section rod element
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Example 10.1:Assembelled Stiffness Matrix
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Example 10.1:Assembelled Mass Matrix
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Example 10.1:Exact Parameter Distributions
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Example 10.1: Convergence Study
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Advanced Vibrations Superaccurate finite element eigenvalue computation
Lecture 26
By: H. [email protected]
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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.
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A Fixed-Free Rod Finite Element Code
Consistent Mass Model
Lumped Mass Model
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Convergence Study of the 1st Mode
Consistent Mass Model
Lumped Mass Model
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Superaccurate finite element eigenvalue computation
Assembly of the linear finite elements over this mesh using the lumped mass matrix leads to:
Provided:
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Superaccurate finite element eigenvalue computation
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Superaccurate finite element eigenvalue computation
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Superaccurate finite element eigenvalue computation
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Superaccurate finite element eigenvalue computation
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.
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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
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Optimal element mass distributionWrite the general finite difference approximation:
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Three-nodes string element
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More details in:
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PARAMETRIC MODELS AND ERROR ANALYSIS FOR RODS
Rod parametric model:
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PARAMETRIC MODELS AND ERROR ANALYSIS FOR RODS
The equation of the ith node in the assembled finite element model
0 /2 .4 1/12
th
nd
th
k EA dxNo new reqθ
⇒ =
⇒
⇒ =
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PARAMETRIC MODELS AND ERROR ANALYSIS FOR BEAMS
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PARAMETRIC MODELS AND ERROR ANALYSIS FOR BEAMS
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PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES
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PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES
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PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES
School of Mechanical EngineeringIran University of Science and Technology
PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES
School of Mechanical EngineeringIran University of Science and Technology
PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES
School of Mechanical EngineeringIran University of Science and Technology
PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES
School of Mechanical EngineeringIran University of Science and Technology
PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES
School of Mechanical EngineeringIran University of Science and Technology
PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES
School of Mechanical EngineeringIran University of Science and Technology
PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES