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Transcript of Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue...
![Page 1: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/1.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions(Lecture 10)
By: H. [email protected]
![Page 2: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/2.jpg)
School of Mechanical EngineeringIran University of Science and Technology
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
![Page 3: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/3.jpg)
School of Mechanical EngineeringIran University of Science and Technology
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.
![Page 4: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/4.jpg)
School of Mechanical EngineeringIran University of Science and Technology
RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS
![Page 5: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/5.jpg)
School of Mechanical EngineeringIran University of Science and Technology
RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS
![Page 6: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/6.jpg)
School of Mechanical EngineeringIran University of Science and Technology
RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS
Ignoring 2nd order term
![Page 7: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/7.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
![Page 8: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/8.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
![Page 9: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/9.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
![Page 10: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/10.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
![Page 11: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/11.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
EOM
BC’s
![Page 12: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/12.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS
![Page 13: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/13.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS
![Page 14: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/14.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS
![Page 15: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/15.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS:EHP
![Page 16: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/16.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS:EHP
![Page 17: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/17.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS:EHP
![Page 18: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/18.jpg)
School of Mechanical EngineeringIran University of Science and Technology
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
![Page 19: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/19.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture 11)
By: H. [email protected]
![Page 20: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/20.jpg)
School of Mechanical EngineeringIran University of Science and Technology
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
![Page 21: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/21.jpg)
School of Mechanical EngineeringIran University of Science and Technology
FREE VIBRATION. THE DIFFERENTIAL EIGENVALUE PROBLEM
![Page 22: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/22.jpg)
School of Mechanical EngineeringIran University of Science and Technology
FREE VIBRATION. THE DIFFERENTIAL EIGENVALUE PROBLEM
On physical grounds
![Page 23: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/23.jpg)
School of Mechanical EngineeringIran University of Science and Technology
FREE VIBRATION. THE DIFFERENTIAL EIGENVALUE PROBLEMThe differential eigenvalue problem
![Page 24: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/24.jpg)
School of Mechanical EngineeringIran University of Science and Technology
FREE VIBRATION. THE DIFFERENTIAL EIGENVALUE PROBLEM
![Page 25: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/25.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Example:
![Page 26: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/26.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Example:
![Page 27: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/27.jpg)
School of Mechanical EngineeringIran University of Science and Technology
The free vibration of beams in bending:
The differential eigenvalue problem:
![Page 28: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/28.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Simply Supported beam:
![Page 29: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/29.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Uniform Clamped Beam:
0 1 0 1
1 0 1 0
− ( )sin β L − ( )cos β L ( )sinh β L ( )cosh β L
− ( )cos β L ( )sin β L ( )cosh β L ( )sinh β L
A
B
C
D
![Page 30: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/30.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Uniform Clamped Beam:
![Page 31: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/31.jpg)
School of Mechanical EngineeringIran University of Science and Technology
The spring supported-pinned beam
Characteristic equation
![Page 32: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/32.jpg)
School of Mechanical EngineeringIran University of Science and Technology
The spring supported-pinned beam =
− ( )sin λ ( )sinh λ
+ 25 ( )sin λ λ3 ( )cos λ − 25 ( )sinh λ λ3 ( )cosh λ
A
C : = f 0
![Page 33: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/33.jpg)
School of Mechanical EngineeringIran University of Science and Technology
ORTHOGONALITY OF MODES. EXPANSION THEOREMConsider two distinct solutions of the string eigenvalue problem:
![Page 34: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/34.jpg)
School of Mechanical EngineeringIran University of Science and Technology
ORTHOGONALITY OF MODES. EXPANSION THEOREM
![Page 35: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/35.jpg)
School of Mechanical EngineeringIran University of Science and Technology
ORTHOGONALITY OF MODES. EXPANSION THEOREM
To demonstrate the orthogonality relations for beams, we consider two distinct solutions of the eigenvalue problem:
![Page 36: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/36.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Orthogonality relations for beams
![Page 37: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/37.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Orthogonality relations for beams
![Page 38: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/38.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Expansion Theorem:Any function Y(x) representing a possible displacement of the system, with certain continuity, can be expanded in the absolutely and uniformly convergent series of the eigenfunctions:
The expansion theorem forms the basis for modal analysis, which permits the derivation of the response to both initial excitations and applied forces.
![Page 39: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/39.jpg)
School of Mechanical EngineeringIran University of Science and Technology
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
![Page 40: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/40.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture 12)
By: H. [email protected]
![Page 41: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/41.jpg)
School of Mechanical EngineeringIran University of Science and Technology
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
![Page 42: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/42.jpg)
School of Mechanical EngineeringIran University of Science and Technology
SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Rod with Tip Mass
![Page 43: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/43.jpg)
School of Mechanical EngineeringIran University of Science and Technology
SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Rod with Tip Mass
By means of the extended Hamilton's principle:
![Page 44: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/44.jpg)
School of Mechanical EngineeringIran University of Science and Technology
SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Rod with Tip Mass
![Page 45: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/45.jpg)
School of Mechanical EngineeringIran University of Science and Technology
SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Rod with Tip Mass
![Page 46: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/46.jpg)
School of Mechanical EngineeringIran University of Science and Technology
SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Beam with Lumped Tip Mass
![Page 47: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/47.jpg)
School of Mechanical EngineeringIran University of Science and Technology
SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Beam with Tip MassBy means of the extended Hamilton's principle:
![Page 48: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/48.jpg)
School of Mechanical EngineeringIran University of Science and Technology
SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Beam with Tip Mass
![Page 49: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/49.jpg)
School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
![Page 50: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/50.jpg)
School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIESThe orthogonality of modes:
![Page 51: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/51.jpg)
School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
![Page 52: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/52.jpg)
School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
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School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
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School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
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School of Mechanical EngineeringIran University of Science and Technology
Example 8.6. The eigenvalue problem for a uniform circular shaft in torsion
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School of Mechanical EngineeringIran University of Science and Technology
Example 8.6. The eigenvalue problem for a uniform circular shaft in torsion
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School of Mechanical EngineeringIran University of Science and Technology
Example 8.7. The eigenvalue problem for a uniform cantilever beam with tip mass
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School of Mechanical EngineeringIran University of Science and Technology
Example 8.7. The eigenvalue problem for a uniform cantilever beam with tip mass
As the mode number increases, the end acts more as a pinned end
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School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIESAny function U(x) representing a possible displacement of the continuous model, which implies that U(x) satisfies boundary conditions and is such that its derivatives up to the order appeared in the model is a continuous function, can be expanded in the absolutely and uniformly convergent series of the eigenfunctions:
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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture 13)
By: H. [email protected]
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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
RAYLEIGH'S QUOTIENT. VARIATIONAL APPROACH TO THE DIFFERENTIAL EIGENVALUE PROBLEM
Cases in which the differential eigenvalue problem admits a closed-form solution are very rare : Uniformly distributed parameters and Simple boundary conditions.
For the most part, one must be content with approximate solutions, Rayleigh's quotient plays a pivotal role.
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School of Mechanical EngineeringIran University of Science and Technology
The strong form of the eigenvalue problemA rod in axial vibration fixed at x=0 and with a spring of
stiffness k at x=L.
An exact solution of the eigenvalue problem in the strong form is beyond reach, The mass and stiffness parameters depend on the
spatial variable x .
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School of Mechanical EngineeringIran University of Science and Technology
The differential eigenvalue problem in a weak form
The solution of the differential eigenvalue problem is in a weighted average sense The test function V(x) plays the role of a
weighting function.The test function V(x) satisfies the geometric
boundary conditions and certain continuity requirments.
Test function
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School of Mechanical EngineeringIran University of Science and Technology
The differential eigenvalue problem in a weak form
Symmetrizing the left side
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School of Mechanical EngineeringIran University of Science and Technology
The differential eigenvalue problem in a weak form: Rayleigh's quotient
We consider the case in which the test function is equal to the trial function:
The value of R depends on the trial functionHow the value of R behaves as U(x) changes?
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School of Mechanical EngineeringIran University of Science and Technology
Properties of Rayleigh's quotient
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School of Mechanical EngineeringIran University of Science and Technology
Properties of Rayleigh's quotient
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School of Mechanical EngineeringIran University of Science and Technology
Properties of Rayleigh's quotient
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School of Mechanical EngineeringIran University of Science and Technology
Properties of Rayleigh's quotient
The trial function U(x) differs from the rth
eigenfunction Ur ( x ) by a small quantity of first order in and Rayleigh's quotient differs from the rth
eigenvalue by a small quantity of second order inRayleigh 's quotient has a stationary value at
an eigenfinction Ur(x), where the stationary value is the associated eigenvalue.
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School of Mechanical EngineeringIran University of Science and Technology
Properties of Rayleigh's quotient
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School of Mechanical EngineeringIran University of Science and Technology
Rayleigh's quotientA fixed-tip mass rod:
A pinned-spring supported beam in bending:
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School of Mechanical EngineeringIran University of Science and Technology
Rayleigh's quotient
Rayleigh's quotient for all systems have one thing in common: the numerator is a measure of the potential
energy and the denominator a measure of the kinetic
energy.
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School of Mechanical EngineeringIran University of Science and Technology
A fixed-spring supported rod in axial vibration
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School of Mechanical EngineeringIran University of Science and Technology
A fixed-spring supported rod in axial vibration
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School of Mechanical EngineeringIran University of Science and Technology
Example 8.8. Estimation of the lowest eigenvalue by means of Rayleigh's principle
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School of Mechanical EngineeringIran University of Science and Technology
Example 8.8: a) The static displacement curve as a trial function
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School of Mechanical EngineeringIran University of Science and Technology
Example 8.8: a) The static displacement curve as a trial function
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School of Mechanical EngineeringIran University of Science and Technology
Example 8.8: b)The lowest eigenfunction of a fixed-free string as a trial function
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School of Mechanical EngineeringIran University of Science and Technology
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
![Page 82: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/82.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture 14)
By: H. [email protected]
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School of Mechanical EngineeringIran University of Science and Technology
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
![Page 84: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/84.jpg)
School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO INITIAL EXCITATIONSVarious distributed-parameter systems exhibit
similar vibrational characteristics, although their mathematical description tends to differ in appearance.Consider the transverse displacement y(x,t) of
a string in free vibration
caused by initial excitations in the form of
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO INITIAL EXCITATIONS
the normal modes
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School of Mechanical EngineeringIran University of Science and Technology
Example:Response of a uniform string to the initial displacement y0(x)and zero initial velocity.
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School of Mechanical EngineeringIran University of Science and Technology
Example:
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO INITIAL EXCITATIONS:Beams in Bending Vibration
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO INITIAL EXCITATIONS:Beams in Bending Vibration
To demonstrate that every one of the natural modes can be excited independently of the other modes we select the initials as:
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO INITIAL EXCITATIONS:Response of systems with tip masses
Boundary conditions
Initial conditions
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO INITIAL EXCITATIONS:Response of systems with tip masses
Observing from boundary condition
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO INITIAL EXCITATIONS:Response of systems with tip masses
Similarly,
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School of Mechanical EngineeringIran University of Science and Technology
Example:Response of a cantilever beam with a lumped mass at the end to the initial velocity:
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School of Mechanical EngineeringIran University of Science and Technology
Example:
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School of Mechanical EngineeringIran University of Science and Technology
Example:
Because initial velocity resembles the 2nd mode
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO EXTERNAL EXCITATIONS
The various types of distributed-parameter systems differ more in appearance than in vibrational characteristics.We consider the response of a beam in
bending supported by a spring of stiffness k at x=0 and pinned at x=L.
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO EXTERNAL EXCITATIONS
Orthonormal modes
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO EXTERNAL EXCITATIONS: Harmonic Excitation
Controls which mode is excited.
Controls the resonance.
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School of Mechanical EngineeringIran University of Science and Technology
RESPONSE TO EXTERNAL EXCITATIONS: Arbitrary Excitation
The developments remain essentially the same for all other boundary conditions, and the same can be said about other systems.
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School of Mechanical EngineeringIran University of Science and Technology
ExampleDerive the response of a uniform pinned-pinned beam to a concentrated force of amplitude F0acting at x = L/2 and having the form of a step function.
Orthonormal Modes
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School of Mechanical EngineeringIran University of Science and Technology
Example
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School of Mechanical EngineeringIran University of Science and Technology
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
![Page 103: Welcome to the new GE PPT template! - iust.ac.ir · Free Vibration: The Differential Eigenvalue Problem Orthogonality of Modes Expansion Theorem Systems with Lumped Masses at the](https://reader031.fdocuments.us/reader031/viewer/2022022601/5b49b01a7f8b9a403d8b97b4/html5/thumbnails/103.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture 15)
By: H. [email protected]
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Stepped Beams
Free Vibrations of Stepped Beams Compatibility Requirements at the Interface Characteristic Equations
Elastically Restrained Stepped Beams Multi-Step Beam with Arbitrary Number of
Cracks Multi-Step Beam Carrying a Tip Mass
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FREE VIBRATION OF STEPPED BEAMS: EXACT SOLUTIONS
As presented by:S. K. JANG and C. W. BERT 1989 Journal of
Sound and Vibration 130, 342-346. Free vibration of stepped beams: exact and numerical solutions.They sought lowest natural frequency of a
stepped beam with two different cross-sections for various boundary conditions.
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FREE VIBRATION OF STEPPED BEAMS: EXACT SOLUTIONS
The governing differential equation for the small amplitude, free, lateral vibration of a Bernoulli-Euler beam is:
Assuming normal modes, one obtains the following expression for the mode shape:
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FREE VIBRATION OF STEPPED BEAMS: EXACT SOLUTIONS
For the shown stepped beam, one can rewrite the governing equation as:
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Boundary Conditions:(1) pinned-pinned,
(2) clamped-clamped,
(3) clamped-free,
(4) clamped-pinned,
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Compatibility Requirements at the Interface
Stress concentration at the junction of the two parts of the beam is neglected.
At the junction, the continuity of deflection, slope, moment and shear force has to be preserved:
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The clamped-clamped beam problem:Introducing the BCs
Yields: C3=-C1, C4=-C2, C7=-C5 . C8=-C6
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The clamped-clamped beam problem:Compatibility Requirements
Let:
Then the compatibility requirements yield:
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Characteristic Equations for Other BCs
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Characteristic Equations for Other BCs
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Characteristic Equations for Other BCs
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Characteristic Equations for Other BCs
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Characteristic Equations for Other BCs
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Exact Solutions:
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HIGHER MODE FREQUENCIES ANDEFFECTS OF STEPS ON FREQUENCYBy extending the computations, higher mode frequencies were found (Journal of Sound andVibration ,1989, 132(1), 164-168):
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Elastically Restrained Stepped Beams
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Boundary Conditions and Compatibility Requirements at the Interface :
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The Characteristic Equation
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Exact Solutions:
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Extension of the Research Work:
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Extension of the Research Work:
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Extension of the Research Work:
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Extension of the Research Work:
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Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture 16)
By: H. [email protected]
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INTRODUCTION
The problem of lateral vibrations of beams under axial loading is of considerable practical interest,Tall buildingsAerospace structuresRotating machinery shafts
Because of its important practical applications, the problem of uniform single-span beams under a constant axial load has been the subject of considerable study.
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BEAM FLEXURE: INCLUDING AXIAL-FORCE EFFECTS
Axial forces acting in a flexural element may have a very significant influence on the vibration behavior of the member, resulting generally in modifications of
frequencies and mode shapes.
The equation of motion, including the effect of a time-invariant uniform axial force throughout its length, is:
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BEAM FLEXURE: INCLUDING AXIAL-FORCE EFFECTS
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BEAM FLEXURE: INCLUDING AXIAL-FORCE EFFECTS
Separating variables:
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BEAM FLEXURE: INCLUDING AXIAL-FORCE EFFECTS
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Example: A simply supported uniform beam
D1=0, D3=0, D4=0.
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BEAM FLEXURE: INCLUDING AXIAL-FORCE EFFECTS
Retaining the constant axial force N, the governing equation can be used to find the static buckling loads and corresponding shapes:
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GALEF Formula E. GALEF 1968 Journal of the Acoustical
Society of America 44, (8), 643. Bending frequencies of compressed beams:
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GALEF FormulaA. BOKAIAN, “NATURAL FREQUENCIES OF BEAMS UNDERCOMPRESSIVE AXIAL LOADS”, Journal of Sound and Vibration (1988) 126(1), 49-65Studied the influence of a constant
compressive load on natural frequencies and mode shapes of a uniform beam with a variety of end conditions.Galef’s formula, previously assumed to be valid
for beams with all types of end conditions, is observed to be valid only for a few.
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GALEF FormulaBOKAIAN showed:The variation of the normalized natural
frequency with the normalized axial force for pinned-pinned, pinned-sliding and sliding-sliding beams is observed to be:
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GALEF FormulaBOKAIAN showed:Galef’s formula, previously
assumed to be valid for beams with all types of end conditions, is observed to be valid only for a few.The effect of end
constraints on natural frequency of a beam is significant only in the first few modes.
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For pinned-pinned, pinned-sliding and sliding-sliding beams, this variation may exactly be expressed as 0 = 1 + i?. This formula may be used for beams with other types
of end constraints when the beam vibrates in a third mode or higher. For beam with other types of boundary conditions, this
approximation may be expressed as 0 = F 1 + yU (y < where the coefficient y depends only on the type of the end constraints.
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The study is an investigation of the combined effects of compressive inertia forces due to a conservative
model of steady thrust and uniform mass depletion on the transverse vibration
characteristics of a single stage variable mass rocket.the effect of the aerodynamic drag in comparison to
the thrust is considered to be negligible and the rocket is structurally modeled as a non-uniform
slender beam representative of practical rocket configurations.
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In the study the typical single stage rocket structure is divided into a number of segments.Within which the
bending rigidity, axial compressive force and the mass distributions can be approximated as constants.
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The non-dimensional equation of motion for the ithconstant beam segment :
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The free-free boundary conditions are:
and the continuity conditions are
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The variation of frequency parameter and cyclic frequency versus the mass depletion parameter Md for the first three modes of vibration of a typical rocket executing a constant acceleration trajectory.
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Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture 17)
By: H. [email protected]
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SYSTEMS WITH EXTERNAL FORCES AT BOUNDARIES
The 2nd of boundary conditions is nonhomogeneous, precludes the use of modal analysis for the
response.
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SYSTEMS WITH EXTERNAL FORCES AT BOUNDARIESWe can reformulate the problem by rewriting the differential equation in the form:
and the boundary conditions as:
Now the solution can be obtained routinely by modal analysis.Any shortcomings?
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SYSTEMS WITH EXTERNAL FORCES AT BOUNDARIES: ExampleObtain the response of a uniform rod, fixed at x=0 and subjected to a boundary force at x=L in the form:
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SYSTEMS WITH EXTERNAL FORCES AT BOUNDARIES: Example
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System Dynamics in the Feed Direction
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System Dynamics in the Cutting Direction
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Modeling Tool as Stepped Beam on Elastic Support: Boundary Conditions
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School of Mechanical EngineeringIran University of Science and Technology
Modeling Tool as Stepped Beam on Elastic Support: The compatibility requirements
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School of Mechanical EngineeringIran University of Science and Technology
Modeling Tool as Stepped Beam on Elastic Support
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School of Mechanical EngineeringIran University of Science and Technology
Modeling Tool as Stepped Beam on Elastic Support
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
VIBRATION OF PLATESLecture 17-1
By: H. [email protected]
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School of Mechanical EngineeringIran University of Science and Technology
VIBRATION OF PLATES
Plates have bending stiffness in a manner similar to beams in bending.In the case of plates one can think of two
planes of bending, producing in general two distinct curvatures. The small deflection theory of thin plates, called
classical plate theory or Kirchhoff theory, is based on assumptions similar to those used in thin beam or Euler-Bernoulli beam theory.
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School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: CLASSICAL PLATE THEORYThe elementary theory of plates is based on the following assumptions: The thickness of the plate (h) is small compared to its lateral
dimensions. The middle plane of the plate does not undergo in-plane
deformation. Thus, the midplane remains as the neutral plane after deformation or bending.
The displacement components of the midsurface of the plate are small compared to the thickness of the plate.
The influence of transverse shear deformation is neglected. This implies that plane sections normal to the midsurface before deformation remain normal to the rnidsurface even after deformation or bending.
The transverse normal strain under transverse loading can be neglected. The transverse normal stress is small and hence can be neglected compared to the other components of stress.
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School of Mechanical EngineeringIran University of Science and Technology
Moment - Shear Force Resultants:
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School of Mechanical EngineeringIran University of Science and Technology
Equation of motion
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School of Mechanical EngineeringIran University of Science and Technology
BOUNDARY CONDITIONS
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School of Mechanical EngineeringIran University of Science and Technology
BOUNDARY CONDITIONS
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School of Mechanical EngineeringIran University of Science and Technology
BOUNDARY CONDITIONS: Free Edge
There are three boundary conditions, whereas the equation of motion requires only two:
Kirchhoff showed that the conditions on the shear force and the twisting moment are not independent and can be combined into only one boundary condition.
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School of Mechanical EngineeringIran University of Science and Technology
BOUNDARY CONDITIONS: Free Edge
Replacing the twisting moment by an equivalent vertical force.
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School of Mechanical EngineeringIran University of Science and Technology
BOUNDARY CONDITIONS
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School of Mechanical EngineeringIran University of Science and Technology
BOUNDARY CONDITIONS
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School of Mechanical EngineeringIran University of Science and Technology
BOUNDARY CONDITIONS
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School of Mechanical EngineeringIran University of Science and Technology
FREE VIBRATION OF RECTANGULAR PLATES
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School of Mechanical EngineeringIran University of Science and Technology
FREE VIBRATION OF RECTANGULAR PLATES
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School of Mechanical EngineeringIran University of Science and Technology
FREE VIBRATION OF RECTANGULAR PLATES
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School of Mechanical EngineeringIran University of Science and Technology
Solution for a Simply Supported Plate
We find that all the constants Ai except A1 and
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School of Mechanical EngineeringIran University of Science and Technology
Solution for a Simply Supported Plate
The initial conditions of the plate are:
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School of Mechanical EngineeringIran University of Science and Technology
Solution for a Simply Supported Plate
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School of Mechanical EngineeringIran University of Science and Technology
Solution for a Simply Supported Plate
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School of Mechanical EngineeringIran University of Science and Technology
Solution for a Simply Supported Plate
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
VIBRATION OF PLATESLecture 17-2
By: H. [email protected]
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular Plates
The functions X(x) and Y(y) can be separated provided either of the followings are satisfied:
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular Plates
These equations can be satisfied only by the trigonometric functions:
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular Plates
Assume that the plate is simply supported along edges x =0 and x =a:
Implying:
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular PlatesThe various boundary conditions can be stated,
SS-SS-SS-SS, SS-C-SS-C, SS-F-SS-F, SS-C-SS-SS, SS-F-SS-SS, SS-F-SS-C
Assuming:
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular Plates
y = 0 and y = b are simply supported:
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular Plates
y = 0 and y = b are simply supported:
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular Plates
y = 0 and y = b are clamped:
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular Plates
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School of Mechanical EngineeringIran University of Science and Technology
Vibrations of Rectangular Plates
Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular platesShahrokh Hosseini Hashemi and M. Arsanjani ,International Journal of Solids
and Structures Volume 42, Issues 3-4, February 2005, Pages 819-853
Exact solution for linear buckling of rectangular Mindlin platesShahrokh Hosseini-Hashemi, Korosh Khorshidi, and Marco Amabili, Journal of
Sound and Vibration Volume 315, Issues 1-2, 5 August 2008, Pages 318-342
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School of Mechanical EngineeringIran University of Science and Technology
FORCED VIBRATION OF RECTANGULAR PLATES
the normal modes
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School of Mechanical EngineeringIran University of Science and Technology
FORCED VIBRATION OF RECTANGULAR PLATES
Using a modal analysis procedure:
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School of Mechanical EngineeringIran University of Science and Technology
FORCED VIBRATION OF RECTANGULAR PLATESThe response of simply supported rectangular plates:
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
VIBRATION OF PLATESLecture 17-2
By: H. [email protected]
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School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach
To develop the strain energy one may assume the state of stress in a thin plate as plane stress:
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School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach
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School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach
Extended Hamilton's principle can be written as:
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School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach
( )∫∫∫ +=
∂∂
−∂∂
CA
dyFdxFdydxyF
xF
2121
Green’s Theorem
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School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach
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School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach
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School of Mechanical EngineeringIran University of Science and Technology
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School of Mechanical EngineeringIran University of Science and Technology
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School of Mechanical EngineeringIran University of Science and Technology
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School of Mechanical EngineeringIran University of Science and Technology
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Approximate Methods
Lecture 18
By: H. [email protected]
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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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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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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
THE RAYLEIGH-RITZ METHOD
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School of Mechanical EngineeringIran University of Science and Technology
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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School of Mechanical EngineeringIran University of Science and Technology
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
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Approximate Methods
Lecture 19
By: H. [email protected]
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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
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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