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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture Ten)
By: H. [email protected]
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
School of Mechanical EngineeringIran University of Science and Technology
FREE VIBRATION. THE DIFFERENTIAL EIGENVALUE PROBLEM
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FREE VIBRATION. THE DIFFERENTIAL EIGENVALUE PROBLEM
On physical grounds
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FREE VIBRATION. THE DIFFERENTIAL EIGENVALUE PROBLEMThe differential eigenvalue problem
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FREE VIBRATION. THE DIFFERENTIAL EIGENVALUE PROBLEM
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The free vibration of beams in bending:
The differential eigenvalue problem:
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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
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The spring supported-pinned beam
Characteristic equation
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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
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ORTHOGONALITY OF MODES. EXPANSION THEOREMConsider two distinct solutions of the string eigenvalue problem:
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ORTHOGONALITY OF MODES. EXPANSION THEOREM
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ORTHOGONALITY OF MODES. EXPANSION THEOREM
To demonstrate the orthogonality relations for beams, we consider two distinct solutions of the eigenvalue problem:
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Orthogonality relations for beams
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Orthogonality relations for beams
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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.
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
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture Eleven)
By: H. [email protected]
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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SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Rod with Tip Mass
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SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Rod with Tip Mass
By means of the extended Hamilton's principle:
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SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Rod with Tip Mass
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SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Rod with Tip Mass
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SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Beam with Lumped Tip Mass
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SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Beam with Tip MassBy means of the extended Hamilton's principle:
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SYSTEMS WITH LUMPED MASSES AT THE BOUNDARIES: Beam with Tip Mass
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EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
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EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIESThe orthogonality of modes:
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EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
School of Mechanical EngineeringIran University of Science and Technology
EIGENVALUE PROBLEM AND EXPANSION THEOREM FOR PROBLEMS WITH LUMPED MASSES AT THE BOUNDARIES
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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Example 8.6. The eigenvalue problem for a uniform circular shaft in torsion
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Example 8.6. The eigenvalue problem for a uniform circular shaft in torsion
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Example 8.7. The eigenvalue problem for a uniform cantilever beam with tip mass
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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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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:
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
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture Twelve)
By: H. [email protected]
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
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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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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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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The differential eigenvalue problem in a weak form
Symmetrizing the left side
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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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Properties of Rayleigh's quotient
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Properties of Rayleigh's quotient
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Properties of Rayleigh's quotient
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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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Properties of Rayleigh's quotient
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Rayleigh's quotientA fixed-tip mass rod:
A pinned-spring supported beam in bending:
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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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A fixed-spring supported rod in axial vibration
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A fixed-spring supported rod in axial vibration
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Example 8.8. Estimation of the lowest eigenvalue by means of Rayleigh's principle
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Example 8.8: a) The static displacement curve as a trial function
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Example 8.8: a) The static displacement curve as a trial function
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Example 8.8: b)The lowest eigenfunction of a fixed-free string as a trial function
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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
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture Thirteen)
By: H. [email protected]
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
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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RESPONSE TO INITIAL EXCITATIONS
the normal modes
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Example:Response of a uniform string to the initial displacement y0(x)and zero initial velocity.
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RESPONSE TO INITIAL EXCITATIONS:Beams in Bending Vibration
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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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RESPONSE TO INITIAL EXCITATIONS:Response of systems with tip masses
Boundary conditions
Initial conditions
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RESPONSE TO INITIAL EXCITATIONS:Response of systems with tip masses
Observing from boundary condition
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RESPONSE TO INITIAL EXCITATIONS:Response of systems with tip masses
Similarly,
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Example:Response of a cantilever beam with a lumped mass at the end to the initial velocity:
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Example:
Because initial velocity resembles the 2nd mode
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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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RESPONSE TO EXTERNAL EXCITATIONS
Orthonormal modes
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RESPONSE TO EXTERNAL EXCITATIONS: Harmonic Excitation
Controls which mode is excited.
Controls the resonance.
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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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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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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
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions(Lecture Fourteen)
By: H. [email protected]
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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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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Modeling Tool as Stepped Beam on Elastic Support: The compatibility requirements
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Modeling Tool as Stepped Beam on Elastic Support
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Modeling Tool as Stepped Beam on Elastic Support
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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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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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Extension of the Research Work:
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Extension of the Research Work:
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Advanced Vibrations
LAGRANGE'S EQUATION FOR DISTRIBUTED SYSTEMS
Lecture Fifteen
By: H. [email protected]
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Distributed-Parameter Systems,The extended Hamilton's principle:
kinetic energy density
potential energy densitypotential energy due to springs at the ends
the distributed force
The effect of lumped boundary masses in translation and rotation
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Distributed-Parameter Systems,The extended Hamilton's principle:The Lagrangian can be expressed as:
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The extended Hamilton's principle:Lagrange differential equation of motion
boundary conditions
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Example 1:Derive the Lagrange equation and boundary
conditions for shown string,
Solution: This is only a second-order system, so that deleting derivatives higher than two the Lagrange equation reduces to:
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Example 1: BC’sThe physics of the problem dictates the BC’s:
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Example 2:Derive the Lagrange equation and boundary conditions for the rotating cantilever beam.
The potential energy consists of two parts, one due to bending and one due to the axial force, where the latter is entirely analogous to the potential energy in a string,
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Example 3: DISTRIBUTED GYROSCOPIC SYSTEMSDerive the boundary-value problem for a rotating elastic shaft simply supported at both ends.
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Example 3: DISTRIBUTED GYROSCOPIC SYSTEMS
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Example 3: DISTRIBUTED GYROSCOPIC SYSTEMS
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Example 4: Rayleigh Beam
Derive the boundary-value problem for the beam in bending vibration of Ex.2 under the assumption that the kinetic energy of rotation is not negligible.
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Example 4: Rayleigh Beam
Boundary conditions at each end:
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BEAM FLEXURE: INCLUDING AXIAL-FORCE EFFECTSThe 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
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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
THE DIFFERENTIAL EIGENVALUE PROBLEM FOR SELF-ADJOINT
SYSTEMS
By: H. [email protected]
School of Mechanical EngineeringIran University of Science and Technology
THE DIFFERENTIAL EIGENVALUE PROBLEM FOR SELF-ADJOINT SYSTEMS
Various systems have many things in common: It is convenient to formulate the differential eigenvalue problem so
as to apply to large classes of systems.
Let w be a function of one or two independent spatial variables x or x and y, and consider the homogeneous differential expression: homogeneous differential expression
of order 2p
Homogeneous differential operator of order 2p
Linear operator
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The Differential Eigenvalue Problem for Self-Adjoint SystemsConsider a generic eigenvalue problem in the operator form:
the domain of definitionstiffness operator of order 2p the boundary
boundary operators
Let two 2p times differentiable trial functions u and v satisfying all the boundary conditions
The differential operator L is self-adjoint if:
Self-adjointness implies certain mathematical symmetry and can be ascertained through integrations by parts with due consideration to the boundary conditions.
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The Differential Eigenvalue Problem for Self-Adjoint SystemsDenote the symmetric form in u and v resulting from
integrations by parts of (u,Lv) by [u,v] and refer to it as an energy inner product.
For one-dimensional domains, the energy inner product has the general expression: Functions of x
The beam with tip spring example:
energy inner product
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The Differential Eigenvalue Problem for Self-Adjoint Systems
Assume that the problem is self-adjoint and consider two distinct solutions:
distinct eigenvalues
Orthogonality of the eigenfunctions is a direct consequence of the system being self-adjoint.
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The Differential Eigenvalue Problem for Self-Adjoint SystemsWe assumed on physical grounds that the eigenvalues and
eigenfunctions are real.
We propose to prove here mathematically that this is indeed so, provided the system is self-adjoint. the complex conjugate pair
the integral is real and positive,
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Example 5:Consider the eigenvalue problem for the string in transverse
vibration, demonstrate that the problem fits one of the generic formulations, check whether the problem is self-adjoint and positive definite and draw conclusions.
p=1
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Example 5:Check for self-adjointness,
Let v = u and obtain the energy norm squared
the operator L, and hence the system, is positive definite; all the eigenvalues are positive.
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Example 6: DISTRIBUTED GYROSCOPIC SYSTEMSDerive the eigen-value problem for a rotating elastic shaft simply supported at both ends.
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Example 6: DISTRIBUTED GYROSCOPIC SYSTEMSThe boundary conditions:
The eigen-functions of the system are:
The free vibration solution takes the form:
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Example 6: DISTRIBUTED GYROSCOPIC SYSTEMS
Projection of equation of motion with the introduced solution on each mode results:
where
2 2 2
2 2 2
2 0.2 0.
jj
jj
ab
λ ω λλ λ ω
′ + −Ω − Ω = ′Ω − −Ω
( ) ( )2 1 2 1 2 2, , ,j j j j j ji iλ λ ω λ λ ω− − ′ ′= ± Ω+ = ± Ω−
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TIMOSHENKO BEAMThe model of a beam including both rotatory inertia
and shear deformation effects is commonly referred to as a Timoshenko beam.
The total deflection w(x,t) of the beam consists of two parts, one caused by bending and one by shear.
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TIMOSHENKO BEAM
Bendingrotation
Sheardistortion
The moment/bending deformation, and shear/ shearing deformation:
Potential energy
Kinetic energy
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TIMOSHENKO BEAMTo formulate the boundary-value problem, we
make use of the extended Hamilton's principle,
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TIMOSHENKO BEAM: The eigenvalueproblemThe boundary-value problem is separable in x and
t, or
where F(t) is harmonic and it satisfies
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TIMOSHENKO BEAM: The eigenvalueproblem
By analogy with the scalar definitions, the problem is self-adjointif
Moreover, the problem is positive definite if
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TIMOSHENKO BEAM: The eigenvalueproblem
The integral on the right side is symmetric in u and v. Hence, all systems for which the boundary term is zero are self-adjoint.
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TIMOSHENKO BEAM: The eigenvalueproblemAssuming that the boundary term is zero and letting v = u, we have
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Advanced Vibrations
VIBRATION OF PLATESLecture 17
By: H. [email protected]
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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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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Strain-Displacement RelationsThe in-plane displacements:
Linear strain-displacement relations:
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Moment-Displacement RelationsAssuming the plate in a state of plane stress:
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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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BOUNDARY CONDITIONS: Free Edge
Replacing the twisting moment by an equivalent vertical force.
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FREE VIBRATION OF RECTANGULAR PLATES
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FREE VIBRATION OF RECTANGULAR PLATES
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FREE VIBRATION OF RECTANGULAR PLATES
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Solution for a Simply Supported Plate
We find that all the constants Ai except A1 and
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Solution for a Simply Supported Plate
The initial conditions of the plate are:
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Solution for a Simply Supported Plate
School of Mechanical EngineeringIran University of Science and Technology
Solution for a Simply Supported Plate
School of Mechanical EngineeringIran University of Science and Technology
Solution for a Simply Supported Plate
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
VIBRATION OF PLATESLecture 18
By: H. [email protected]
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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Vibrations of Rectangular Plates
These equations can be satisfied only by the trigonometric functions:
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Vibrations of Rectangular Plates
Assume that the plate is simply supported along edges x =0 and x =a:
Implying:
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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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Vibrations of Rectangular Plates
y = 0 and y = b are simply supported:
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Vibrations of Rectangular Plates
y = 0 and y = b are simply supported:
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Vibrations of Rectangular Plates
y = 0 and y = b are clamped:
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Vibrations of Rectangular Plates
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FORCED VIBRATION OF RECTANGULAR PLATES
the normal modes
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FORCED VIBRATION OF RECTANGULAR PLATES
Using a modal analysis procedure:
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FORCED VIBRATION OF RECTANGULAR PLATESThe response of simply supported rectangular plates:
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Advanced Vibrations
VIBRATION OF PLATESLecture 19
By: H. [email protected]
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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EQUATION OF MOTION: Variational Approach
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EQUATION OF MOTION: Variational ApproachExtended Hamilton's principle can be written as:
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EQUATION OF MOTION: Variational Approach
( )∫∫∫ +=
∂∂
−∂∂
CA
dyFdxFdydxyF
xF
2121
Green’s Theorem
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EQUATION OF MOTION: Variational Approach
School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach
School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach
School of Mechanical EngineeringIran University of Science and Technology
EQUATION OF MOTION: Variational Approach