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![Page 1: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/1.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations Lecture One
Elements of Analytical Dynamics
By: H. [email protected]
![Page 2: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/2.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Elements of Analytical DynamicsNewton's laws were formulated for a single particleCan be extended to systems of particles.The equations of motion are expressed in terms of
physical coordinates vector and force vector. For this reason, Newtonian mechanics is often referred to
as vectorial mechanics.The drawback is that it requires one free-body diagram for each of the masses, Necessitating the inclusion of reaction forces and
interacting forces. These reaction and constraint forces play the role of
unknowns, which makes it necessary to work with a surplus of equations of motion, one additional equation for every unknown force.
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School of Mechanical EngineeringIran University of Science and Technology
Elements of Analytical DynamicsAnalytical mechanics, or analytical dynamics, considers the system as a whole: Not separate individual components,This excludes the reaction and constraint
forces automatically.This approach, due to Lagrange, permits the formulation of problems of dynamics in terms of: two scalar functions the kinetic energy and the
potential energy, and an infinitesimal expression, the virtual work
performed by the nonconservative forces.
![Page 4: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/4.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Elements of Analytical DynamicsIn analytical mechanics the equations of motion are formulated in terms of generalized coordinates and generalized forces: Not necessarily physical coordinates and forces. The formulation is independent of any special system
of coordinates.
The development of analytical mechanics required the introduction of the concept of virtual displacements, Ied to the development of the calculus of variations. For this reason, analytical mechanics is often referred
to as the variational approach to mechanics.
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School of Mechanical EngineeringIran University of Science and Technology
6 Elements of Analytical Dynamics
6.1 DOF and Generalized Coordinates
6.2 The Principle of Virtual Work
6.3 The Principle of D'Alembert
6.4 The Extended Hamilton's Principle
6.5 Lagrange's Equations
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School of Mechanical EngineeringIran University of Science and Technology
6.1 DEGREES OF FREEDOM AND GENERALIZED COORDINATESA source of possible difficulties in using Newton's equations is use of physical coordinates, which may not always be independent.
Independent coordinates
The generalized coordinates are not unique
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School of Mechanical EngineeringIran University of Science and Technology
6.2 THE PRINCIPLE OF VIRTUAL WORKThe principle of virtual work, due to Johann Bernoulli, is basically a statement of the static equilibrium of a mechanical system.We consider a system of N particles and define the virtual displacements, as infinitesimal changes in the coordinates.The virtual displacements must be consistent with the system constraints, but are otherwise arbitrary.The virtual displacements, being infinitesimal, obey the rules of differential calculus.
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School of Mechanical EngineeringIran University of Science and Technology
THE PRINCIPLE OF VIRTUAL WORK
applied force constraint forceresultant force on each particle
The virtual work performed by the constraint forces is zero
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School of Mechanical EngineeringIran University of Science and Technology
THE PRINCIPLE OF VIRTUAL WORKWhen ri are independent,
If not to switch to a set of generalized coordinates:
Generalized forces
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School of Mechanical EngineeringIran University of Science and Technology
THE PRINCIPLE OF D'ALEMBERT
The virtual work principle can be extended to dynamics, in which form it is known as d'Alembert's principle.
Lagrange version of d'Alembertls principle
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School of Mechanical EngineeringIran University of Science and Technology
Example1:,
,
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School of Mechanical EngineeringIran University of Science and Technology
Example2:Derive the equation of motion for the systems of Example 1 by means of the generalizedd'Alembert's principle.
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
The virtual work of all the applied forces,
The kinetic energy of particle mi
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
It is convenient to choose
Extended Hamilton's principle
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
where V is the potential energy
Or in terms of the independent generalized coordinates
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School of Mechanical EngineeringIran University of Science and Technology
A Note:
The work performed by the force F in moving the particle m from position r1 to position r2 is responsible for a change in the kinetic energy from TI to T2.
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School of Mechanical EngineeringIran University of Science and Technology
A Note (continued):
The work performed by conservative forces in moving a particle from r1 to r2 is equal to the negative of the change in the potential energy from V1 to V2
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School of Mechanical EngineeringIran University of Science and Technology
A Note (continued):
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School of Mechanical EngineeringIran University of Science and Technology
Example3:
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School of Mechanical EngineeringIran University of Science and Technology
Example3: cont.
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School of Mechanical EngineeringIran University of Science and Technology
Example3: cont.
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School of Mechanical EngineeringIran University of Science and Technology
6 Elements of Analytical Dynamics
6.1 DOF and Generalized Coordinates
6.2 The Principle of Virtual Work
6.3 The Principle of D'Alembert
6.4 The Extended Hamilton's Principle
6.5 Lagrange's Equations
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations Lecture Two:
LAGRANGE'S EQUATIONS
By: H. [email protected]
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
LagrangianHamilton's principle
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
Use the extended Hamilton's principle to derive the equations of motion forthe two-degree-of-freedom system.
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
Only the virtual displacements are arbitrary.
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
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School of Mechanical EngineeringIran University of Science and Technology
THE EXTENDED HAMILTON'S PRINCIPLE
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School of Mechanical EngineeringIran University of Science and Technology
LAGRANGE'S EQUATIONS
For many problems the extended Hamilton's principle is not the most efficient method for deriving equations of motion: Involves routine operations that must be
carried out every time the principle is applied, The integrations by parts.
The extended Hamilton's principle is used to generate a more expeditious method for deriving equations of motion, Lagrange's equations.
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School of Mechanical EngineeringIran University of Science and Technology
LAGRANGE'S EQUATIONS
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School of Mechanical EngineeringIran University of Science and Technology
LAGRANGE'S EQUATIONS
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School of Mechanical EngineeringIran University of Science and Technology
LAGRANGE'S EQUATIONS
Derive Lagrange's equations of motion for the system
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School of Mechanical EngineeringIran University of Science and Technology
LAGRANGE'S EQUATIONS
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School of Mechanical EngineeringIran University of Science and Technology
LAGRANGE'S EQUATIONS
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School of Mechanical EngineeringIran University of Science and Technology
Example of a non-natural systemThe system consists of a mass m connected to a rigid ring through a viscous damper and two nonlinear springs. The mass m is subjected to external damping forces proportional to the absolute velocities X and Y, where the proportionality constant is h.
The Rayleigh dissipation function:
Two nonlinear springs
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School of Mechanical EngineeringIran University of Science and Technology
The Kinetic Energy:
When T2 = T, T1 = T0 = 0, the system is said to be natural
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School of Mechanical EngineeringIran University of Science and Technology
The potential energy/The Lagrangian:
U=V-T0
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School of Mechanical EngineeringIran University of Science and Technology
The Rayleigh dissipation function:
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School of Mechanical EngineeringIran University of Science and Technology
Lagrange's equations of motion
The gyroscopic matrix
The damping matrixThe circulatory matrix
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School of Mechanical EngineeringIran University of Science and Technology
Final word:
Lagrange's equations are more efficient, the extended Hamilton principle is more versatile.
In fact, it can produce results in cases in which Lagrange's equations cannot, most notably in the case of distributed-parameter systems.
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations Lecture Three:
MULTI-DEGREE-OF-FREEDOM SYSTEMS
By: H. [email protected]
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School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
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School of Mechanical EngineeringIran University of Science and Technology
7.1 EQUATIONS OF MOTION FOR LINEAR SYSTEMS
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School of Mechanical EngineeringIran University of Science and Technology
7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTSThe stiffness coefficients can be obtained by other means, not necessarily involving the equations of motion. The stiffness coefficients are more properly
known as stiffness influence coefficients, and can be derived by using its definition.
There is one more type of influence coefficients, namely, flexibility influence coefficients. They are intimately related to the stiffness
influence coefficients.
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School of Mechanical EngineeringIran University of Science and Technology
7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTS
We define the flexibility influence coefficient aijas the displacement of point xi, due to a unit force, Fj = 1.
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School of Mechanical EngineeringIran University of Science and Technology
7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTS
The stiffness influence coefficient kij is the force required at xi to produce a unit displacement at point xj, and displacements at all other points are zero. To obtain zero displacements at all points the
forces must simply hold these points fixed.
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School of Mechanical EngineeringIran University of Science and Technology
7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTS
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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
7.3 PROPERTIES OF THE STIFFNESS AND MASS COEFFICIENTSThe potential energy of a single linear spring:
By analogy the elastic potential energy for a system is:
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School of Mechanical EngineeringIran University of Science and Technology
Symmetry Property:
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School of Mechanical EngineeringIran University of Science and Technology
Maxwell's reciprocity theorem:
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School of Mechanical EngineeringIran University of Science and Technology
7.4 LAGRANGE'S EQUATIONS LINEARIZED ABOUT EQUILIBRIUM
Rayleigh's dissipation function
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School of Mechanical EngineeringIran University of Science and Technology
7.4 LAGRANGE'S EQUATIONS LINEARIZED ABOUT EQUILIBRIUM
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School of Mechanical EngineeringIran University of Science and Technology
7.4 LAGRANGE'S EQUATIONS LINEARIZED ABOUT EQUILIBRIUM
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School of Mechanical EngineeringIran University of Science and Technology
7.4 LAGRANGE'S EQUATIONS LINEARIZED ABOUT EQUILIBRIUM
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School of Mechanical EngineeringIran University of Science and Technology
7.5 LINEAR TRANSFORMATIONS. COUPLING
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School of Mechanical EngineeringIran University of Science and Technology
Derivation of the matrices M' and K' in a more natural manner
![Page 59: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/59.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
![Page 60: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/60.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations Lecture Four:
Multi-Degree-of-Freedom Systems (Ch7)
By: H. [email protected]
![Page 61: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/61.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
![Page 62: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/62.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM
synchronous motion
![Page 63: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/63.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM
![Page 64: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/64.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM
In general, all frequencies are distinct, except:In degenerate cases, They cannot occur in one-dimensional structures; They can occur in two-dimensional symmetric
structures.
characteristic polynomial
![Page 65: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/65.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM
The shape of the natural modes is unique but the amplitude is not.
A very convenient normalization scheme consists of setting:
![Page 66: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/66.jpg)
School of Mechanical EngineeringIran University of Science and Technology
THE SYMMETRIC EIGENVALUE PROBLEM: LINEAR CONSERVATIVE NATURAL SYSTEMS
Standard eigenvalue problem
![Page 67: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/67.jpg)
School of Mechanical EngineeringIran University of Science and Technology
THE SYMMETRIC EIGENVALUE PROBLEM: LINEAR CONSERVATIVE NATURAL SYSTEMS
Eigenvalues and eigenvectors associated with real symmetric matrices are real.
To demonstrate these properties, we consider these eigenvalue, eigenvector are complex:
Norm of a vector is a positive number , therefore:
The eigenvalues of a real symmetric matrix are real.
As a corollary, the eigenvectors of a real symmetric matrix are real.
![Page 68: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/68.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM
![Page 69: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/69.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Free vibration for the initial excitations
![Page 70: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/70.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Free vibration for the initial excitations
![Page 71: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/71.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Free vibration for the initial excitations
![Page 72: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/72.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Free vibration for the initial excitations
![Page 73: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/73.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Free vibration for the initial excitations
![Page 74: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/74.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.7 ORTHOGONALITY OF MODAL VECTORS
![Page 75: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/75.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.7 ORTHOGONALITY OF MODAL VECTORS
![Page 76: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/76.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.8 SYSTEMS ADMITTING RIGID-BODY MOTIONS
![Page 77: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/77.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.8 SYSTEMS ADMITTING RIGID-BODY MOTIONS
The orthogonality of the rigid-body mode to the elastic modes is equivalent to the preservation of zero angular momentum in pure elastic motion.
![Page 78: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/78.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.8 SYSTEMS ADMITTING RIGID-BODY MOTIONS
![Page 79: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/79.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.8 SYSTEMS ADMITTING RIGID-BODY MOTIONS
![Page 80: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/80.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.8 SYSTEMS ADMITTING RIGID-BODY MOTIONS
![Page 81: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/81.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
![Page 82: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/82.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Lecture Five
By: H. [email protected]
![Page 83: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/83.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
![Page 84: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/84.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.9 Decomposition of the Response in Terms of Modal Vectors
![Page 85: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/85.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.9 Decomposition of the Response in Terms of Modal Vectors
The modal vectors are orthonormal with respect to the mass matrix M,
![Page 86: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/86.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.10 Response to Initial Excitations by Modal Analysis
![Page 87: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/87.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.10 Response to Initial Excitations by Modal Analysis
Modal Coordinates
![Page 88: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/88.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.10 Response to Initial Excitations by Modal Analysis
![Page 89: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/89.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.10 Response to Initial Excitations by Modal Analysis
We wish to demonstrate that each of the natural modes can be excited independently of the other;
![Page 90: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/90.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix
![Page 91: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/91.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue Problem
![Page 92: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/92.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue Problem
![Page 93: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/93.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue ProblemSolving the eigenvalue problem by finding the principle axes of the ellipse.
![Page 94: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/94.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue Problem
Transforming to canonical form implies elimination of cross products:
![Page 95: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/95.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue Problem
![Page 96: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/96.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue Problem
Obtaining the angle, one may calculate the eigenvalues and eigenvectors:
![Page 97: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/97.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue Problem
Example: Solving the eigenvalue problem by finding the principal axes of the corresponding ellipse.
![Page 98: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/98.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue Problem
![Page 99: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/99.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.12 Geometric Interpretation of the Eigenvalue Problem
![Page 100: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/100.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
![Page 101: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/101.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Lecture Six
By: H. [email protected]
![Page 102: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/102.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
![Page 103: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/103.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES
Rayleigh's quotient
![Page 104: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/104.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES
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School of Mechanical EngineeringIran University of Science and Technology
7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES
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School of Mechanical EngineeringIran University of Science and Technology
7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES
![Page 107: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/107.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES
Of special interest in vibrations is the fundamental frequency.
Rayleigh's quotient is an upper bound for the lowest eigenvalue.
![Page 108: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/108.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIESExample:
Simulates gravity loading
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School of Mechanical EngineeringIran University of Science and Technology
7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES
Exact solution
![Page 110: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/110.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES
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School of Mechanical EngineeringIran University of Science and Technology
7.14 RESPONSE TO HARMONIC EXTERNAL EXCITATIONS
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School of Mechanical EngineeringIran University of Science and Technology
7.14 RESPONSE TO HARMONIC EXTERNAL EXCITATIONS
This approach is feasible only for systems with a small number of degrees of freedom.
For large systems, it becomes necessary to adopt an approach based on the idea of decoupling the equations of motion.
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School of Mechanical EngineeringIran University of Science and Technology
7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Undamped systems
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School of Mechanical EngineeringIran University of Science and Technology
7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS
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School of Mechanical EngineeringIran University of Science and Technology
7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS Harmonic excitation
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School of Mechanical EngineeringIran University of Science and Technology
7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Transient Vibration
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School of Mechanical EngineeringIran University of Science and Technology
7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Systems admitting rigid-body modes
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School of Mechanical EngineeringIran University of Science and Technology
7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Systems with proportional damping
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School of Mechanical EngineeringIran University of Science and Technology
7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Harmonic excitation
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School of Mechanical EngineeringIran University of Science and Technology
7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Transient Vibration
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School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Lecture Seven
By: H. [email protected]
![Page 123: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/123.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping7.17 Discrete-Time Systems
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
Nonsymmetric
The eigenvalues/vectors are in general complex.
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Orthogonality
Left eigenvectors Right eigenvectors
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
The right eigenvectors xi are biorthogonal to the left eigenvectors yj.
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
The bi-orthogonality property forms the basis for a modal analysis for the response of systems with arbitrary viscous damping.
Biorthonormality Relations
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
Assume an arbitrary 2n-dimensional state vector:
The expansion theorem forms the basis for a state space modal analysis:
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Harmonic Excitations
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Harmonic Excitations
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Arbitrary Excitations
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
The state transition matrix
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School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPINGExample 7.12. Determine the response of the system to the excitation:
![Page 138: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/138.jpg)
School of Mechanical EngineeringIran University of Science and Technology
7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING
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School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping7.17 Discrete-Time Systems
![Page 140: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/140.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Discrete-Time Systems
Lecture Eight
By: H. [email protected]
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School of Mechanical EngineeringIran University of Science and Technology
7.Multi-Degree-of-Freedom Systems7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients7.4 Lagrange's Equations Linearized about Equilibrium7.5 Linear Transformations : Coupling7.6 Undamped Free Vibration :The Eigenvalue Problem7.7 Orthogonality of Modal Vectors7.8 Systems Admitting Rigid-Body Motions7.9 Decomposition of the Response in Terms of Modal Vectors7.10 Response to Initial Excitations by Modal Analysis
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 7.12 Geometric Interpretation of the Eigenvalue Problem7.13 Rayleigh's Quotient and Its Properties7.14 Response to Harmonic External Excitations 7.15 Response to External Excitations by Modal Analysis 7.15.1 Undamped systems 7.15.2 Systems with proportional
damping7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems
![Page 142: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/142.jpg)
School of Mechanical EngineeringIran University of Science and Technology
NON CONSERVATIVE SYSTEMSExample 1:Solve the eigenvalue problem for the linearized model of shown system about the trivial equilibrium(X0 = Y0 = 0), for the parameter values:
m= 1 kg, Ω = 2 rad/s,
c =0.1 N.s/m, h = 0.2 N.s/m
kx = 5N/m, ky = 10 N/m,
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School of Mechanical EngineeringIran University of Science and Technology
Example 1:THE NONSYMMETRIC EIGENVALUE PROBLEM
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School of Mechanical EngineeringIran University of Science and Technology
Example 1:THE NONSYMMETRIC EIGENVALUE PROBLEM
The response may beevaluated in the modal domain:
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School of Mechanical EngineeringIran University of Science and Technology
Response Evaluation in Time Domain
An nth power approximation
The transition matrix
Semigroup property
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School of Mechanical EngineeringIran University of Science and Technology
Convergence Criteria:
The rate of convergence depends on t×max IλiI, in which max IλiI denotes the magnitude of the eigenvalue of A of largest modulus (the semigroup property, can be used to expedite the convergence of the series).
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School of Mechanical EngineeringIran University of Science and Technology
DISCRETE-TIME SYSTEMS
The discrete-time equivalent of above equation is given by the sequence:
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School of Mechanical EngineeringIran University of Science and Technology
DISCRETE-TIME SYSTEMS,Example 2:Compute the discrete-time response sequence of the system represented by matrix A to an initial impulse applied in the 1st
DOF.
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School of Mechanical EngineeringIran University of Science and Technology
Example 2 (cont.) :The largest eigenvalue modulus is 4.7673 and if
we choose t=0.05s and an accuracy of 10-4, then the transition matrix can be computed with five terms.
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School of Mechanical EngineeringIran University of Science and Technology
Example 2 (cont.) :
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School of Mechanical EngineeringIran University of Science and Technology
Example 2 (cont.) :
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School of Mechanical EngineeringIran University of Science and Technology
Defective SystemsExample 3:
1 2 3 1 2 1 2, , 0.m m m m k k k c c= = = = = = =1 0 0 1 1 0 0 0 00 1 0 , 1 2 1 , 0 0 00 0 1 0 1 1 0 0
M m K k Cc
− = = − − = −
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School of Mechanical EngineeringIran University of Science and Technology
State Space Form
Note: A is defective as it fails to have a linearly independent set of 6 eigenvectors (has repeated eigenvalues).
( )
1 1
5 4 3 2 2 2
0 0 0 1 0 00 0 0 0 1 0
0 0 0 0 0 0 1,
0 0 0 02 0 0 0
0 0 0
4 3 3 0.
kI mA
M K M C cm
α
α α βα α α
α α β
λ λ βλ αλ αβλ α λ α β
− −
=
= = − − − = −
− −
+ + + + + =
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School of Mechanical EngineeringIran University of Science and Technology
Undamped Systems with Rigid-Body Modes
There is only one rigid body mode.Zero eigen-values must occur with
multiplicity of 2The generalized eigen-problem is
defective.For each regular state rigid-body
mode, there will be a corresponding generalized state rigid-body mode
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School of Mechanical EngineeringIran University of Science and Technology
Generalized Eigenvectors: Jordan Form
Using a linearly independent set of generalized eigenvectors A is transformed into block-diagonal Jordan form:
,A X X J=
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School of Mechanical EngineeringIran University of Science and Technology
Undamped Systems with Rigid-Body Modes
[ ] [ ] ( )1 2 1 2
2 2
1, 0
0
1 0 1 01 0 1 01 0 1 00 0 0 10 0 0 10 0 0 1
rr
r
A X X X X
A X X
λλ
λ
= =
= ⇒ =
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School of Mechanical EngineeringIran University of Science and Technology
Undamped Eigenvalues c=0.( )5 4 3 2 2 2
1,2 3,4 5,6
5 63 4
5 6
5 63 4
5 63 4
5 6
5 63 4
4 3 3 0.
0 0.,0., , 3
1 11 0 1 1 1 02 2 2 2
1 0 0 0 1 1 1 0 0 01 11 0 1 1 1 02 2 2 2,
0 12 2
0 1 0 0
0 12 2
i i
X Y
λ λ βλ αλ αβλ α λ α β
β λ λ α λ α
λ λλ λ
λ λλ λλ λ
λ λλ λ
λ λλ λλ λ
+ + + + + =
= ⇒ = = ± = ±
− − − − − − − −
− − − − = =
− − − − − −
.1 10 1 1 12 2
0 1 0 0 1 11 10 1 1 12 2
− − − − − −
Right Eigenvectors Left Eigenvectors
![Page 158: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/158.jpg)
School of Mechanical EngineeringIran University of Science and Technology
Damped Eigenvalues c≠0.( )5 4 3 2 2 2
1 2
3,4
5,6
1,22
2
2
4 3 3 0.
0., - 0.0672,11, -0.0500 0.9962 ,5
0.0164 1.7299 .
1 11 1.00451 1.013600 1.00450 1.0136
ii
X
λ λ βλ αλ αβλ α λ α β
λ λα β λ
λ
λλλ
+ + + + + =
= == = ⇒ = ± = ±
=
-
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Response of Damped System to Initial Excitation
( ) (0), (0) [0,0,1,0,0,0] 'Atx t e x x= =The state transition matrix
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School of Mechanical EngineeringIran University of Science and Technology
Response of Damped System to Initial Excitation
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School of Mechanical EngineeringIran University of Science and Technology
Advanced Vibrations
Distributed-Parameter Systems: Exact Solutions
(Lecture Nine)
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 163: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/163.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.
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School of Mechanical EngineeringIran University of Science and Technology
RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS
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School of Mechanical EngineeringIran University of Science and Technology
RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS
![Page 166: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/166.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 167: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/167.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
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School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
![Page 169: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/169.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
![Page 170: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/170.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
![Page 171: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/171.jpg)
School of Mechanical EngineeringIran University of Science and Technology
DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE
EOM
BC’s
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School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS
![Page 173: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/173.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS
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School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS
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School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS:EHP
![Page 176: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/176.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS:EHP
![Page 177: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/177.jpg)
School of Mechanical EngineeringIran University of Science and Technology
BENDING VIBRATION OF BEAMS:EHP
![Page 178: Advanced Vibrations1).pdf · School of Mechanical Engineering Iran University of Science and Technology Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian](https://reader034.vdocuments.mx/reader034/viewer/2022042919/5f63453c83899c07240f6f2f/html5/thumbnails/178.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