Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One

Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian ahmadian@iust.ac.ir

Elements of Analytical Dynamics Newton's laws were formulated for a single particle Can 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.

Elements of Analytical Dynamics Analytical 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.

Elements of Analytical Dynamics In 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.

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

6.1 DEGREES OF FREEDOM AND GENERALIZED COORDINATES A 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

6.2 THE PRINCIPLE OF VIRTUAL WORK The 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.

THE PRINCIPLE OF VIRTUAL WORK resultant force on each particle applied force constraint force The virtual work performed by the constraint forces is zero

THE PRINCIPLE OF VIRTUAL WORK When r i are independent, If not to switch to a set of generalized coordinates: Generalized forces

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

, Example1:,

Example2: Derive the equation of motion for the systems of Example 1 by means of the generalized d'alembert's principle.

THE EXTENDED HAMILTON'S PRINCIPLE The virtual work of all the applied forces, The kinetic energy of particle m i

THE EXTENDED HAMILTON'S PRINCIPLE It is convenient to choose Extended Hamilton's principle

THE EXTENDED HAMILTON'S PRINCIPLE where V is the potential energy Or in terms of the independent generalized coordinates

A Note: The work performed by the force F in moving the particle m from position r 1 to position r 2 is responsible for a change in the kinetic energy from T I to T 2.

A Note (continued): The work performed by conservative forces in moving a particle from r 1 to r 2 is equal to the negative of the change in the potential energy from V 1 to V 2

A Note (continued):

Example3:

Example3: cont.

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

Advanced Vibrations Lecture Two: LAGRANGE'S EQUATIONS By: H. Ahmadian ahmadian@iust.ac.ir

THE EXTENDED HAMILTON'S PRINCIPLE Hamilton's principle Lagrangian

THE EXTENDED HAMILTON'S PRINCIPLE Use the extended Hamilton's principle to derive the equations of motion for the two-degree-of-freedom system.

THE EXTENDED HAMILTON'S PRINCIPLE

THE EXTENDED HAMILTON'S PRINCIPLE Only the virtual displacements are arbitrary.

THE EXTENDED HAMILTON'S PRINCIPLE

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.

LAGRANGE'S EQUATIONS

LAGRANGE'S EQUATIONS Derive Lagrange's equations of motion for the system

LAGRANGE'S EQUATIONS

Example of a non-natural system The 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. Two nonlinear springs The Rayleigh dissipation function:

The Kinetic Energy: When T 2 = T, T 1 = T 0 = 0, the system is said to be natural

The potential energy/the Lagrangian: U=V-T 0

The Rayleigh dissipation function:

Lagrange's equations of motion The gyroscopic matrix The damping matrix The circulatory matrix

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.

Advanced Vibrations Lecture Three: MULTI-DEGREE-OF-FREEDOM SYSTEMS By: H. Ahmadian ahmadian@iust.ac.ir

7.Multi-Degree-of-Freedom Systems 7.1 Equations of Motion for Linear Systems 7.2 Flexibility and Stiffness Influence Coefficients 7.3 Properties of the Stiffness and Mass Coefficients 7.4 Lagrange's Equations Linearized about Equilibrium 7.5 Linear Transformations : Coupling 7.6 Undamped Free Vibration :The Eigenvalue Problem 7.7 Orthogonality of Modal Vectors 7.8 Systems Admitting Rigid-Body Motions 7.9 Decomposition of the Response in Terms of Modal Vectors 7.10 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 Problem 7.13 Rayleigh's Quotient and Its Properties 7.14 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 damping 7.16 Systems with Arbitrary Viscous Damping 7.17 Discrete-Time Systems

7.1 EQUATIONS OF MOTION FOR LINEAR SYSTEMS

7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTS The 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.

7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTS We define the flexibility influence coefficient a ij as the displacement of point x i, due to a unit force, F j = 1.

7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTS The stiffness influence coefficient k ij is the force required at x i to produce a unit displacement at point x j, and displacements at all other points are zero. To obtain zero displacements at all points the forces must simply hold these points fixed.

7.2 FLEXIBILITY AND STIFFNESS INFLUENCE COEFFICIENTS

Example:

7.3 PROPERTIES OF THE STIFFNESS AND MASS COEFFICIENTS The potential energy of a single linear spring: By analogy the elastic potential energy for a system is:

Symmetry Property:

Maxwell's reciprocity theorem:

7.4 LAGRANGE'S EQUATIONS LINEARIZED ABOUT EQUILIBRIUM Rayleigh's dissipation function

7.4 LAGRANGE'S EQUATIONS LINEARIZED ABOUT EQUILIBRIUM

7.5 LINEAR TRANSFORMATIONS. COUPLING

Derivation of the matrices M' and K' in a more natural manner

Advanced Vibrations Lecture Four: Multi-Degree-of-Freedom Systems (Ch7) By: H. Ahmadian ahmadian@iust.ac.ir

7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM synchronous motion

7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM

7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM characteristic polynomial 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.

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:

THE SYMMETRIC EIGENVALUE PROBLEM: LINEAR CONSERVATIVE NATURAL SYSTEMS Standard eigenvalue problem

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.

7.6 UNDAMPED FREE VIBRATION. THE EIGENVALUE PROBLEM

Free vibration for the initial excitations

7.7 ORTHOGONALITY OF MODAL VECTORS

7.8 SYSTEMS ADMITTING RIGID-BODY MOTIONS

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.

7.8 SYSTEMS ADMITTING RIGID-BODY MOTIONS

Advanced Vibrations Lecture Five By: H. Ahmadian ahmadian@iust.ac.ir

7.9 Decomposition of the Response in Terms of Modal Vectors

7.9 Decomposition of the Response in Terms of Modal Vectors The modal vectors are orthonormal with respect to the mass matrix M,

7.10 Response to Initial Excitations by Modal Analysis

7.10 Response to Initial Excitations by Modal Analysis Modal Coordinates

7.10 Response to Initial Excitations by Modal Analysis

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;

7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix

7.12 Geometric Interpretation of the Eigenvalue Problem

7.12 Geometric Interpretation of the Eigenvalue Problem Solving the eigenvalue problem by finding the principle axes of the ellipse.

7.12 Geometric Interpretation of the Eigenvalue Problem Transforming to canonical form implies elimination of cross products:

7.12 Geometric Interpretation of the Eigenvalue Problem

7.12 Geometric Interpretation of the Eigenvalue Problem Obtaining the angle, one may calculate the eigenvalues and eigenvectors:

7.12 Geometric Interpretation of the Eigenvalue Problem Example: Solving the eigenvalue problem by finding the principal axes of the corresponding ellipse.

7.12 Geometric Interpretation of the Eigenvalue Problem

Advanced Vibrations Lecture Six By: H. Ahmadian ahmadian@iust.ac.ir

7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES Rayleigh's quotient

7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES

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.

7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES Example: Simulates gravity loading

7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES Exact solution

7.13 RAYLEIGH'S QUOTIENT AND ITS PROPERTIES

7.14 RESPONSE TO HARMONIC EXTERNAL EXCITATIONS

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.

7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Undamped systems

7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS

7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS Harmonic excitation

7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Transient Vibration

7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Systems admitting rigid-body modes

7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Systems with proportional damping

7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Harmonic excitation

7.15 RESPONSE TO EXTERNAL EXCITATIONS BY MODAL ANALYSIS: Transient Vibration

Advanced Vibrations Lecture Seven By: H. Ahmadian ahmadian@iust.ac.ir

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING Nonsymmetric The eigenvalues/vectors are in general complex.

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Orthogonality Left eigenvectors Right eigenvectors

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING The right eigenvectors x i are biorthogonal to the left eigenvectors y j.

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING Biorthonormality Relations The bi-orthogonality property forms the basis for a modal analysis for the response of systems with arbitrary viscous damping.

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:

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Harmonic Excitations

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING: Arbitrary Excitations

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING The state transition matrix

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING Example 7.12. Determine the response of the system to the excitation:

7.16 SYSTEMS WITH ARBITRARY VISCOUS DAMPING

Advanced Vibrations Discrete-Time Systems Lecture Eight By: H. Ahmadian ahmadian@iust.ac.ir

NON CONSERVATIVE SYSTEMS Example 1: Solve the eigenvalue problem for the linearized model of shown system about the trivial equilibrium(x 0 = Y 0 = 0), for the parameter values: m= 1 kg, Ω = 2 rad/s, c =0.1 N.s/m, h = 0.2 N.s/m k x = 5N/m, k y = 10 N/m,

Example 1:THE NONSYMMETRIC EIGENVALUE PROBLEM

Example 1:THE NONSYMMETRIC EIGENVALUE PROBLEM The response may be evaluated in the modal domain:

Response Evaluation in Time Domain The transition matrix An n th power approximation Semigroup property

Convergence Criteria: The rate of convergence depends on t max Iλ i I, in which max Iλ i I denotes the magnitude of the eigenvalue of A of largest modulus (the semigroup property, can be used to expedite the convergence of the series).

DISCRETE-TIME SYSTEMS The discrete-time equivalent of above equation is given by the sequence:

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 1 st DOF.

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.

Example 2 (cont.) :

Defective Systems Example 3: m = m = m = m, k = k = k, c = c = 0. 1 2 3 1 2 1 2 1 0 0 1 1 0 0 0 0 M = m 0 1 0, K k 1 2 1, C 0 0 0 = = 0 0 1 0 1 1 0 0 c

State Space Form A 0 0 0 1 0 0 0 0 0 0 1 0 α = 0 I 0 0 0 0 0 1 = =, α α 0 0 0 0 β = α 2α α 0 0 0 0 α α 0 0 β 1 1 M K M C c ( 5 4 3 2 2 2 ) λ λ + βλ + 4αλ + 3αβλ + 3α λ + α β = 0. Note: A is defective as it fails to have a linearly independent set of 6 eigenvectors (has repeated eigenvalues). k m m

Undamped Systems with Rigid-Body Modes There is only one rigid body mode. Zero eigen-values must occur with multiplicity of 2 The generalized eigen-problem is defective. For each regular state rigid-body mode, there will be a corresponding generalized state rigid-body mode

Generalized Eigenvectors: Jordan Form Using a linearly independent set of generalized eigenvectors A is transformed into blockdiagonal Jordan form: AX = X J,

Undamped Systems with Rigid-Body Modes λ 1 A X 1 X 2 X 1 X 2, r 0 0 λ r 1 0 1 0 1 0 1 0 1 0 1 0 A X 2 = X 2 = 0 0 0 1 0 0 0 1 0 0 0 1 [ ] [ ] r = ( λ = )

Undamped Eigenvalues c=0. X 5 4 3 2 2 2 ( ) λ λ + βλ + 4αλ + 3αβλ + 3α λ + α β = 0. β = 0 λ = 0.,0., λ =± i α, λ =± i 3α 1,2 3,4 5,6 1 1 λ5 λ6 1 0 1 1 1 0 3 4 2 2 λ λ 2 2 1 0 0 0 1 1 1 0 0 0 λ5 λ6 1 1 λ5 λ 6 1 0 1 1 1 0 λ3 λ4 2 2 =, Y = 2 2. λ5 λ 6 1 1 0 1 λ3 λ4 0 1 1 1 2 2 2 2 0 1 0 0 λ5 λ6 0 1 0 0 1 1 λ5 λ 6 1 1 0 1 λ3 λ 4 0 1 1 1 2 2 2 2 Right Eigenvectors Left Eigenvectors

Damped Eigenvalues c 0. ( 5 4 3 2 2 2 ) λ λ + βλ + 4αλ + 3αβλ + 3α λ + α β = 0. λ1 = 0., λ2 = - 0.0672, 1 α = 1, β = λ3,4 = -0.0500 ± 0.9962 i, 5 λ = 5,6-0.0164 ± 1.7299 i. X 1,2 1 1 1 1.0045 1 1.0136 = 0 λ2 0 1.0045λ 2 0 1.0136λ2

Response of Damped System to Initial Excitation At x( t) = e x(0), x(0) = [0,0,1,0,0,0]' The state transition matrix

Response of Damped System to Initial Excitation

Advanced Vibrations Distributed-Parameter Systems: Exact Solutions (Lecture Nine) By: H. Ahmadian ahmadian@iust.ac.ir

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

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.

RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS

RELATION BETWEEN DISCRETE AND DISTRIBUTED SYSTEMS: TRANSVERSE VIBRATION OF STRINGS Ignoring 2 nd order term

DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE

DERIVATION OF THE STRING VIBRATION PROBLEM BY THE EXTENDED HAMILTON PRINCIPLE EOM BC s

BENDING VIBRATION OF BEAMS

BENDING VIBRATION OF BEAMS:EHP