Part 1: Discrete systems

•Introduction•Single degree of freedom oscillator•Convolution integral•Beat phenomenon•Multiple degree of freedom discrete systemsp g y•Eigenvalue problem•Modal coordinates•DampingDamping•Anti‐resonances

1

Why suppress vibrations ?

FailureFailureBuilding response to earthquakes (excessive strain)Wind on bridges (flutter instability)FatigueFatigue

ComfortCar suspensionsCar suspensionsNoise in helicoptersWind-induced sway in buildings

Operation of precision devicesDVD readersWafer steppersWafer steppersTelescopes & interferometers

2

How ?

Vibration damping:Reduce the resonance peaks

Vibration isolation: Prevent propagation of disturbances to sensitive payloads

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d l Active damping in civil engineering structures

TMD: Tuned Mass Damper = DVA: Dynamic Vibration AbsorberAMD: Active Mass Damper

4

Single degree of freedom (d.o.f.) oscillator

Free body diagramy g

Free response:p

Characteristic equation:

Solution ?

Characteristic equation:Eigenvalues:

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6

(A ,B, A1, B1 depend on initial conditions)

Impulse responseImpulse response

Spring and damping forces Have finite amplitudes

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Impulse response for various damping ratios

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Convolution IntegralConvolution Integral

Linear system

For a causal system:

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Harmonic response

1. Undamped oscillator

Dynamic amplification

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Harmonic response

2. Damped oscillator

Dynamic amplification

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Bode plots Quality factor

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Nyquist plot

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Frequency Response Function (FRF)Frequency Response Function (FRF)

Harmonic excitation:

FRF:

The FRF is the Fourier transform of the impulse response

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Fourier transform

Convolution integral (linear systems):

Parseval theorem:

= energy spectrum of f(t)

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Transient response (Beat)Transient response (Beat)

Undamped oscillator starting from rest:

[ ]

Modulating functionModulating function

At resonance

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Transient response Beat

Steady stateSteady-state amplitude:

The beat is a transientPhenomenon !

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State Space form(system of first order differential equations)

Oscillator:

State variables:State variables:

Alternative choice Of state variables:Of state variables:

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Problem 1: Find the natural frequency of the single story building

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Problem 2: write the equation of motion of the hinged rigid bar

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Multiple degree of freedom systems

In matrix form:

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Mass matrix Stiffness matrix Damping matrix

Symmetric & semi positive definite

[]

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Eigenvalue problemEigenvalue problem

Free response of the conservative system (C=0)

A non trivial solution exists if Eigenvalueproblem

The eigenvalues s are solutions of

Because M and K are symmetric and ysemi-positive definite, the eigenvaluesare purely imaginary:

Natural frequency Mode shape

Two-mass system:

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Example: Two-mass system:

Natural frequencies:

Mode shapes:

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Orthogonality of the mode shapesOrthogonality of the mode shapes

Upon permuting i and j,

Subtracting:

The mode shapes corresponding to distinct natural frequencies are gorthogonal with respect to M and K

Modal mass(or generalized mass)

[Can be selected freely]

Rayleigh quotient:25

Orthogonality relationships in matrix form: withOrthogonality relationships in matrix form: with

Notes:(1) Multiple natural frequencies:

If several modes have the same natural frequency they form a subspaceIf several modes have the same natural frequency, they form a subspaceand any vector in this subspace is also solution of the eigenvalue problem.

(2) Rigid body modes:

They have no strain energy:

They also satisfy which means that they are solutions of theThey also satisfy which means that they are solutions of the eigenvalue problem with

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Free response from initial conditionsFree response from initial conditions

2n constants to determine from the initial conditions.Using the orthogonality conditions,g g y ,

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If there are rigid body modes (i=0)

Rigid body modes Flexible modesRigid body modes Flexible modes

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Problem: write the mass and stiffness matrix of the 5 storey building

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Building with n identical floors

Natural frequencies:

Mode h

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shapes:

Modal coordinates

Orthogonality relationships

xx

Assumption of modal damping:

Set of decoupled equations of single d.o.f. oscillators:

Mode i:Mode i:Work of the external

forces on mode i

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Modal truncation

Mode i:

The modes within the bandwidth of f respond dynamically; y yThose outside the bandwidth respond in a quasi-static manner.

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Modal truncation

If

The response may be split into two groups of modes:

Responding dynamically

Responding in a quasi-static manner

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Damping

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Passive damping of very lightlydamped Structures (0.0002)

with shunted PZT patches

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Rayleigh damping

and are free parameters that and are free parameters that Can be selected to match the

Damping of two modes.

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Dynamic flexibility matrixDynamic flexibility matrix

Harmonic response of:

[ ]

Modal expansion of G( ):Modal expansion of G():

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M d l t tiModal truncation

Dynamic flexibility matrix

[m<<n]

Dynamic flexibility matrix

Dynamicamplification

of mode i

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Modal truncationModal truncation

Residual modesmodes

Dynamic part restricted tothe low frequency modes

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Anti resonancesAnti-resonances

Diagonal terms of the dynamicflexibility matrix:

collocated

If the system is undamped,Gkk is purely real:

All the residues are positive and Gkk is a monotonously increasing function of

Gkk () = 0

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Gkk () = 0

1. There is always one anti-resonance frequency between two resonance frequencies.(i t l i )(interlacing)

2. The anti-resonances depend on the location of the collocated actuator/sensor pair.42

P l tt f t t ith ll t d t t / iPole-zero pattern of a structure with collocated actuator/sensor pair

With damping

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Alternative form of the open-loop transfer function of collocated systems:Alternative form of the open loop transfer function of collocated systems:

undamped damped

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Single degree of freedom oscillator

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N i t & B d l t f ll t d tNyquist & Bode plots of collocated systems

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Anti resonances and constrained systemAnti-resonances and constrained system

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Part 2 Lagrangian dynamics

•Principle of virtual work•D’Alembert principle•Hamilton’s principle•Hamilton s principle•Lagrange’s equations•Examples•First integrals of Lagrange’s equationsFirst integrals of Lagrange s equations

1

L i D iLagrangian Dynamics

Newton (1642‐1727) introduced the equation of dynamics in vector formHamilton (1805 –1865) wrote them in variational form which is more generalHamilton (1805  1865) wrote them in variational form, which is more generalBecause it can be extended to distributed and electromechanical systems.

Generalized coordinates qi: set of coordinates describing the kinematics of the system.If minimum, they are independent.  If not, they are connected by kinematic constraints.

2

Kinematic constraints:

The number of d.o.f. of a system is the minimum number of coordinatesNecessary to provide a full geometric description of the kinematics. 

If the number of coordinates is not minimum, they are not independent. They are related by constraints:

Holonomic constraints: 

Scleronomic :Scleronomic :(independent of time)

Non‐holonomic: or(non integrable)

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Example of  non‐integrable constraints

The  four coordinates (x,y,may takearbitrary values, but the differential motion y ffmust satisfy the constraints:

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Vi t l di l t I fi it i l h f di t i t t t tiVirtual displacements: Infinitesimal change of coordinates occuring at constant time,and consistent with the kinematic constraints of the system (but otherwise arbitrary).

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Consider a particle constrained to move on a smooth surface: 

The virtual displacements must satisfy the constraint equation: 

(The virtual displacement must be in the plane tangent to the surface)

If the system is smooth and frictionless (reversible), the reaction force is alsoNormal  to the surface: 

The virtual work of the constraint forces on any virtual displacement is zero

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Principle of virtual work (static equilibrium)Principle of virtual work (static equilibrium)

Consider a system of N particles with position vectors: 

The static equilibrium implies that the resultant force applied on each particle is R =0The static equilibrium implies that the resultant force applied on each particle is Ri=0It follows that:   and therefore:

For all virtual displacements.  If Ri is decomposed into the external force and the constraint (reaction) forces :

=0 (virtual work of constraint forces)

Finally,

The virtual work of external applied forces on the virtual displacementscompatible with the kinematics is zero.

1. The constraint forces do not appear any more2. The principle may be written in generalized coordinates

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Principle of virtual work (example)

Find the relationship between f and w at the static equilibrium

The static equilibrium problem is transformed into kinematics:

Principle of virtual work:

Virtual displacements:

0=0

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D’Alembert’s principle(Extension of the principle of virtual work to dynamics)

The inertia forces are added:

The virtual work of the effective forces on the virtual displacementsCompatible with the constraints is zero.

D’Alembert’s principle is most general, but it is difficult to apply because it refersto vector quantities expressed in inertial frame; it cannot be transformeddirectly in generalized coordinates This achieved with Hamilton’s principledirectly in generalized coordinates. This achieved with Hamilton s principle.

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Conservation of the Total Energy

(1) If t does not appear explicitly in the constraints, the virtual displacements are possible and the actual displacements may be used in d’Alembert’s principle:

(2) If the force depend of a potential which does not depend explicitely on t (conservative field)(2) If the force depend of a potential which does not depend explicitely on t (conservative field)

Since

Under the two conditions above, there is conservation of the total energy

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Hamilton’s principle

D’Alembert:

Virtual work of external forces:

ThusThus,

And d’Alembert becomes: This term is a total time derivativeWhich may be eliminated by integration

Work and energy functions

y y gOver some interval assumingThat the configutation is fixed at t1 and t2.

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Hamilton’s Principle (2)

L iLagrangian:

The actual path is that which cancels the variational indicator V I with respectThe actual path is that which cancels the variational indicator V.I. with respect To all arbitrary variations of the path between t1 and t2, compatible with the Kinematic constraints, and such that

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does not measure the displacements on the truepath, but the separation between the true path anda perturbed one at a given time .p g

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E lExample:

Hamilton:

is eliminated by integrating by parts

(differential equation of the pendulum)14

Lagrange’s equation

Hamilton’s principle contains only scalar work and energy quantities.Does not refer to any specific coordinate system and the system configuration may beexpressed In terms of generalized coordinates:expressed In terms of generalized coordinates: We assume an explicit dependency on time which is important for gyroscopic systems.

Kinetic energyKinetic energy:

General formsof T and V:

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Lagrange’s equation (2)

Virtual work of non conservative forces:

Generalized force associated to qi :

Hamilton:

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Lagrange’s equation (3)

Example 1: Example 2:

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Vibration of a linear discrete non‐gyroscopic systemVibration of a linear discrete non gyroscopic system

Lagrange

Dissipation function

All the forces non already included in Dalready included in D

Viscous damping: 

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Example 3: Pendulum with a sliding massp g

Gravity Spring elasticenergy

Note: Try to obtain these results by writing the absolute acceleration in moving frame andapplying Newton’s law. Which way is easier ? 19

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Example 4: Pendulum with a sliding disk

Example 3: The rod is a uniform bar of length l and mass M

Two additional terms:

Kinetic energy of the rod:

Potential energy of the rod:

[ ]

Potential energy of the rod:

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Example 5: Rotating pendulumExample 5: Rotating pendulum

T2 T0

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Example 6: Rotating spring‐mass system(constant rotation speed)

T2 T02 0

Lagrange:Lagrange:

The system becomes unstable when

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Example 7: Two‐axis oscillator with anisotropic stiffness(constant rotation speed)(constant rotation speed)

Generalized coordinates: position (x,y) in the rotating frame.

Absolute velocity in rotating frame:

Kinetic energy:

Potential energy:

[T1 is responsible forNon conservative forces:

[T1 is responsible for the gyroscopic effects]

or

Note: This model is representative of a « Jeffcott rotor »  with anisotropic shaft 24

Coupling between x and y

Anti‐symmetric matrixof gyroscopic forcesof gyroscopic forces, which couples the motionin the two directions

Modified potential:

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In the particular case:(i i & d d)(isotropic & undamped) 

[ ]

To study the stability, we assume a solution

Characteristic equationCharacteristic equation:

The solutions are purely imaginaryThe solutions are purely imaginary: 

Campbell diagram

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Anisotropic stiffness, undamped

Characteristic equation:

This term is negative for:This term is negative for:The system is unstable (Routh‐Hurwitz)

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Vibrating angular rate sensor

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Vibrating angular rate sensor (2)

Assuming:

Harmonic excitation:

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« First integrals » of the Lagrange equations:

1. Jacobi integral

If the system is conservative and if the Lagrangian does not depend explicitly on timeIf  the system is conservative and  if the Lagrangian does not depend explicitly on time

Total time derivative of L:

Lagrange equation:

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Euler theorem on homogeneous functions:

If Tn is homogeneous of degree n in some variables qi,

It satisfies the identity:It satisfies the identity:

Since

Jacobi integral or Painlevé integral

If T=T2, it reduces to the Conservation of  the total energy:

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2 Ignorable coordinate2. Ignorable coordinate

If the Lagrangian does not depend explicitly on some coordinate qs, the coordinate is ignorable:

Lagrange equation:

The generalized momentum associated with an ignorable coordinate is conserved.

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Example: the spherical pendulump p p

=T2

is ignorable:

[Conservation of the angular momentum about Oz][ g ]

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Part 3: Vibration alleviation

•Dynamic Vibration Absorber•Vibration isolator•Relaxation isolator•Six‐axis isolator•Isolation by kinematic coupling

1

Why suppress vibrations ?

FailureFailureBuilding response to earthquakes (excessive strain)Wind on bridges (flutter instability)FatigueFatigue

ComfortCar suspensionsCar suspensionsNoise in helicoptersWind‐induced sway in buildings

Operation of precision devicesDVD readersWafer steppersWafer steppersTelescopes & interferometers

2

How ?

Vibration damping:Reduce the resonance peaks

Vibration isolation: Prevent propagation of disturbances to sensitive payloads

3

Active damping in civil engineering structures

TMD: Tuned Mass Damper = DVA: Dynamic Vibration AbsorberAMD: Active Mass Damper

4

Dynamic Vibration Absorber (DVA)Tuned Mass Damper (TMD)

5

DVA -TMD AMD - HMD

6

Taipei 101 (509 m)

730 T Tuned Mass Damper

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Yokohama Landmark TowerYokohama Landmark Tower

A ti M DActive Mass Damper

8

(from K. Seto)

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(from K. Seto)

10

Multiple tower with active control bridges (Seto)

11

D i Vib ti Ab b (DVA)Dynamic Vibration Absorber (DVA)

In Laplace form:

Response:

with

12

1. Harmonic disturbanceThe response is also harmonic:

If the frequency of theIf the frequency of the harmonic disturbanceis constant, the best solution is to create a transmissionzero atzero at

The tuning may be based on

By exciting the structure atand tuning in such a way that

13

X1 and X2 be 90° out of phase.

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2. Wide band disturbance

2 design2 design parameters

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For all values of the damping, The curves cross each otherThe curves cross each otherin P and Q.

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Equal peak design (Den Hartog, 1929)

The relative amplitude of the points P and Qis controlled by the frequency ratio. P and Q have equal amplitude for:P and Q have equal amplitude for:

P and Q constitute the maxima of the dynamic amplification curve for:

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0.01

0.05

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Multiple D.O.F.

One mode approximation in the vicinity of the targeted frequency:For frequencies close to mode k, the response is dominated by mode kp y

Mass m to take intoMass m1 to take intoAccount in the design

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Wide-band isolator: Various isolator architectures for spacecraft

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James Webb Space Telescope ( ~ 2015?)James Webb Space Telescope ( 2015?)

1 HzIsolator

RWAIsolatorIsolator

7 Hz21

Effect of the isolator on the transmissibility of disturbancesy

Reaction wheel speedRange 10-100 Hz Hz

10 100Hz

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Various isolation concepts

« Classical »i

Sky-hookD

Relaxation i l tpassive

isolatorDamper(active)

isolator

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Linear Isolator

24

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Relaxation Isolator

The poles are solutions of the characteristic equation:

26

The characteristic equation may be rewritten

27

28

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Six-axis isolator

Kinetic energy:

Jacobian of the isolator(depens on the topology):

Strain energy:

Eigenvalue problem:Eigenvalue problem:

(6 isolator modes)normalized according to

30

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If the springs k are replaced by relaxation isolators theIf the springs k are replaced by relaxation isolators, thespring stiffness must be replaced by the dynamic stiffness:

Upon eliminating x1 from

Dynamic stiffness of one leg:

(pure spring) (relaxation isolator)

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(pure spring) (relaxation isolator)

Relaxation isolator:

Change of coordinates (using the modes of the isolator with pure springs)

One finds a set of decoupled equations:decoupled equations:

With the notation:With the notation:

(identical to a single axis isolator.

or

However, there is a single scalarparameter: k1/c)

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Isolation by kinematic coupling

Harmonic disturbance w with a constant frequency :Control strategy = introducing a transmission zeroat the excitation frequency.

Kinematics:

Lagrange equation:

Transfer function:

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The system is tuned in such a waythat the frequency z of the zeroMatches that of the disturbance.

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Part 4Continuous structures: beam, bar and string

1

Vibration of beams (Euler‐Bernoulli)

Vertical equilibrium:

Rotational equilibrium:

2

Kinematic assumptions:

The fibers are in a uniaxial state of stress and strain:

(No axial loading)Bending moment:

Partial differential equation

3

Alternative derivation from Hamilton’s principle

StrainEEnergy:

KineticEnergy: Virtual work:Energy:

Hamilton:

4

Finally:

PDE

BoundaryConditions:At x=0 , x=L

PDE

5

Free vibration of a beam

A solution of the form exists if:

Introducing: 

Eigenvalue Non dimensional

= 0

EigenvalueProblem:

Non‐dimensionalFrequency:

Characteristic equation:Characteristic equation:

General solution:General solution:

6

Decoupling the boundary conditionsDecoupling the boundary conditions

We define:

Alternative form of the general solution:

Decoupled !!7

Example 1: Simply supported beam

Solutions: (natural frequency)

(mode shape)

8

Simply supported beam (modes)

9

Example 2: Free‐free beamp

10

Double root at1. Rigid body modes:Double root at

Boundary conditions:

2 Flexible modes:2. Flexible modes:

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For any given value of µFor any given value of µ,

12

Orthogonality relationships

Mode i satisfies:

and

13

Orthogonality relationships (2)

Modal massModal mass

Rayleigh quotient:Rayleigh quotient:

Compare to similar results for discrete systems:

14

Modal decomposition

(integrating by parts)

(using the orthogonality relations)

(work of p on mode k)

15

Vibration of a string

Free vibration: 

Assuming:

General solution:

Boundary conditions at x=0 and x=L:

16

Axial vibration of a bar

Free vibration:S d f dSpeed of sound

General solution:

17

Boundary conditions: fixed at x=0, free at x=L

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