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Vibrations and Acoustics 2019-2020 4. MDOF systems

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4.Multiple degree of freedom systems

Vibrations and Acoustics

Arnaud Deraemaeker ([email protected])

Multiple degree of freedom systems in real life

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A test-case based learning of vibrations in mechanical engineering

Case study 2 : Passenger comfort in cars (suspension)

• Source of excitation• Effects• Design methodology• Remedial measures

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One dof representation of a car


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Multiple dof representation of a car

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Sources of excitation

[Barbosa 2011]


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Resonance, discomfort ?

Equivalent mass and stiffness

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Reduction of a system to a mdof system

Reduction of a system to a mdof system


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Reduction to a sdof system

-Equivalent stiffness?-Equivalent mass ?-Validity (frequency band) ?

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Equivalent stiffness

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General principle :

F in the direction of excitation(and motion) of the mass

Computation :- simplified model- numerical approximation (finite elements)


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Equivalent stiffness: bar in traction

Displacement of the attachmentpoint (in the direction of excitation):

Constitutive equation: Stress is constant

Static equilibrium

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Direction of excitation

Attachment point

Equivalent stiffness : cantilever beam in bending

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Direction of excitation

Attachmentpoint

From tables


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Equivalent stiffness : of beams in bending

Equivalent stiffness of beams in bending can be computed using formulas fromresistance of materials or using tables :

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Equivalent stiffness: portal frame


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Equivalent stiffness: portal frame

From tables (internet)

Equivalent mass : energy method

Kinetic energy of a mass-spring system

Kinetic energy of the spring

Additional mass

Example 1 :

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Equivalent mass : energy method

Kinetic energy of a mass-spring system

Kinetic energy of the bar

Additional mass ma

Example 2 :

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Frequency limits for one dof systems

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Continuous model

SDOF model 1 SDOF model 2

m=100 kg, E =210 GPa, =7800 kg/m3, L=1m;A = 0.0004 m2


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Frequency limits of SDOF systems for real continuous systems

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k =EA/L = 8.4 107 N/mM = 100 kg

fn=145.87 Hz (mass M)

fn=145.11 Hz (mass M+ma)ma = AL/3 = 1.04 kg <<M

SDOF hypothesisnot valid for frequencies > 2000 Hz

Frequency limits for one dof systems

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Continuous model

SDOF model 1 SDOF model 2

m=100 kg, E =210 GPa, =78000 kg/m3, L=1m;A = 0.0004 m2


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Frequency limits of SDOF systems for real continuous systems

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k =EA/L = 8.4 107 N/mM = 100 kg

fn=145.87 Hz (mass M)

fn=138.83 Hz (mass M+ma)ma = AL/3 = 10.4 kg <M

SDOF hypothesisnot valid for frequencies > 500 Hz

ma should be taken into account

Response of a mdof system

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Conservative system : equations of motion

Mass matrix Stiffness matrix

Equations of motion: general solution

Admits a non trivial solution if

r2 is negative (K and M are positive definite matrices)


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General solution: eigen frequencies and modeshapes

If the system has n degrees of freedom, there exist n values of -2 for which this equation is satisfied. These are the n eigenvalues whichcorrespond to n eigenfrequencies

n eigen vectors are associated to these eigenfrequencies. Theycorrespond to the n mode shapes of the structure

Generalized eigenvalue problem (-2)

The general solution is written in the form:

Premultiply (1) by ,(2) by and substract taking into account

symmetry of K ( ) and M ( )

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Orthogonality of the modeshapes

Proof :

Property :


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Orthogonality of the modeshapes

Define

Matrix notation

Example : two degrees of freedom system

Second order equation in 2


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Eigen frequencies and modeshapes



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Mode 1 Mode 2

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General solution

Assume the following initial conditions


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Particular solution : projection of the solution in the modal basis

Projection on the modal basis

N independent equations of the type

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The solution can be obtained by solving a set of n independent equations of the type

This equation corresponds to the equation of motion of a sdof system with

Particular solution : projection of the solution in the modal basis


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Particular solution : harmonic excitation

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Harmonic excitation: modal basis solution

sdof oscillator solution

or


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The solution is the sum of sdof oscillators :


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Example : two dofs system


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Equations of motion : modal basis solution

(m=1kg, k = 1N/m)

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(m=1kg, k = 1N/m)

Equations of motion : modal basis solution


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Forced excitation videos

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Damped equations of motion

Damping matrix


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Damped equations of motion : general solution

Non trivial solution if

•Complex roots of the characteristic equation-> Oscillatory functions with exponential envelope

•Complex eigen vectors = complex modeshapes-> Not often used in practice in vibrations

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Mode shapes of conservative system :

Projection on the real modal basis:

In general is not diagonal and the equations remain coupled but …

or

Equations of motion : modal basis solution – real mode shapes


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•Rayleigh damping:

Often used as a simplifying assumption to decouple the equations but doesnot have a physical meaning

•Modal damping

When damping is small, off-diagonal terms can be neglected leading to:

is the modal damping of mode i


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This equation corresponds to the equation of motion of a sdof system with

n independent equations of the type



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Link between Rayleigh damping and modal damping

•Only two parameters to define the damping of all modes->Overestimation at low and high frequencies->Represents accurately the damping of two modes only

Rayleigh damping Modal damping

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Particular solution : harmonic excitation


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Projection on modal basis

Modal damping hypothesis (small damping)

Sum of damped sdof oscillators

n decoupledequations


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k=1 N/m, m=1kg, b=0.04 Ns/m

Never goes to zero

Damped resonances

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Validity of modal damping hypothesis

Neglect off-diagonal terms


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k=1 N/m, m=1kg, b=0.04 Ns/m

Comparison of exact (coupled-dotted line) and approached (uncoupled) responses

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k=1 N/m, m=1kg, b=0.2 Ns/m

The modal damping hypothesis is not valid for high values of damping


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Example : 5 dofs structure frequency response function computation

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Example : 5 dofs building : projection in the modal basis


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5 dofs structure : mode shapes and eigenfrequencies

Mode 1 (5.41 Hz) Mode 2 (15.8 Hz) Mode 3 (24.9 Hz) Mode 4 (32.0 Hz) Mode 5 (36.5 Hz)

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5 dofs structure : modeling of damping

Frequency response function


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5 dofs structure : FRF for top excitation

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Example : 5 dofs impulse response


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1 mode impulse response

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Example : 5 dofs structure excited by random force – time domain response

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Modal excitation

5 dofs structure : time domain response


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Impulse response

Convolution

Expansion

5 dofs structure : time domain response

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5 dofs building : time domain response

FFT


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Base excitation of mdof systems

Equations of motion:

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Base excitation of mdof systems

Matrix notations:

All developments for force excitation apply


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Multiple dof representation of a car

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A test-case based learning of vibrations in mechanical engineering

Case study 1 : Payload comfort in space launchers

• Source of excitation• Effects• Design methodology• Remedial measures

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