Dynamics of structures - Personal...

Dynamics of Structures 2017-2018 4. MDOF systems

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4.Multiple degree of freedom systems

Dynamics of structures

Arnaud Deraemaeker ([email protected])

*Multiple degree of freedom systems in real lifeHypothesisExamples

*Response of a multiple degree of freedom (mdof) systemFree responseForced responseInfluence of the damping

*MDOF application : The tuned mass damperMass-spring TMDPendulum TMD

Outline of the chapter

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Multiple degree of freedom systems in real life

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Reduction of a system to a mdof system


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Response of a mdof system

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Conservative system : equations of motion

Mass matrix Stiffness matrix


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Equations of motion: general solution

Admits a non trivial solution if

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

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:


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=

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 :

(1)

(2)

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Orthogonality of the modeshapes

Define =

Matrix notation


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Example : two degrees of freedom system

Second order equation in 2

Eigen frequencies and modeshapes

for

for


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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

*mass (modal mass)

*stiffness

* angular eigenfrequency

*excitation (modal excitation)

Particular solution : projection of the solution in the modal basis

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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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 …

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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 does not 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

*mass (modal mass)

*stiffness

*angular eigenfrequency

* damping coefficient (modal damping)

*excitation (modal excitation)

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 decoupled equations


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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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MDOF application :The tuned mass damper (TMD)

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Application example : the tuned mass damper

Equations of motion:


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Application example : the tuned mass damper

Undamped vibration absorber (b=0)

for

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Undamped tuned mass damper

DVA (dynamic vibration absorber = TMD) tuned to eigenfrequency of primary system-> Reduces vibrations in a narrow band around eigenfrequency-> Amplification outside of this narrow band

=0.03=1


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Pendulum tuned mass damper

Inertial coupling of the two systems

small

neglected

*

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for


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=1

•Tuning of the PTMD based on the length of the pendulum

•Effect of the mass mainly on the spreading of the peaks

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Undamped tuned mass damper

The building is excited at its natural frequency (2.62 Hz)


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Damped tuned mass damper

-Reduction of vibration is lower around eigenfrequency with b increasing-Reduces the amplification outside of the narrow frequency band-Existence of P and Q : points where all curves cross

=0.03=1

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Optimal design of tuned mass dampers

Optimum damping is given by

P and Q are at equal height for


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Steps to follow to design a TMD

- The maximum mass of the device is decided fixing

- Based on this value, the frequency of the TMD is tuned :

- Which allows to compute the stiffnes of the TMD

- And finally the optimal damping is computed

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Tuned mass damper in action


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Example of tuned mass dampers in structures

Millenium bridge, London

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Tuned mass damper in action on a bridge


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John Hancock Tower (Boston-1976)

Two TMDs of 2700 kN (approx 5.2x5.2x1m steel blocks)

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City Corp Center (New York - 1977)

Tuned mass damper

-400 Tons block installed at the top(2% of effective mass of first mode)

- 279m high- Fundamental period = 6.5s


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City Corp Center (New York - 1977)

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Chiba Port Tower (Japan - 1986)

Tuned mass damper : 15 tonsCan slide in two directions


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Damped pendulum tuned mass damper

small

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Damped pendulum tuned mass damper

Let us define :

TMD frequency


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Damped PTMD : optimal parameters

with

=0.03

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Pendulum tuned mass damper example

Tai Pei (Taiwan)

Dampers


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Pendulum tuned mass damper example – Taipei 101

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Pendulum tuned mass damper example – Taipei 101


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Pendulum tuned mass damper example

Pendulum motion during earthquake, May 12, 2008

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