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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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Dynamics of Structures 2017-2018 4. MDOF systems
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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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Reduction of a system to a mdof system
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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Eigen frequencies and modeshapes
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Eigen frequencies and modeshapes
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 :
Harmonic excitation: modal basis solution
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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 …
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 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
Equations of motion : modal basis solution – real mode shapes
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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
Equations of motion : modal basis solution – real mode shapes
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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
Harmonic excitation: modal basis solution
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Example : two dofs system
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Example : two dofs system
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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Validity of modal damping hypothesis
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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Validity of modal damping hypothesis
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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Pendulum tuned mass damper
for
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Pendulum tuned mass damper
=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