R t Rotor-structure t t I t Interaction ti in i Helicopter H li t
Image quality of a telescope
Point Source
Diffraction
Aberration
Effect of a vibration with amplitude /4 on image quality
250 nm
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Part 1: Discrete systems •Introduction •Single degree of freedom oscillator •Convolution integral •Beat phenomenon B h •Multiple degree of freedom discrete systems •Eigenvalue problem •Modal coordinates •Modal coordinates •Damping •Anti‐resonances
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Why suppress vibrations ? Failure Building B ildi response tto earthquakes th k ((excessive i strain) t i ) Wind on bridges (flutter instability) Fatigue
Comfort Car suspensions Noise in helicopters Wi d i d Wind-induced d sway iin b buildings ildi
Operation of precision devices DVD readers Wafer steppers Telescopes & interferometers
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How ? Vibration damping: Reduce the resonance peaks Vibration isolation: Prevent propagation of disturbances to sensitive payloads
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Active damping in civil engineering structures
TMD: Tuned Mass Damper = DVA: Dynamic Vibration Absorber AMD: Active Mass Damper 4
Single degree of freedom (d.o.f.) oscillator
Free body diagram
Free response: Solution
?
Characteristic equation: Eigenvalues:
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(A ,B, A1, B1 depend on initial conditions)
Impulse response
Spring and damping forces Have finite amplitudes p
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Impulse response for various damping ratios
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Convolution 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)
Harmonic excitation:
FRF:
The FRF is the Fourier transform of the impulse response 14
Fourier transform
Convolution integral (linear systems):
Parseval theorem:
= energy spectrum of f(t) 15
Transient response (Beat) Undamped oscillator starting from rest:
[
]
Modulating function
At resonance
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Transient response
Beat
Steady-state amplitude:
The beat is a transient Phenomenon !
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State Space form (system of first order differential equations)
Osc ato Oscillator: State variables:
Alternative choice 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 hinge rigid bar
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Multiple degree of freedom systems
In matrix form:
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Stiffness matrix
Mass matrix
Damping matrix
Symmetric & semi positive definite
[ ]
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Eigenvalue problem Free response of the conservative system (C=0)
A non trivial solution
Eigenvalue Ei l problem
exists if
The eigenvalues s are solutions of Because M and K are symmetric and semi-positive definite, the eigenvalues are purelyy imaginary: g y
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 shapes
Upon permuting i and j, Subtracting:
The mode shapes corresponding to distinct natural frequencies are orthogonal with respect to M and K Modal mass (or generalized mass) [Can be selected freely] Rayleigh quotient: 25
Orthogonality g y relationships p in matrix form: with
Notes: (1) Multiple natural frequencies: If several modes have the same natural frequency, they form a subspace and any vector in this subspace is also solution of the eigenvalue problem. (2) Rigid body modes: They have no strain energy: They also satisfy eigenvalue problem with
which means that they are solutions of the
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Free response p from initial conditions
2n constants to determine from the initial conditions. Using the orthogonality conditions,
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If there are rigid body modes (i=0)
Rigid body modes
Flexible modes
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Orthogonality relationships
Modal coordinates
x
Assumption of modal damping:
Set of decoupled equations of single d.o.f. oscillators: Mode i: Work of the external forces on mode i
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Modal truncation M d i: Mode i The modes within the bandwidth of f respond dynamically; Those 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 lightly damped Structures (0.0002) with shunted PZT patches
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R l i h damping Rayleigh d i
and are free parameters that Can be selected to match the Damping of two modes.
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Dynamic y flexibility y matrix Harmonic response of:
