REDUCTION TO SDOF SYSTEMS
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Vibrations : Reduction to SDOF systems 1 Arnaud Deraemaeker REDUCTION TO SDOF SYSTEMS [email protected] EQUIVALENT MASS AND STIFFNESS APPROACH 1 2
Transcript of REDUCTION TO SDOF SYSTEMS
Vibrations : Reduction to SDOF systems
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Arnaud Deraemaeker
REDUCTION TO SDOFSYSTEMS
EQUIVALENT MASS AND STIFFNESS APPROACH
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Vibrations : Reduction to SDOF systems
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-Equivalent stiffness?-Equivalent mass ?-Validity (frequency band) ?
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Portal frame
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General principle :
F in the direction of excitation (and motion) of the mass
Computation : - simplified model- numerical approximation (finite elements)
Equivalent stiffness computation
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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 : bar in traction
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Direction of excitation
Attachmentpoint
From tables
Equivalent stiffness : bending cantilever beam
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Equivalent stiffness of beams in bending can be computed using formulas from resistance of materials or using tables :
Equivalent stiffness of beams
http://home.eng.iastate.edu/~shermanp/STAT447/STAT%20Articles/Beam_Deflection_Formulae.pdf
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Equivalent stiffness of beams
http://home.eng.iastate.edu/~shermanp/STAT447/STAT%20Articles/Beam_Deflection_Formulae.pdf
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Portal frame example
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Portal frame example
https://www.sprecace.com/node/43
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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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Equivalent mass : energy method
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Continuous model
SDOF model 1 SDOF model 2
m=100 kg, E =210 GPa, r=7800 kg/m3, L=1m;A = 0.0004 m2
Limits of the approximation
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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 = rAL/3 = 1.04 kg <<M
SDOF hypothesis not valid for frequencies > 2000 Hz
Limits of the approximation
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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 = rAL/3 = 10.4 kg <M
SDOF hypothesis not valid for frequencies> 500 Hz
ma should be taken into account
Limits of the approximation
Multiply mass of bar by 10
SINGLE MODEAPPROXIMATION
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Displacement of the beam(first mode shape approximation)
Approach for continuous systems
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Approach for continuous systems
Strain energy
Kinetic energy
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19Vibration problems in structures, H. Bachman, 1995
Approach for continuous systems
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Example of a building
Equivalent 1 DOFsystem
Full modelCollocated transfer function
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Single mode approximation
Particular case of a point load at reference position
Finite element approach
Position dependent scaling
Equation of motion projectedin the modal domain
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Mode 5 (0.89 Hz)Mode 4 (0.79 Hz)
Mode 3 (0.40 Hz)Mode 2 (0.27 Hz)
Mode 1 (0.24 Hz)
Mode shapes are generally ‘mass normalized’ :
Building mode shapes
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Reference point
Mode 1 (0.24 Hz)
Exemple of a building
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Example of a building
Equivalent 1 DOFsystem
Full model
Collocated transfer function
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