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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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Sources of excitation
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Sources of excitation
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Sources of excitation
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
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
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 :
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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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
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
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 decoupledequations
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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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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Example : 5 dofs impulse response
1 mode impulse response
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Example : 5 dofs 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