Mechanische trillingen

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Mechanische trillingen

• LES 8 MODALE ANALYSE

Patrick Guillaume E-mail patrick.guillaume_at_vub.
ac.be Tel. [phone]
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Signal vs. System Analysis

• Signal Analysis / Fourier Analysis
• Only the responses (due to the unknown operating
  forces) are measured
• System Analysis / Modal Analysis
• The system is stimulated with known forces (in
  laboratory conditions)
• Applied forces and resulting responses are
  measured

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Basic Equations for a 2-DOF System

• Forces acting on mass 1

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Basic Equations for a 2-DOF System

• Time domain
• Frequency domain (Laplace)
• Matrix notations
• Dynamic stiffness matrix
• Transfer function matrix

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FRF of 2-DOF System
0
-180
-360
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Multiple Degree of Freedom (MDOF)

• Dynamic stiffness matrix
• Transfer function matrix

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FRF and IRF of MDOF Systems
POLES (GLOBAL PARAMETERS)
IMPULSE RESPONSE FUNCTION
FREQUENCY RESPONSE FUNCTION
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Modal Decomposition
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Eigenvalues and Eigenvectors

• No Damping

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Modal Mass and Stiffness

• No Damping
• Modal mass and stiffness are not unique !

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Modal Coordinates
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Modal Model
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Scaling of the Mode Shapes
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Modal Model Modal Parameters
Modal participation factors
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Modal Model Modal Parameters
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Modal Model Modal Parameters
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Mode Shape Vector

• Mode shape is a deflection-pattern associated
  with a modal frequency or pole
• Relative displacements
• Mode shapes are continuous functions of position
• Sampled mode shapes mode shape vector
• Spatial resolution and Spatial aliasing

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Normal and Complex Modes

• Normal modes (or real modes)
• All points are moving in phase or out of phase
• Node points
• Complex modes
• Mode shape vectors are complex
• Node points are not well defined anymore
• Warning Poor measurements

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Cantilever Beam
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FRF in respectievelijk knooppunt 1 en 2
Mode 1 2 3
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Real-Imaginary Plot Peak Picking (normal modes)
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Simple Structure Mode Shapes
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Peak Picking
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Shaker Testing - Fixed FRF Measurements
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Impact Testing - Roving FRF Measurements
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Roving or Fixed FRF Measurements
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Noise in the Output Measurement

• Force measurement
• Electrical noise
• Response measurement
• Electrical noise
• Machines, footsteps, wind, sound, will result
  in mechanical noise (process noise)
• Least-Squares Estimation
• Minimize the effect of output noise

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Noise in the Output Measurement Multivariable
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Noise in the Input Measurement

• At its natural frequencies the structure becomes
  very compliant
• Least-Squares Estimation
• Minimize the effect of input noise

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The Coherence Function

• Degree of linearity
• Smaller than 1 when
• Noise in the measurements
• Nonlinearities

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Noise in the Input and Output Measurements

• Choice of optimal FRF estimator
• H1
• Under estimation
• H2
• Over estimation
• H3, Hv, Hiv,

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Implementing the Excitation

• Attached exciters
• Electrodynamic exciter
• Hydraulic exciter
• Eccentric rotating masses
• More exotic devices such as rockets,
• Non-attached exciters
• Hammers, impactors
• Acoustic excitation

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Force Measurements

• Piezo-electric force transducer
• Small size and mass
• Linearity
• Wide dynamic range
• Wide frequency band
• Exciter attachment
• Stringer
• High axial stiffness
• Low transverse and rotational stiffness
• Mechanical fuse
• Protection against destructive overload

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Random Excitation

• Random sequence with Gaussian distribution
• Random amplitudes and phases
• Averaging is needed
• Converge to flat amplitude spectrum
• Remark Force is not flat in general due to
  interaction with the structure
• Signal processing errors (leakage errors)
• Effect of nonlinearities is reduced by averaging

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Random Noise
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Uniform Window
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Hanning Window
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Burst Random
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Burst Random with Uniform Window
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Exponential Window
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Periodic Random / Pseudo-Random Excitation

• Periodic random
• No leakage errors (periodic signal) in
  steady-state conditions
• Averaging is required
• Pseudo random
• Constant amplitudes and random phases
• No leakage errors (periodic signal) in
  steady-state conditions
• Averaging is not required

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Impact Excitation

• Force transducer and tip
• Advantages
• Easy to use
• No interaction with structure
• Force is flat in useful frequency range
• Relatively inexpensive

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Impact Excitation

• Disadvantages
• Large crest factor
• Nonlinearities
• Limited control of amplitude spectrum

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Force Window

• Force window (or transient window)
• Remove noise
• Effect of Double Hits

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Response Window

• Lightly damped structure
• Leakage errors
• Heavily damped structure
• Remove noise