Random Eigenvalue Problems in Structural Dynamics: An Experimental - - PowerPoint PPT Presentation

Random Eigenvalue Problems in Structural Dynamics: An Experimental Investigation

• S. Adhikari, A. Srikantha Phani and D. A. Pape

School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/∼adhikaris

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Outline of the presentation

A Brief Overview of Random Eigenvlaue Problems Random Eigenvalues of a Fixed-Fixed Beam Random Eigenvalues of a cantilever plate System Model and Experimental Setup Experimental methodology Eigenvalue Statistics Experimental results Monte Carlo simulation Conclusions & future directions

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Ensembles of structural dynamical systems

Many structural dynamic systems are manufactured in a production line (nominally identical sys- tems)

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A complex structural dynamical system

Complex aerospace system can have millions of degrees of freedom and signifi- cant ‘errors’ and/or ‘lack of knowledge’ in its numerical (Finite Element) model

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Sources of uncertainty

(a) parametric uncertainty - e.g., uncertainty in geometric parameters, friction coefficient, strength of the materials involved; (b) model inadequacy - arising from the lack of scientific knowledge about the model which is a-priori unknown; (c) experimental error - uncertain and unknown error percolate into the model when they are calibrated against experimental results; (d) computational uncertainty - e.g, machine precession, error tolerance and the so called ‘h’ and ‘p’ refinements in finite element analysis, and (e) model uncertainty - genuine randomness in the model such as uncertainty in the position and velocity in quantum mechanics, deterministic chaos.

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Overview of Random Eigenvalue Problems

EVP of Undamped or proportionally damped systems: Kφj = λjMφj (1) λj: Eigenvalue (natural frequency squared) φj: Eigenvector (modeshape) M &K are symmetric and P .D random matrices ⇒ λj real and positive. M = M + δM and K = K + δK. (2) (•): Nominal (deterministic) of of (•) δ(•): Random parts of (•).

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Randomness

M = M + δM and K = K + δK. δM and δK are zero-mean random matrices. Small randomness assumption that preserve symmetry and P .D of M and M . No assumptions on the type of randomness: need not be Gaussian, for example Fixed-Fixed beam with random placement of equal masses gives δM = 0 δK = 0 Cantilever plate with random placement of random

• scillators gives δM = 0 δK = 0

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Fixed-Fixed Beam: Experiments

The test rig for the fixed-fixed beam Actuator: Shaker, Sensors: Accelerometers

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Fixed-Fixed Beam: Experiments

Attached masses (magnets) at random locations. 12 masses, each weighting 2g, are used.

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Fixed-Fixed Beam: Properties

Beam Properties Numerical values Length (L) 1200 mm Width (b) 40.06 mm Thickness (th) 2.05 mm Mass density (ρ) 7800 Kg/m3 Young’s modulus (E) 2.0 × 105 MPa Total weight 0.7687 Kg Material and geometric properties of the beam.

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Shaker as an Impulse Hammer

pulse rate: 20s & pulse width: 0.01s. Eliminate input uncertainties. brass plate (2g) takes impact.

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Experiments: Protocol

Arrange the masses along the beam at random locations (computer generated) Measure impulse response at: 23 cm (Point1) 50 cm (Point2, also the actuation point) and 102 cm (Point3) from the left end of the beam in a 32 channel LMSTM system Transform to frequency domain to estimate frequency response function (FRF). Curvefit the FRF to estimate the natural frequencies ωn and damping factors Qn Rational Fraction Polynomial (RFP) method Nonlinear Leastsquares method Calculate the statistics of natural frequencies

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Experiments: FRF at Point 1 (23 cm from the left end)

100 200 300 400 500 600 700 800 900 1000 −20 −10 10 20 30 40 50 60 70 80 Frequency (Hz) Log amplitude (dB) of H

(2,1) (ω)

Baseline Ensemble mean 5% line 95% line

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Experiments: FRF at point 2 (the driving point FRF, 50 cm from the left end)

100 200 300 400 500 600 700 800 900 1000 −20 −10 10 20 30 40 50 60 70 80 Frequency (Hz) Log amplitude (dB) of H

(2,2) (ω)

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Experiments: FRF at point 3 (102 cm from the left end)

100 200 300 400 500 600 700 800 900 1000 −20 −10 10 20 30 40 50 60 70 80 Frequency (Hz) Log amplitude (dB) of H

(2,3) (ω)

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Ensemble Mean

5 10 15 20 25 30 35 500 1000 1500 2000 2500 3000 3500 4000 Mode number Mean (Hz) MCS Response point 1 Response point 2 Response point 3 5 10 15 20 25 30 35 500 1000 1500 2000 2500 3000 3500 4000 Mode number Mean (Hz) MCS Response point 1 Response point 2 Response point 3

Left: RFP; Right: Nonlinear Leastsquares

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Standard Deviation

5 10 15 20 25 30 35 10 10

1

10

2

10

3

Mode number Standard deviation (Hz) MCS Response point 1 Response point 2 Response point 3 5 10 15 20 25 30 35 10 10

1

10

2

10

3

Mode number Standard deviation (Hz) MCS Response point 1 Response point 2 Response point 3

Left: RFP; Right: Nonlinear Least-squares

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PDFs

−5 5 0.5 1 1.5 Eigenvalue number: 5 Normalized pdf

MCS Response point 1 Response point 2 Response point 3

−5 5 0.5 1 1.5 Eigenvalue number: 10 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 20 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 30 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 5 Normalized pdf

MCS Response point 1 Response point 2 Response point 3

−5 5 0.5 1 1.5 Eigenvalue number: 10 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 20 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 30 Normalized pdf

Left: RFP; Right: Nonlinear Least-squares

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Cantilever plate

The test rig for the cantilever plate: Front View

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Cantilever plate

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Cantilever plate: Properties

Plate Properties Numerical values Length (L) 998 mm Width (b) 530 mm Thickness (th) 3.0 mm Mass density (ρ) 7800 Kg/m3 Young’s modulus (E) 2.0 × 105 MPa Total weight 12.38 Kg

Table 1: Material and geometric properties of the can- tilever plate considered for the experiment

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Attached Oscillators

Attached oscillators at random locations. The spring stiffness varies

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Properties of Attached Oscillators

Oscillator Number Spring stiffness (×104 N/m) Natural Frequency (Hz) 1 1.6800 59.2060 2 0.9100 43.5744 3 1.7030 59.6099 4 2.4000 70.7647 5 1.5670 57.1801 6 2.2880 69.0938 7 1.7030 59.6099 8 2.2880 69.0938 9 2.1360 66.7592 10 1.9800 64.2752

Table 2:

Stiffness of the springs and natural frequency of the oscillators used to simulate unmodelled dynamics (the mass of the each oscillator is 121.4g).

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Experiments: Protocol

Attach the oscillators at random locations (computer generated) Measure impulse response at: Point 1: (4,6), Point 2: (6,11), Point 3: (11,3), Point 4: (14,14), Point 5: (18,2), Point 6: (21,10) Transform to frequency domain to estimate frequency response function (FRF). Curvefit the FRF to estimate the natural frequencies ωn and damping factors Qn Rational Fraction Polynomial (RFP) method Nonlinear Leastsquares method Calculate the statistics of natural frequencies

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Experiments: FRF at Point 1

100 200 300 400 500 600 −60 −40 −20 20 40 60 Frequency (Hz) Log amplitude (dB) of H

(1,1) (ω)

Baseline Ensemble mean 5% line 95% line

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Experiments: FRF at point 3

100 200 300 400 500 600 −60 −40 −20 20 40 60 Frequency (Hz) Log amplitude (dB) of H

(1,3) (ω)

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Ensemble Mean

5 10 15 20 25 30 35 40 50 100 150 200 250 300 350 400 450 Mode number Mean (Hz) MCS Response point 1 Response point 2 Response point 3 5 10 15 20 25 30 35 40 50 100 150 200 250 300 350 400 450 Mean (Hz) Mode number

Left: RFP; Right: Nonlinear Leastsquares

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Standard Deviation

5 10 15 20 25 30 35 10

−1

10 10

1

10

2

10

3

Mode number Standard deviation (Hz) MCS Response point 1 Response point 2 Response point 3 5 10 15 20 25 30 35 10

−1

10 10

1

10

2

10

3

Mode number Standard deviation (Hz) MCS Response point 1 Response point 2 Response point 3

Left: RFP; Right: Nonlinear Least-squares

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PDFs

−5 5 0.5 1 1.5 Eigenvalue number: 5 Normalized pdf MCS Response point 1 Response point 2 Response point 3 −5 5 0.5 1 1.5 Eigenvalue number: 10 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 20 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 30 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 5 Normalized pdf MCS Response point 1 Response point 2 Response point 3 −5 5 0.5 1 1.5 Eigenvalue number: 10 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 20 Normalized pdf −5 5 0.5 1 1.5 Eigenvalue number: 30 Normalized pdf

Left: RFP; Right: Nonlinear Least-squares

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Conclusions

The ensemble statistics such as mean and standard deviation for natural frequencies vary with the spatial location of the measured FRFs and the type of the system identification technique chosen to estimate the natural frequencies. Whilst a reasonable predictions for the mean and the standard deviations may be obtained using the Monte Carlo Simulation, higher moments, and hence the pdfs can be significantly different. In some cases, the differences in pdfs arising from different points and different identification methods can be more than those obtained from the Monte Carlo Simulation.

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