Modal analysis for impulse-type stimulated structures · Modal analysis for impulse-type stimulated...

Modal analysis for impulse-type

stimulated structures

Bastian Kanning

September 17, 2009

Zentrum fürTechnomathematik Modal analysis for impulse-type stimulated structures

Fachbereich 03Mathematik/Informatik

Outline

1 Introduction

2 Modal analysis

3 Instationary excitation

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http://www.math.uni-bremen.de/zetem

http://www.uni-bremen.de



Introduction

In production processes the quality of the emerging product and theeconomy of the process itself is constrained to

reliability of machine tools,

satisfactory environmental behavior,

high accuracy and e�ciency.

Although there will always be workpieces that do not need to bemanufactured with high accuracy, the demand for the highest accuracypossible in some processes is the motivation of ongoing research. Thedeviation from the intended work process is an indicator for accuracy.Insu�cient static and dynamic sti�ness of a machine tool can lead tounsatisfactory results.

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Insu�cient static and dynamic sti�ness of a machine tool can lead tounsatisfactory results.

Determination of static sti�ness of a system possible with satisfyingresult during construction.

Analysis of dynamic sti�ness still challenging.

Imbalanced dynamic behavior of machines causes vibrations. The result:

bad surface quality of the workpiece,

higher abrassive wear,

broken die [1].

Improvement of machine tools and the safety of buildings and vehiclesare motivations for extensive analyses of vibrations.

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Modal analysis

Modal analysis still is the most common tool for the estimation of thedynamical behavior of structures, such as buildings or productionmachines. With the classical modal analysis the eigenfrequencies,eigenvectors (modes) and compliance of vibrating systems are determinedvia experiments.

Figure: A modal hammer, an electromagnetic shaker and one possible structureto be analyzed.

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Sensors, e.g. attached to the structure (cf. �g.1) and inside the hammermeasure the exciting force and the resulting acceleration.

Figure: A measured impulse-type excitation and the resulting acceleration.

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The compliance represents the inverse sti�ness of a structure at eachfrequency and may vary for di�erent positions of the acceleration sensor.It is computed by a Fast Fourier Transform (FFT) of the accelerationdata pointwise divided by a FFT of the excitation data.

Figure: A FFT of the measured acceleration and excitation and the resultingcompliance.

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For given mass m and measurements of the exciting force Em and theresulting acceleration Am of the structure in one spatial direction, we canestimate the structures sti�ness s by

s =2πω2m√1− c2

, (1)

where the resonance frequency ω is taken as the maximum of thecompliance and c is an estimation of the damping This may be obtainede.g. by the so called Square Root of Two Method (SRTM).

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The structures of interest are usually referred to as damped mass-springsystems and are mathematically modelled by a system of linear ordinarydi�erential equations (ODE)

Mu(t) + Du(t) + Su(t) = f (t) , (2)

which is, according to Gasch and Knothe, the result of discretizing thepartial di�erential equation of motion in time with the Finite ElementMethod (FEM) [2].

Unfortunately, the matrices conditions can be arbitrarily bad, and thedamping properties are rarely known and sometimes di�cult to evaluateprecisely [3, 4].

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The structures of interest are usually referred to as damped mass-springsystems and are mathematically modelled by a system of linear ordinarydi�erential equations (ODE)

Mu(t) + Du(t) + Su(t) = f (t) , (2)

which is, according to Gasch and Knothe, the result of discretizing thepartial di�erential equation of motion in time with the Finite ElementMethod (FEM) [2].Unfortunately, the matrices conditions can be arbitrarily bad, and thedamping properties are rarely known and sometimes di�cult to evaluateprecisely [3, 4].

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An algorithm to simulate oscillating behaviour was implemented inMATLAB and tested on a simple structure (cf. �g.1). By (1) a sti�ness of40950.388N/m was computed and a damping ratio of 0.034 wasestimated by the SRTM.The strucure is considered an oscillating point mass, since the masses ofthe spring (0.014kg) and sensor (0.004kg) are low compared to the massof the qube (0.25kg). Additionally, the aluminium qube is consideredrigid. Thus, (2) reads

0.268u(t) + 1.134u(t) + 40950.388u(t) = f (t) . (3)

The relation of the two di�erent representations of the damping is

D = 2mωc .

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An impulse-type excitation with force f0 at the time t0 may be modelledby an appropriate Gauss curve, e.g.

f (t) = f0e−

(t−t0)2

σ . (4)

The parameter σ alters the width of the curve and should be chosen suchthat the energy of the modelled excitation equals the energy of themeasured one.

The 2nd order ODE is now transferred into a system of order one andthen solved with the Runge-Kutta Method (RKM).On the next slides we compare measurement and simulation.

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An impulse-type excitation with force f0 at the time t0 may be modelledby an appropriate Gauss curve, e.g.

f (t) = f0e−

(t−t0)2

σ . (4)

The parameter σ alters the width of the curve and should be chosen suchthat the energy of the modelled excitation equals the energy of themeasured one.The 2nd order ODE is now transferred into a system of order one andthen solved with the Runge-Kutta Method (RKM).On the next slides we compare measurement and simulation.

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The measured and simulated acceleration and a close up on a measuredand the simulated impulse.

Figure: Acceleration and impulse.

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A FFT of the measurement and simulation in m/s2 over frequency Hz.

Figure: Frequency spectra.

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Motivation

For economical and technical reasons modern production facilities areincreasingly optimized to have the highest possible production output.Due to that, they often have irregular and quickly changing loadtransmissions. These kind of processes lead to wide-band excitations ofthe systems and can also circumvent complete transient oscillation.

Figure: Rotary swaging

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A double impulse-type excitation with forces fi at times ti , i = 1, 2 isalso modelled by an appropriate Gauss curve, e.g.

f (t) = f1e−

(t−t1)2

σ1 + f2e−

(t−t2)2

σ2 .

Still, the parameters σi alter the width of the curve and should be chosensuch that the energy of the modelled excitation equals the energy of themeasured one.

After the 2nd order ODE is transferred into a 1st order ODE system it issolved with the RKM on several subintervals, depending on how manyimpulses are considered.The next slides show again a comparison of measurement and simulation.

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A double impulse-type excitation with forces fi at times ti , i = 1, 2 isalso modelled by an appropriate Gauss curve, e.g.

f (t) = f1e−

(t−t1)2

σ1 + f2e−

(t−t2)2

σ2 .

Still, the parameters σi alter the width of the curve and should be chosensuch that the energy of the modelled excitation equals the energy of themeasured one.After the 2nd order ODE is transferred into a 1st order ODE system it issolved with the RKM on several subintervals, depending on how manyimpulses are considered.The next slides show again a comparison of measurement and simulation.

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Figure: The measured and simulated excitation in N over time in s.

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Figure: The measured and simulated acceleration in m/s2 over time in s.

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Figure: A FFT of the measurement and simulation in m/s2 over frequency Hz.

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Weck, M., Brecher, C., Werkzeugmaschinen - Messtechnische

Untersuchung und Beurteilung, dynamische Stabilität, 7. Au�age,Springer, 2006.

Gasch, R. Knothe, K., Strukturdynamik Bd.2 Kontinua und ihre

Diskretisierung, Springer Berlin, 1989.

Gasch, R. Knothe, K., Strukturdynamik Bd.1 Diskrete Systeme,Springer Berlin, 1989.

Tisseur, F., Meerenberg, K., The quadratic eigenvalue problem,SIAM Rev., Vol. 43, No. 2, pp. 235-286, 2001.

Singer, N. C., Residual Vibration Reduction in Computer ControlledMachines, PhD Thesis, Department of Mechanical Engineering,

MIT, 1988.

Tropp, J. A., Greed is good: Algorithmic results for sparseapproximation, IEEE Trans. Inform. Theory, Vol. 50, No. 10, pp.2231-2242, 2004.

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Modal analysis for impulse-type stimulated structures · Modal analysis for impulse-type stimulated...

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