41514 – Dynamics of Machinery – Theory, Experiment, Phenomenology and Industrial Applications –

Ilmar Ferreira Santos

1. Recapitulation – Mathematical Modeling & Steps

2. Eigenvalues Problem – Physical Meaning

3. Introduction Signal Analysis & Fourier Transformation & Applications

4. Experimental Examples

* Mass Elements M Particle Rigid Body Distributed * Spring Elements K Elasticity Theory & Material Magnetism Fluid Mechanics * Damping Elements D Fluid Mechanics Contact Mechanics (friction)

Mathematical Model

Mechanical Model

Physical System

Assumptions (simplifications)

Newton, Euler, D‘Alembert, Lagrange, Hamilton, Jourdain

(principles & axioms)

Static Equilibrium Position (Linearization) (structure)

(machine)

Solution:

(eigenvalues)

(eigenvectors)

1. Recapitulation – Mathematical Modeling & Steps

2. Eigenvalue Problem – Linear Model

[U,λ] = eig (–B, A) diagonal matrix lambda of generalized eigenvalues λ full matrix U whose columns are the corresponding eigenvectors so λ *A*U = –B*U

…

2. Eigenvalue Problem – Physical Meaning

λ= 0 +48.8527i 0 0 0 0 0 -48.8527i 0 0 0 0 0 + 7.3517i 0 0 0 0 0 - 7.3517i

U = 1.0000 1.0000 -0.3300 -0.3300 -0.3300 -0.3300 -1.0000 -1.0000 0 - 0.0205i 0 + 0.0205i 0 + 0.0449i 0 - 0.0449i 0 + 0.0068i 0 - 0.0068i 0 + 0.1360i 0 - 0.1360i

3. Introduction to Signal Analysis & Fourier Transform

Fourier series is a way to represent a (wave-like) function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series.

3. Introduction to Signal Analysis & Fourier Transform

3. Introduction to Signal Analysis & Fourier Transform

3. Introduction to Signal Analysis & Application

Aliasing – Distortion caused in analog-to-digital conversion by a too low rate of data sampling. For audio signals, it usually results in flutter (rapid change in the signal) and, for video signals, in stairsteps-like jagged edges (called 'jaggies') that appear on an image of curved or slanted lines. Filtering – Anti-aliasing solution techniques are based on analog filtering or super-sampling.

3. Introduction to Signal Analysis & Application

Analyzer

Leakage – In reality, signals are of time-limited nature and nothing can be known about the signal beyond the measured interval. The FFT implicitly assumes that the signal essentially repeats itself after the measured interval and hence the Fourier Transform assumes the signal to be continuous (conceptually, juxtapose the measured signal repetitively. When the measurement time is purposefully made to be a non-integral multiple of the actual signal rate, these sharp discontinuities will spread out in the frequency domain leading to spectral leakage. Windowing – Anti-leakage solution techniques are based on windowing

3. Introduction to Signal Analysis & Application

Windowing – Anti-leakage solution techniques are based on windowing. The window function should be applied the time signal by multiplication eliminating the discontinuities at the extremities of the sampled signal.

3. Introduction to Signal Analysis & Application

Windowing – Anti-leakage solution techniques are based on windowing. The window function should be applied the time signal by multiplication eliminating the discontinuities at the extremities of the sampled signal.

3. Introduction to Signal Analysis & Application

Overlapping & Averaging – But what do we do about the data that is being missed at the beginning and end of each record? Data is being reduced and/or set to zero over one half the time record - How do we assure events happening in the region of reduced amplitude areas? A processing technique exists called “overlap” processing. By applying this technique, the events occurring at or near the beginning and ending of the time records are enhanced by using overlap processing.

3. Introduction to Signal Analysis & Application

Fast Fourier Transform (FFT) of transient vibrations (acceleration signal – response of mass 4 due to an impulsive excitation on mass 5) a) frequency rate – number of points per second: 200 Hz (Rate) b) block – number of points per block: 5000

4. Experimental Examples (demonstration during the class)

mass 4

mass 5