Mechanical Engineering - Measurement

Oral exam

Questions & Answers for the exam Measurement exam – Tarabini Sampling Describe what is the aliasing problem and how can we avoid it. Aliasing is a problem related to the choice of the wrong fs, if the fs is not high enough the signal cannot be properly reconstructed starting from its samples: - In time it can be explained by S hannon’s theorem: fs > 2 fmax - In frequency it can be explained through convolution: sampling in time is equal to multiplying by a train of unitary pulses at distance dT, so in frequency it is equivalent of a convolution of the signal’s spectrum wit h a train of pulses at distance 1/dT=fs. If fs is n ot high enough (>2 fmax) the convolution of the spectrum will overlap Describe what is the leakage problem and how can we avoid it. The leakage is an error of the spectrum due to the wrong choice of the observation time T. - In time it can be explained by the circular convolution, because the signal has to be made periodic, but if the length of the observation is not equal to the period of the signal (or an integer multiple) the circular convolution of the signal will reconstruct a signal different from the original one, that will show different frequency component - In frequency it can be explained by convolution, because observing the signal for a period T means multiply it by a rectangular window of length Tw, which in frequency corresponds to convoluti ng the signal with the spectrum of the window. If the period of the signal T is not an integer multiple of the frequency resolution (=1/Tw), then the zero crossing of the window’s spectrum won’t correspond with the spectral lines and there will be a distor tion of the spectrum à to avoid leakage problem we can use different types of windows like Hanning window Short time leakage/long time leakage Digitalization of analog signals: describe the meaning of A/D conversion and sampling . Compute the LSB of a 12 - bit converter with full range 0 - 10 V Digitalization consist in quantization and sampling of the signal. Sampling = discretization of the time (dT, fs, N) Quantization = discretization of the amplitudes of the signal, depending on the range available and the number of bits to store the information LSB = range / 2 n = 10V / 2 12 You have to acquire a square wave with a frequency of 127.34 Hz. Please determine the sampling frequency, the number of samples and the windows to be used. Is an antialiasing filter necessary or not? Justify your answers Fs = 2 f max for example 400 Hz The length of the window (and N) must be chosen to have a frequency resolution such that the frequency of the signal is a multiple integer of th e frequency resolution. However f of the signal is non rational, so we won’t have an exact multiple and there will be leakage. We can choose delta_f = 1/T = 2.5 Hz T = 0.4 s T = N/fs à N = T fs = 160 There will be leakage à hanning window The fs chosen is high enough so there won’t be the necessity of an anti - aliasing filter N.B. filtro anti - aliasing = filtro passa basso che restringe la banda del segnale per essere sicuri che la frequenza di campionamento soddisfi il teorema di Shannon ( fs > 2 fmax) Time Domain Definition of auto - and cross - correlation (biased and unbiased). What are the implications for time big time delays ? Describe a practical usage of these two functions in a real case. Scrivere def… For t tending to infinity the auto correlation of a random signal tends to zero or to a constant equal to the mean value of the signal (if there is a deterministic component) The cross correlation for big time delays tends to zero if the signals are unrelated Auto correlation à find periodicity of the signals and extract periodic components (es: gait) Cross correlation à find delay between functions re lated to the same phenomenon (es: radar ) Plot the autocorrelation of the signal: [0;0;0;0;0;1;1;1;1;0;0;0;0;0] Disegnare.. Plot the cross - correlation between the signals X= [0;0;0;1;1;1;1;0;0;0], Y=[ 0;0;0;1;2;2;1;0;0;0] for time delays between - 3 and +3 samples Disegnare.. Plot the biased auto - correlation of the signals X= [0;0;0;1;2;2;1;0;0;0] for time delays between - 3 and +3 samples Disegnare.. Compute and plot the auto - correlation for posit ive and negative time lags of a square wave with amplitude 7 pW/m 2 and period 1 s Scrivere .. Is there any link between the autocorrelation and the RMS? The auto - correlation of the signal for tau=0 is equal to the RMS 2 of the signal. If there are other tau for with the auto correlation is equal to RMS 2 it means that the signal is repeating itself so it is periodic à auto - correlation used to find periodicity or extract periodic components Definition of the probability density function. Sketch qualitatively the probability density function of a square wave, of a sine wave, of a triangle wave, of the Gaussian noise and of the uniform noise. Scrivere def e d isegnare.. Which are the differences between the RMS and the standard deviation? Definition and of Crest Factor and typical uses in experimental mechanics. Definition and of Skewness and Kurtosis and their uses in experimental mechanics. How can we check the periodicity of a signal in time domain? Scrivere def .. CF à presence of spikes Skewness à symmetry of the signal Kurtosis à flatness/presence of spikes Periodicity in time à auto - correlation of the signal à find the value of delay for which the auto - correlation is again equal to the RMS 2 When a certain phenomenon is said to be stationary? A signal is stationary if the mean value and the other statistical parameters are constant in time Which is the RMS of a sine wave whose zero - to - peak amplitude is 10 V? RMS=Rad(2) for every harmonic Which is the RMS of a square wave whose amplitude is 3.2 V? RMS = A = 3.2 sine wave goes from A to - A s o the RMS is A and not A/2 Which is the RMS of a DC signal whose constant amplitude is 1.32 V? RMS = 1.32 Frequency Domain Define, if there is any, the relationship between autocorrelation and autospectrum of the same signal. The cross correlation between the signal and itself is equal to the autocorrelation of the signal Demonstrate that the Fourier transform of the autocorrelation is a real function Scrivere dim .. Demonstrate that the Fourier transform of an odd signa l is an imaginary even function Scrivere dim .. Demonstrate that the elements at negative frequencies of the FT are the complex conjugate of the positive ones. Scrivere dim .. (vectors with counter - current rotations) Compare the two approaches of the counter - rotating vectors and of the single vector in the Fourier analysis Counter rotating vectors à each sine is represented as the sum of two counter rotating vector with the same frequency (but opposite sign) with amp litude A/2 and opposite phase à the spectrum will be double sided, the amplitudes will be symmetric and half of the original amplitude, the phase will be odd Single vector à each sine can be represented as a single rotating vector with amplitude A and ph ase and a certain frequency à the spectrum will be only positive Linear Systems, Dirac and Convolution Definition of linearity, homogeneity and causality of a system. To be linear a system must be homogeneous, additive, causal, time - invariant and stable. A linear system is a system in which there is a proportional relationship between input and output (that can vary in time/frequency), described by the FRF of the system. Scrivere def .. Describe system response through convolution, by a step - by - step calculation (scaling ad delayed impulse response) Superposition integral: output in time equal to the convolution between the input and the response to the impulse à product in frequency domain between the FRF and the spectrum of the input Scrivere dim .. Describe leakage through convolution. The leakage is an error of the spectrum due to the wrong choice of the observation time T. In frequency it can be explained by convolution, because observing the signal for a period T means multiply it by a rectangular window of length Tw, which in frequency corresponds to convoluting the signal with the spectrum of the window. If the period of the signal T is not an integer multiple of the frequency resolution (=1/Tw), then the zero crossing of the window’s spectrum won’t correspond with the spectral lines and there will be a distortion of the spectrum Describe aliasing through convolution. Aliasing is a problem related to the choice of the wrong fs, if the fs is not high enough the signal cannot be properly reconstructed starting from its samples . In frequency it can be explained through convolution: sampling in time is equal to multiplying by a train o f unitary pulses at distance dT, so in frequency it is equivalent of a convolution of the signal’s spectrum wit a train of pulses at distance 1/dT=fs. If fs is not high enough (>2 fmax) the convolution of the spectrum will overlap Demonstrate the sampli ng property of the Dirac function . Scrivere dim .. Compute the Fourier Transform of the Dirac function Scrivere dim .. Graphically describe the response of a system to a ramp using the convolution Disegnare .. Windows Are there any situations in which the rectangular window to observe a signal is the best solution? And which are the main differences between the rectangular and the Hanning windows? Explain advantages and drawbacks of the Hanning window if compared to the rectangular one . Rectangular window better when we have synchronization between the signal and the window length (no leakage at all) or when the signal has very close frequencies (small short time leakage) Hanning window is better to reduce long term leakage when the length of the window is not synchronized with the signal Hanning window à main lobe of 4/T, higher side lobes attenuation (60 dB/dec), higher short term leakage Rectangular window à main lobe of 2/T, lower side lobes attenuation (20 dB/dec), lower short term leakage, non - stationary or transient signals Which is the main difference between the Fourier transform of a rectangular window and of an hanning window? Hanning window à main lobe of 4/T, higher side lobes attenuation (60 dB/dec) Rectangular window à main lobe of 2/T, lower side lobes attenuation (20 dB/dec) Frequency Response, Coherence and Estimators Which are the stimuli that are commonly used to identify the frequency response function of a system? - Impulse - Harmonic – sweep - Harmonic – step - by - step - Random noise - Quasi - periodic signals Describe a practical use of the frequency response function on (a car, a motorbike, a bike, a kite surfer, your stereo system, the calibration of a sensor) Predict the output of a system give a generic input and its frequency content (= energy content) For example in a bike the vibrations transmitted by the wheel to the biker ??? Which are the actuators that can be used to identify the vibration response of mechanical systems? In which force range they are usually employed? - Impact hammer – wide ran ge of frequencies (depending on the hardness of the tip), force varies depending on the mass - Electromagnetic shaker – high frequencies, low forces - Hydraulic shaker – low frequencies, high forces - Piezoelectric shaker – high frequencies, high force also with small displacements - Vibrodynes – only sinusoidal Compare the advantages and the drawbacks of the sweep versus the white noise stimuli for the estimation of the FRF of a mechanical system S ine sweep à time - saving method to identify a large band of frequencies, the frequency has to be kept for a time period long enough to register correctly the response of the system (depending on its transitory) , sweep rate of different types , possible presence of leakage White noise à FRF has to be studied statistically ( averages and H1 and H2 estimators ) , investigation of a large set of frequencies , no presence of leakage à P ag 193 libro Exp lain the difference between random and pseudo - random signals Random signals are non - deterministic signals, that could be stationary or non - stationary and cannot be described by a mathematical expression. They have a continuous spectrum . Pseudo - random noise are observations of a random noise for a certain period T, that is then repeated periodically in time, so the signal still contains a random component but it is periodic, therefore its spectrum is discrete . Supposing that a 1DOF mechanical system has a natural frequency of 732 Hz and has a lumped mass of 3.5 kg: • Outli ne two excitation sources that can be used to derive the FRF • Plot for each of the selected excitation sources a time history and its spectrum of the stimulus that you would use to identify the frequency response function. - Impact hammer – impulse – flat spectrum - Electromagnetic shaker – sinusoidal – frequency lines Plots … Describe the procedure (position of the accelerometers, impact point, sampling frequency, window, ecc ) that you would adopt to identify the first two bending vibration modes of the structure in the figure below with two accelerometers and a dynamometric hammer. ??? ??? ??? Coherence function and correlation coefficient: definitions, analogies and e xamples of practical use. Coherence: Gamma 2 = H1/H2 = |Sxy| 2 / (Sxx Syy) It is an index of the quality of the FRF estimation. 0< Gamma 2