Question 3 Flashcards
(17 cards)
Briefly describe the advantages and disadvantages of using an impact hammer for modal analysis, compared to using an electrodynamic shaker. How might the bandwidth of the impact signal be controlled?
Advantages are: no ongoing contact with structure, hence no intrusion on modal properties, and it easy to apply force to different points on the structure.
The disadvantage is that only the impulse signal can be applied. The bandwidth is increased through the use of a harder tip.
Explain what is meant by leakage in data acquisition and explain its effect on the resulting frequency response function (FRF). Suppose a signal consists of two sinusoidal frequencies of 2.5 Hz and 4.0 Hz; what is the minimum time period that the signal must be sampled to avoid leakage?
Leakage – errors in calculating FFTs because the signal is not periodic in the sample window. Tends to spread peaks (single frequencies or resonant responses) – the signal energy ‘leaks’ into adjacent frequencies. The time period of the 2.5Hz component is 0.4s, and the time period for the 4Hz component is 0.25s – the minimum time to get full periods of both signals is 2s.
Figure Q3.1 (on page 5) shows a receptance measured on a cantilever beam plotted on a Nyquist plot (complex plane). The receptance is measured every 0.2 Hz (denoted by a circle), and the given frequencies are also marked by cross (every 1Hz). The damping can be assumed to be structural damping. Estimate the two natural frequencies and damping ratios of this system using the circle fit method.
Natural frequencies given when the sweep rate around the circle is greatest – in this case
32.1 Hz and 201.5 Hz.
For the damping ratio, to make life easier estimate the frequencies for q=180 degrees. For
mode 1 this is about 31.3Hz and 32.9Hz. Then, from data sheet, for the first mode
ζ = (wb-wa)/(2Wd)
Similarly for the second mode, frequencies are about 200.3Hz and 202.4Hz, and so the
damping ratio is about 0.0052.
Figure Q3.2 (on page 6) shows three receptances for the cantilever beam around
the second mode (the receptance for accelerometer 3 is the same data as Figure
Q3.1). The receptance is measured every 0.2 Hz (denoted by a circle), and the
given frequencies are also marked by cross (every 1Hz). Estimate the second
mode shape for the beam at the measured locations.
he mode shapes are essentially given by the diameter of the circles in the Nyquist plots.
Measurement of the diameters gives 2.8, 2.6 and 4.8 (all times 10-5). Note that the first two
circles are in the negative imaginary half plane, and the third in the positive imaginary half
plane; thus the mode shape for accelerometer 3 has the opposite sign to the other two. Hence
the (non-normalised) mode shape can be written as {2.8 2.6
−4.8} (3x1 Matrix form)
If we want to normalise the mode shapes (not required in question), then the point receptance
(accelerometer 1) has amplitude 𝑢# given by (note natural frequency in rad/s)
𝑢1^2 = 2𝜉2𝜔2^2 ×2.8×10^-5 = 0.685
Scaling the other elements in the same way gives the normalised mode shape as {
0.685
0.637
−1.175
} (3x1 Matrix form)
Briefly explain the phenomena of aliasing. What measure is taken to reduce the effects of aliasing?
Alaising – sample rate is not fast enough compared to frequency content of signal. High frequency components cannot be distinguished from those at lower frequencies. Use an anti-aliasing filter, which is essentially a low pass filter (with cut off frequency below the Nyquist frequency), to remove the higher frequency components.
If you are using a signal analyser that supports data sample rates of 1024 Hz,
2048 Hz, 4096 Hz or 8192 Hz, what is the minimum signal sample rate that
must be chosen for an FFT where the maximum frequency of interest is
2000 Hz? How many samples will there be in the time series if the FFT is to
have a resolution of 0.25 Hz?
[4 marks]
Max frequency is 2000Hz so the Nyquist frequency must be 2000Hz, or higher. Nyquist
frequency is Fs/2 so we need Fs 4000 or higher. The lowest available sample frequency to meet
this is 4096Hz. The length of the sample must be 1/delta f = 4s. The number of samples is 4*fs =16384.
Figures Q3.1 to Q3.3 (on pages 7 and 8) shows receptances measured by
three accelerometers on a cantilever beam plotted in terms of real and
imaginary parts. The crosses denotes the FRF (frequency response function)
every 0.5 Hz. Estimate the two natural frequencies and damping ratios of this
system using the circle fit method.
Natural frequencies given when the sweep rate around the circle is greatest – in this case
(pick any of the accelerometers) about 32 Hz and 201.5 Hz.
For the damping ratio, to make life easier estimate the frequencies for Theta=180 degrees. For mode 1 this is about 31.3Hz and 32.9Hz. Then, from data sheet, for the first mode.
ζ = (wb-wa)/(2Wd)
Similarly for the second mode, frequencies are about 200.3Hz and 202.4Hz, and so the
damping ratio is about 0.0052.
If conducting a modal test on a parked car, why might it be important to check the tyre
pressures?
The tyres will have a significant compliance which will affect the dynamics of the car, and
hence the natural frequencies will change with the tyre pressure.
Why might a displacement sensor be a more appropriate choice than an accelerometer for
a modal test for a structure with very low natural frequencies?
For a given displacement amplitude the acceleration increases as a function of ω2. There-
fore the low frequency response of a structure often has a large displacement but small
acceleration, meaning that the acceleration measurement would be susceptible to noise.
Furthermore, because of charge leakage piezoelectric sensors perform poorly at very low
frequencies. Since displacements tend to be high at low frequencies, they are relatively
easy to measure.
Why is it often important to use soft mounts for a modal test on a component? An FEA model has predicted the first modal frequency for a component that you are testing to be 8Hz. The component has a mass of 2.5kg, and is suspended by two elastic chords,
each with an estimated spring rate of 5N/mm. What experimental factor might cause the agreement with FEA to be poor?
Often it is difficult to mount the structure in the configuration in which it would operate,
and in these case approximating free boundaries is a convenient approach.
If the structure
is required to be tested free-free, then the rigid body modes need to be significantly lower
than the flexible modes of the structure. In this case, the support stiffness is 10N/mm,
and hence the rigid body mode would have a frequency of sqrt(10 ×(10^3/2.5))= 63.25rad/s
or 10.07Hz.
Since this is larger than the expected flexible natural frequency in free-free
conditions, the support springs are too stiff.
What is the principal disadvantage of stepped sine testing?
Stepped since testing is very slow to set up and very slow in taking the measurements as
each frequency must be applied in turn, and the transients allowed to decay, before the
force and response measurement can be acquired.
The sampling rate of audio data on a CD is 44100Hz. The highest frequency detectable by
the human ear is generally accepted to be 20000Hz. How are these two facts connected?
The Nyquist criterion states that the signal must be sampled at least twice the largest
frequency of interest. Hence a signal at 20000Hz would have to be sampled at 40000Hz,
and in the case of audio signals the sample rate is slightly higher than this.
You ask a colleague to perform some modal test measurements. The next day, they return
with the time series data files for you to analyse on memory stick. However, the colleague
reports that at the end of the test, they realised that they had forgotten to use anti-
aliasing filters when recording the data. To remedy this, they post-processed the data in
Matlab, using a digital low pass filter. Is the data acceptable?
No! However the digital filter in Matlab works, it will have no way of knowing what is a
genuine frequency component and what is due to aliasing.
What is the minimum number of samples required to capture the DFT of a signal, where
the maximum frequency of interest is 50kHz, to a resolution of 0.2Hz? If the data acqui-
sition system only supports fixed sample rates of 32kHz, 64kHz, 128kHz or 256kHz, what
is the minimum time duration required to acquire the DFT?
The maximum frequency is 50kHz, and so we must sample at at least 2 ×50 = 100kHz,
hence maximum sample period is 1/100 = 0.01ms. The required frequency resolution
is 0.2Hz, so the measurement must be 1/0.2 = 5s long. So 5/(0.01 ×10−3) = 500,000
samples are required. The minimum time duration is driven by the required frequency
resolution, and so it remains unchanged in the second part of the question (note that if
the required sample rate did not give an integer number of samples in the required time
period, the time period would have to rounded up slightly ).
Why are windows often applied to data before the FRF is taken?
The windows are applied to ensure the signal is periodic in the time window sampled.
Why would the given window signal be a poor choice to use with tap test data?
The window shown would be suitable for continuous signals, such as random sig-
nals. The problem for the impact test is that the force is applied very early in the
time window where the window function is small. Thus the window function will
significantly attenuate the force and response signals when they are at their highest
values, and hence significantly distort the FFTs of the windowed signals. The correct
window for an impact response is an exponential window.
A datapoint at 12Hz has a magnitude of 3.45 in a receptance FRF. What is the magnitude
of the equivalent datapoint in the corresponding accelerance FRF?
Since acceleration is the second derivative of displacement, we must multiply the magnitude of the receptance by ω2 to obtain the magnitude of the accelerance. In this case
the frequency is 12Hz, or 12 ×2π = 75.40rad/s (note in all response type calculations
the frequency should be in rad/s). Thus, assuming the units of receptance are m/N, the
magnitude of the accelerance is 3.45 ×75.402 = 19614m/(Ns2).
