Extract time-domain order waveforms from vibration signal
the time-domain waveforms corresponding to a specified set of orders
present in an input signal,
xrec = orderwaveform(
measured at a set
rpm of rotational speeds expressed
in revolutions per minute.
fs is the measurement
sample rate in Hz. The vector
the desired orders, whose waveforms are returned in the corresponding
xrec. The function uses the Vold-Kalman
filter for the computation.
Create a simulated signal sampled at 600 Hz for 5 seconds. The system that is being tested increases its rotational speed from 10 to 40 revolutions per second (or, equivalently, from 600 to 2400 revolutions per minute) during the observation period.
Generate the tachometer readings.
fs = 600; t1 = 5; t = 0:1/fs:t1; f0 = 10; f1 = 40; rpm = 60*linspace(f0,f1,length(t));
The signal consists of four harmonically related chirps with orders 1, 1/2, √2, and 2. The amplitudes of the chirps are 1, 1/2, √2, and 2, respectively. To generate the chirps, use the trapezoidal rule to express the phase as the integral of the rotational speed.
ord = [1 0.5 sqrt(2) 2]; amp = [1 0.5 sqrt(2) 2]; ph = 2*pi*cumtrapz(rpm/60)/fs; x(1,:) = amp(1)*cos(ord(1)*ph); x(2,:) = amp(2)*cos(ord(2)*ph); x(3,:) = amp(3)*cos(ord(3)*ph); x(4,:) = amp(4)*cos(ord(4)*ph); xsum = sum(x);
Reconstruct the time-domain waveforms that compose the signal.
xrec = orderwaveform(xsum,fs,rpm,ord);
Visualize the results. Zoom in on a time interval occurring after the transients have decayed.
for kj = 1:4 subplot(2,2,kj) plot(t,x(kj,:),t,xrec(:,kj)) title(['Order = ' num2str(ord(kj))]) xlim([2 3]) end
Create a simulated vibration signal consisting of two crossing orders corresponding to two different motors. The signal is sampled at 300 Hz for 3 seconds. The first motor increases its rotational speed from 10 to 100 revolutions per second (or, equivalently, from 600 to 6000 rpm) during the measurement. The second motor increases its rotational speed from 50 to 70 revolutions per second (or 3000 to 4200 rpm) during the same period.
fs = 300; nsamp = 3*fs; rpm1 = linspace(10,100,nsamp)'*60; rpm2 = linspace(50,70,nsamp)'*60;
The measured signal is of order 1.2 and amplitude 2√2 with respect to the first motor. With respect to the second motor, the signal is of order 0.8 and amplitude 4√2.
x = [2 4]*sqrt(2).*cos(2*pi*cumtrapz([1.2*rpm1 0.8*rpm2]/60)/fs);
Make the first motor excite a resonance at the middle of the frequency range.
y = [1+1./(1+linspace(-10,10,nsamp).^4)'/2 ones(nsamp,1)].*x; x = sum(y,2);
Visualize the orders using
Reconstruct the time-domain waveforms that compose the signal. Use the Vold-Kalman algorithm to decouple the crossing orders.
xrec = orderwaveform(x,fs,[rpm1 rpm2],[1.2 0.8],[1 2],'Decouple',true);
Plot the original and reconstructed waveforms.
for kj = 1:2 figure(kj) subplot(2,1,1) plot((0:nsamp-1)/fs,y(:,kj)) legend('Original') title(['Motor ' int2str(kj)]) subplot(2,1,2) plot((0:nsamp-1)/fs,xrec(:,kj)) legend('Reconstructed') end
orderlist— List of orders
List of orders, specified as a vector.
not have values larger than
fs/(2 × max(
rpmrefidx— RPM column indices
RPM column indices, specified as a vector of the same size as
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
'Decouple',true,'FilterOrder',2extracts the specified order waveforms simultaneously and uses a second-order Vold-Kalman filter.
'FilterOrder'— Vold-Kalman filter order
Vold-Kalman filter order, specified as the comma-separated pair
'FilterOrder' and either
'Bandwidth'— Approximate half-power bandwidth
fs/100 (default) | real scalar | real vector
Approximate half-power bandwidth, specified as the comma-separated
pair consisting of
'Bandwidth' and either a real
scalar or a real vector with the same number of elements as
Smaller values of
'Bandwidth' produce smooth, narrowband
output. However, this output might not accurately reflect rapid changes
in order amplitude.
'Decouple'— Mode decoupling option
Mode decoupling option, specified as the comma-separated pair
'Decouple' and a logical value. If
this option is set to
order waveforms simultaneously, enabling it to separate closely spaced
or crossing orders.
'SegmentLength'— Length of overlapping segments
Length of overlapping segments, specified as the comma-separated
pair consisting of
'SegmentLength' and an integer.
If you specify a segment length, then
the input signal into segments. It then computes the reconstructed
waveforms for each segment and combines the results to produce the
output. If the segments are too short, the function might not properly
capture localized events such as crossing orders.
xrec— Reconstructed time-domain order waveforms
Reconstructed time-domain order waveforms, returned as a matrix with one waveform in each column.
 Feldbauer, Christian, and Robert Höldrich. “Realization of a Vold-Kalman Tracking Filter — A Least Squares Problem.” Proceedings of the COST G-6 Conference on Digital Audio Effects (DAFX-00). Verona, Italy, December 7–9, 2000.
 Vold, Håvard, and Jan Leuridan. “High Resolution Order Tracking at Extreme Slew Rates Using Kalman Tracking Filters.” Shock and Vibration. Vol. 2, 1995, pp. 507–515.