Track and extract order magnitudes from vibration signal
mag = ordertrack(
mag, that contains time-dependent root-mean-square
(RMS) amplitude estimates of a specified set of orders,
present in input signal
measured at a set
rpm of rotational speeds expressed
in revolutions per minute.
fs is the measurement
sample rate in Hz.
ordertrack(___) with no output
arguments plots in the current figure the time-dependent orders and
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, 0.5, 4, and 6. 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.
o1 = 1; o2 = 0.5; o3 = 4; o4 = 6; a1 = 1; a2 = 0.5; a3 = sqrt(2); a4 = 2; ph = 2*pi*cumtrapz(rpm/60)/fs; x = [a1 a2 a3 a4]*cos([o1 o2 o3 o4]'*ph);
Extract and visualize the magnitudes of the orders.
ordertrack(x,fs,rpm,[o1 o2 o3 o4])
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 revolutions per minute) during the measurement. The second motor increases its rotational speed from 50 to 70 revolutions per second (or 3000 to 4200 revolutions per minute) 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.
rs = [1+1./(1+linspace(-10,10,nsamp).^4)'/2 ones(nsamp,1)]; x = sum(rs.*x,2);
Visualize the orders using
Compute the order magnitudes for both motors as a function of RPM. Use the Vold-Kalman algorithm to decouple the crossing orders.
ordertrack(x,fs,[rpm1 rpm2],[1.2 0.8],[1 2],'Decouple',true)
Analyze simulated data from an accelerometer placed in the cockpit of a helicopter.
Load the helicopter data. The vibrational measurements,
vib, are sampled at a rate of 500 Hz for 10 seconds. Inspection of the data reveals that it has a linear trend. Remove the trend to prevent it from degrading the quality of the order estimation.
load('helidata.mat') vib = detrend(vib);
Compute the order-RPM map. Specify an order resolution of 0.005.
[map,order,rpm,time,res] = rpmordermap(vib,fs,rpm,0.005);
Compute and plot the average order spectrum of the signal. Find the three highest peaks of the spectrum.
[spectrum,specorder] = orderspectrum(map,order); [~,pkords] = findpeaks(spectrum,specorder,'SortStr','descend','Npeaks',3); findpeaks(spectrum,specorder,'SortStr','descend','Npeaks',3)
Track the amplitudes of the three highest peaks.
x— Input signal
Input signal, specified as a row or column vector.
a sinusoid embedded in white Gaussian noise.
fs— Sample rate
Sample rate, specified as a positive scalar expressed in Hz.
rpm— Rotational speeds
Rotational speeds, specified as a vector or matrix of positive values expressed in revolutions
per minute. If
rpm is a vector, it must have the same
rpm is a matrix,
rpmrefidx is specified, then
rpm must have at least two columns, and each column
must have as many elements as
100:10:3000 specifies that a system
rotates initially at 100 revolutions per minute and runs up to 3000
revolutions per minute in increments of 10.
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
The presence of this argument specifies that the Vold-Kalman algorithm
is to be used.
map— Order-RPM map
Order-RPM map, specified as a matrix. Use
rpmordermap to compute order-RPM maps.
order— Orders in order-RPM map syntax
Orders in order-RPM map syntax, specified as a vector. The length
order must equal the number of rows in
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,'Amplitude','peak'extracts the specified orders simultaneously and returns the peak amplitude of each order.
'Amplitude'— Amplitude type
Amplitude type, specified as the comma-separated pair consisting
'Amplitude' and one of
'rms' — Returns the root-mean-square
amplitude for each estimated order.
'peak' — Returns the peak
amplitude for each estimated order.
'power' — Returns the power
level for each estimated order.
'Scale'— Magnitude scaling
Magnitude scaling, specified as the comma-separated pair consisting
'Scale' and either
'linear' — Returns magnitude
values scaled in linear units.
'dB' — Returns magnitude
values scaled logarithmically and expressed in decibels.
'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
orderlist. Smaller values of
'Bandwidth' produce smooth, narrowband output.
However, this output might not accurately reflect rapid changes in order
amplitude. This argument applies only to the Vold-Kalman
'Decouple'— Mode decoupling option
Mode decoupling option, specified as the comma-separated pair consisting of
'Decouple' and a logical value. If this option is
extracts order magnitudes simultaneously, enabling it to separate
closely spaced or crossing orders. This argument applies only to the
'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
ordertrack divides the input
signal into segments. It then computes the order magnitudes 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. This argument applies only to the Vold-Kalman
mag— Order-magnitude matrix
Order-magnitude matrix, returned as a matrix.
rpm— Rotational speeds
Rotational speeds, returned as a vector of positive values expressed in revolutions per minute.
time— Time instants
Time instants, returned as a vector.
 Brandt, Anders. Noise and Vibration Analysis: Signal Analysis and Experimental Procedures. Chichester, UK: John Wiley & Sons, 2011.
 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.
 Tůma, Jiří. “Algorithms for the Vold-Kalman Multiorder Tracking Filter.” Proceedings of the 14th International Carpathian Control Conference (ICCC), 2013, pp. 388–94. https://doi.org/10.1109/CarpathianCC.2013.6560575.