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Insight • Vol 63 • No 8 • August 2021 457
VIBRATION CM
l Submitted 13.01.21 / Accepted 27.05.21
S Lahdelma, Professor Emeritus, DSc (Tech), is with the University of
Oulu, Finland. Email: sulo.lahdelma@gmail.com
DOI: 10.1784/insi.2021.63.8.457
On the use of jerk and snap in condition
monitoring of machinery – review and case studies
The time derivatives of acceleration oer a great advantage in detecting impact-causing faults at an early stage
in condition monitoring applications. Defective rolling bearings and gears are common faults that cause impacts.
This article is based on extensive real-world measurements, through which large-scale machines have been studied.
Numerous laboratory experiments provide additional insight into the matter. A practical solution for detecting faults with
as few features as possible is to measure the root mean square (RMS) velocity according to the standards in the frequency
range from 10 Hz to 1000 Hz and the peak value of the second time derivative of acceleration, ie snap. Measuring
snap produces good results even when the upper cut-o frequency is as low as 2 kHz or slightly higher. This is valuable
information when planning the mounting of accelerometers.
Keywords: derivative of acceleration, jerk, snap, crackle, pop, cavitation, slowly rotating bearings,
condition monitoring, vibration analysis.
S Lahdelma
1. Introduction
1.1 Displacement
e rst measurement parameter used in vibration measurements
for machine condition monitoring was displacement x(t), which
was measured using mechanical or optical devices.
In 1939, Rathbone[1] published a groundbreaking vibration severity
chart, which is presented using a log-log scale in the frequency range
from 1 Hz to 120 Hz. e severity chart is based on extensive practical
measurements of vibrations from a variety of machines[1-3]. Rathbone
uses the following terms to describe how smoothly a machine runs:
very smooth; good; fair; slightly rough; rough; and very rough. Among
other early displacement research, the works of Yates (1949)[3-5],
Federn (1958)[6] and Blake (1964)[2,7] could be mentioned.
e VDI 2056 standards were an important milestone in the
evaluation of vibration severity. e rst edition, published in 1958,
includes the peak values of displacement and velocity as features.
e edition is covered in more detail in[6,8]. e second edition
of VDI 2056 from 1964[9] is renowned. According to the second
edition, measurements are made in the frequency range from
10 Hz to 1000 Hz and the peak value of velocity is replaced by the
root mean square value of velocity (VRMS).
Furthermore, the 1964 edition of VDI 2056 divides machines
into six groups (Gruppe) based on their size and how exible their
support is. It also includes VRMS limit values describing the running
of a machine: good (gut); usable (brauchbar); still acceptable (noch
zulässig); and not acceptable (unzulässig). Rathbone’s research was
used when VDI 2056 was compiled[4,6].
Lastly, the IRD Mechanalysis ‘General machinery vibration
severity chart’ from 1964[20,21] should be mentioned, in which the
peak-to-peak value of displacement and the peak value of velocity
are measured. e measuring range is from 1000 to 100,000 r/min,
ie from 1.67 Hz to 1667 Hz. Additional information on the use of
displacement can be found, for example, in[10-19].
1.2 Velocity
In the past, vibration velocity was measured mostly with velocity
transducers[20,21]. In the present day, however, mainly accelerometers
are used to form a velocity signal by applying either analogue or
numeric integration to its output. On the other hand, a velocity
spectrum can be easily calculated directly from an acceleration
spectrum[22-24]. Vibration velocity is usually measured in the
frequency range from 10 Hz to 1000 Hz. is is carried out, for
example, according to VDI 2056[9], as well as to the ISO 2372 and
ISO 3945 standards[22,25].
In vibration measurements, velocity was already being used as a
measurement parameter in the 1940s and the 1950s[3,4,6]. VDI 2056
(1964) was a milestone in the promotion of velocity measurements.
It contains the idea that when using velocity as a measurement
parameter in the frequency range from 10 Hz to 1000 Hz, equally
large vibration components are of equal value when considering the
severity of a fault. In particular, this makes time-domain features
easier to use.
It must be pointed out, though, that in some cases vibration
components in the frequency range from 10 Hz to 1000 Hz can
increase noticeably without it showing clearly in the VRMS. An
example would be an oil whirl in a sleeve bearing, which is something
that the author has experience of[26]. Also, even though ball pass
frequencies of rolling bearings[27,28] are within the frequency range
from 10 Hz to 1000 Hz, VRMS measurements still do not always
reveal bearing faults early enough. ere has been experience of
this in the Finnish paper industry since the 1980s.
VDI 2056 (1964) is based on extensive real-world experiments
and theoretical discussions. e basic work was performed by
Rathbone (1939), Yates (1949) and Federn (1958), among others.
VDI 2056 has been the foundation for most standards, such as the
ISO 2372, BS 4675, ISO 3945[22,25,29] and PSK standards[30], to name
just a few.
In[12], equally high vibration levels are considered to be
of the same severity grade when measuring the peak value
of velocity in the frequency range from 30 Hz to 1000 Hz.
VIBRATION CM
Other vibration-velocity-based severity charts are found in[2-4,21,31-33].
Examples of applying vibration velocity to solve practical problems
are found, for example, in[18,33,34].
1.3 Acceleration
Even though there is little displacement and low velocity, there
can be high accelerations at higher frequencies[35,36], which cause
high inertia forces that can damage machines. For example, if
a sine waveform vibration has a frequency of 1591.5 Hz and the
displacement amplitude X0 is only 1 μm, the acceleration amplitude
X2 is 100 m/s2. Generally, the use of acceleration is recommended
when the vibration frequency is over 1 kHz[35,37].
It was already recognised in some papers in the 1960s that
acceleration measurement was suitable for examining gears and
rolling bearings[35]. Even though accelerometers have mainly
been used for research since the 1960s, they were already being
manufactured in the 1940s[20,38].
Acceleration measurements became common in the 1970s. At
that point, their analysis was performed mostly in the frequency
domain[39-42]. In the 1980s, analysers and data loggers became more
common. eir fast analogue-to-digital (A/D) converters enabled
an easier and more exact analysis of acceleration signals in the time
domain[43].
Many methods have been developed for detecting impact-
causing faults at an early stage, utilising the resonance vibrations
of bearings or other machine parts[44-46]. Examples of these are the
shock pulse method (SPM)[18,47-49], envelope analysis [14,18,50-52], spike
energy[53,54] and PeakVue analysis[54,55]. Traditionally, these methods
use acceleration signals.
ere is much more information available on allowable
displacement and velocity levels than on allowable acceleration
levels. In[11], there is a severity chart for acceleration covering the
frequency range from 300 Hz to 10 kHz. According to the chart,
equally large peak values of acceleration in the frequency range
from 3 kHz to 10 kHz are of equal weight when considering the
severity of a fault. Based on their running, machines are divided
into nine categories ranging from extremely smooth to very rough.
In[12], equally high peak values of acceleration are considered to
be of the same severity grade in the frequency range from 1 kHz
to 4 kHz. Above the 4 kHz limit, the allowable peak values for
acceleration are considerably lower. Based on this severity chart, the
running of a machine can be determined to be normal, problematic,
abnormal or dangerous.
e ISO 10816-6:1995(E) standard[56] includes vibration severity
grades for piston machines of over 100 kW, in the frequency range
from 2 Hz to 1000 Hz, using displacement, velocity and acceleration
RMS values.
2. Background on the use of jerk,
snap, crackle and pop
e rst time derivative of acceleration, ie the third time derivative
of displacement, is known as jerk. A common application for it
has been research into the ride comfort of lis and vehicles[57-59]. A
publication from 1960[57] discusses the ideal acceleration changes
for a li cabin. Allowable jerk levels, when accelerating and
decelerating a li, are examined in[58,59].
e second time derivative of acceleration, ie the fourth time
derivative of displacement, is known as snap. A study of amusement
park roller coasters from 2016[60] mentions that ‘the human
tolerance to jerk and snap is not well understood’.
Kirchho (1824-1887)[61] states in his book from 1876,
Vorlesungen über mathematische Physik: Mechanik, regarding the
time derivatives of acceleration: ‘[…] so könnten auch die dritten
und höheren eingeführt werden’. He thought that the third and even
higher time derivatives of displacement could be used but did not
consider them necessary for describing movement[61,62].
Routh (1831-1907)[63] expresses ideas similar to those of
Kirchho in his book from 1898, A Treatise on Dynamics of a
Particle. In it he states: ‘If the third dierential coecient were
required, we should use some such name as the hyper-acceleration,
but this extension is not necessary to dynamics’.
It should be pointed out, however, that jerk and snap are of use
even in the dynamics. e author has had good results in active
vibration control, when forces proportional to jerk and snap have
been created with a force generator. In a study from 1995[26], the
RMS value of velocity was reduced from 37.2 mm/s to 0.3 mm/s
in several repeated tests when using control forces proportional
to jerk. e best result in this two-mass vibration system was
0.2 mm/s.
On the other hand, Feynman (1918-1988) et al[64] write in their
book about the self-force on an accelerating electron, stating that
a force is also dependent on jerk and even higher time derivatives.
Rathbone[65], who was a pioneer of vibration measurements in
condition monitoring, stated the following about jerk in 1965: ‘is
term has been applied to the rate of change of acceleration, or the
third derivative of the displacement. Its measurement would be
of little interest in connection with machines that rotate only; it is
more useful with apparatus of the sudden jar type, such as drop-
hammers, etc.’
In[17], there is an extensive summary of both real-world
measurements in industry and laboratory experiments, covering
rotational frequencies from 0.0254 Hz to 525 Hz. ey clearly show
that jerk and snap oer a great benet in condition monitoring
of rotating machinery across a very wide scale of rotational
frequencies.
In 1982, Smith[66] published a study in which jerk is used to
examine faults in slowly rotating rolling bearings.
In the author’s publication from 1992[62], snap is used for
detecting the unstable running of an electric motor. e snap
signal was obtained from an acceleration signal using numerical
dierentiation twice. e publication also presents indices that
describe the condition of a machine using even higher time
derivatives than snap.
In 1997, an article[67] followed in which the unstable running of
an electric motor was examined using snap, crackle and pop. e
snap signal was obtained from an acceleration signal by analogue
dierentiation using a Mitsol D-94 vibration meter. Crackle and
pop, ie the h and sixth time derivatives of displacement, were
obtained from the snap signal using numerical dierentiation.
Unstable running, which was also detected with a stroboscope, was
most evident in the pop signal.
3. Practical measurements using jerk
and snap
First, three cases are examined in which the peak values of jerk
and snap were used in condition monitoring of large-scale
machinery. e measurements in Tables 1 and 2 were performed by
dierentiating acceleration signals once or twice with a Mitsol D-94
vibration meter, which also contains a detector for peak values. Its
frequency range is from 2 Hz to 2000 Hz. For comparison purposes,
458 Insight • Vol 63 • No 8 • August 2021
Insight • Vol 63 • No 8 • August 2021 459
VIBRATION CM
the RMS values of velocity were measured in the frequency range
from 10 Hz to 1000 Hz, according to the ISO 2372 standard. e
values in Table 3 were obtained using numerical dierentiation and
integration.
3.1 Case 1: Lime kiln
A lime kiln is an essential machine for the manufacturing process in
a pulp mill. e lime kiln in question was 97.5 m long and had eight
supporting rolls. One rotation of the lime kiln took 42-45 s. e
diameter of supporting roll 2 was 1098.5 mm and its rotation time
was 12.2-13.1 s. For supporting roll 2, the peak values of jerk and
snap were distinctly dierent from the values of other supporting
rolls[67]. When the unevenness of the supporting roll was ground,
the jerk and snap peak values (Table 1) clearly decreased, but
there was no change in the VRMS values. is means that VRMS
measurements did not detect the unevenness of the supporting roll.
Table 1. Measurements on a lime kiln supporting roll: peak values
of jerk and snap and VRMS values, before and after grinding
Jerk peak
(km/s3)
Snap peak
(Mm/s4)
VRMS
(mm/s)
Before grinding 19.3 567 0.2
Aer grinding 4.0 33 0.2
e ratios 19.3/4.0 = 4.8 and 567/33 = 17.2 show that the
condition of the surface of a supporting rolls can be assessed
using the peak values of jerk and snap. Snap was a more sensitive
parameter than jerk. e results were similar when misalignment
was present in the lime kiln. is is just some of the research that the
author has undertaken on the lime kiln over a period of six years.
3.2 Case 2: Cavitation of a Kaplan water
turbine
A Kaplan water turbine with a rotational speed of 115 r/min is
examined in[69]. ere were three distinct cavitation points in the
whole power range of the turbine. ey were easily revealed with
the help of jerk and snap, but only one was revealed through VRMS
measurements according to the ISO 2372 standard.
Another Kaplan water turbine[68,70] had a rotational speed of
88.2 r/min. It had a severe cavitation problem in the power range
from 37 MW to 43.2 MW. When the old runner was replaced with
one of a new construction, the cavitation ceased, which is evident
from the results shown in Table 2[68,70]. A Mitsol D-94 vibration
meter was also used in this study.
Table 2. Cavitation measurements on a Kaplan water turbine’s
supporting bearing, before and after the runner was replaced
Jerk peak
(km/s3)
Snap peak
(Gm/s4)
VRMS
(mm/s)
Old runner 946 25.4 1.3
New runner 106 2.52 0.6
e ratios 946/106 = 8.9, 25.4/2.52 = 10.1 and 1.3/0.6 = 2.2
show that the cavitation was clearly detected using jerk and snap.
In[71], acceleration, jerk and snap measurements are compared for
the detection of cavitation over a variety of power and frequency
ranges. In these studies, the jerk and snap signals proved to be
better for detecting cavitation than the acceleration signals. Time-
domain signals related to the cavitation of a Kaplan water turbine
are examined in[71,74].
3.3 Case 3: Pulp washer
e pulp washer in question had a sha diameter of 725 mm.
One rotation of the pulp washer took 39.36 s. Table 3 shows the
measurement results from both a defective rolling bearing and a new
rolling bearing
[72,73]
. e related snap signals are shown in Figure 1.
e crest factor and the peak value responded well to the fault.
e RMS value, which was calculated from the whole length of the
signal, was not suitable for detecting a bearing fault. e velocity
signal was not suitable either, which was to be expected based on
the author’s past experience. e results improved when changing
from acceleration to jerk. When the velocity x(1), acceleration x(2),
jerk x(3) and snap x(4) signals were compared, snap gave the best
results.
3.4 Summary of the applications of jerk and
snap
Tables 4 and 5 contain references concerning the use of jerk and snap
signals for the detection of faults, both in real-world measurements
and in laboratory experiments.
Table 3. Features analysed from a very slowly rotating rolling bearing of a pulp washer, when the rotation time was 39.36 s[72,73]
Frequency range
10-2000 Hz
Defective bearing
10-2000 Hz
New bearing Ratio
10-3000 Hz
Defective bearing
10-3000 Hz
New bearing Ratio
Crest factor
x(1) 2.57 4.51 0.57 2.57 4.50 0.57
x(2) 21.28 4.71 4.52 34.60 4.73 7.32
x(3) 96.59 4.85 19.92 157.69 5.00 31.54
x(4) 121.42 4.40 27.60 188.26 4.61 40.84
RMS
x(1) 0.32 mm/s 0.25 mm/s 1.28 0.32 mm/s 0.25 mm/s 1.28
x(2) 0.11 m/s20.21 m/s20.52 0.11 m/s20.21 m/s20.52
x(3) 0.19 km/s30.51 km/s30.37 0.27 km/s30.52 km/s30.52
x(4) 1.51 Mm/s42.90 Mm/s40.52 3.65 Mm/s43.28 Mm/s41.11
Peak value
x(1) 0.81 mm/s 1.14 mm/s 0.71 0.81 mm/s 1.14 mm/s 0.71
x(2) 2.32 m/s20.99 m/s22.34 3.80 m/s20.99 m/s23.84
x(3) 18.34 km/s32.47 km/s37.43 43.22 km/s32.61 km/s316.56
x(4) 183.4 Mm/s412.78 Mm/s414.35 687.3 Mm/s415.13 Mm/s445.43
VIBRATION CM
Table 5. Jerk and snap measurements in a laboratory
Unstable running of an
electric motor [26,62,67]
Bearing faults [26,66-69,72,
84-90,93,94,98]
Bent sha [88]
Misalignment [88,91,92]
Cavitation [95-97]
Poor lubrication [91]
A laboratory experiment is examined in which automatic
diagnostics were used with acceleration and jerk signals. e
outer race of a rolling bearing contained the faults shown in
Table 6[69,87,94]. e rotational frequency of the sha varied from 5 Hz
to 40 Hz. An algorithm automatically selected the best feature subset
for a given number of features. Available features[94 p 132] included
the most common time-domain features such as RMS value, crest
factor and kurtosis, as well as more advanced features. When ve
dierent features were extracted from the acceleration signals, the
classication was incorrect in 11% of the cases. However, when the
jerk signals were used, the misclassication rate was only 1%.
Table 6. Bearing fault classification in a laboratory experiment
using automatic diagnostics with acceleration and jerk
signals
Defect width (mm) Condition
Class 1 0.00 Good
Class 2 0.25 Small defect
Class 3 0.50 Medium-sized defect
Class 4 1.00 Clear defect
Crackle and pop have been useful in condition
monitoring[17,67,97,99]. eir eects on the human body are discussed
in[60,100].
When designing a piston engine, it is important to get the
camsha working smoothly. To guarantee this, higher time
derivatives than acceleration have to be taken into account[60,101].
e jerk and snap signals can also be used for the stress evaluation
of machinery[102-106]. ere are several studies that deal with jerk and
snap in chaotic systems[107-110]. In these studies, electrical circuits
have also been used to create jerk and snap signals[111]. Finally,
jerk and snap can be used for vibration damping[26,112-115] and in
cosmology[116,117].
4. Conclusions
Faults causing low-frequency vibrations
can be detected eectively by means
of displacement and velocity signals.
An example of this kind of fault is unbalance.
With faults causing impacts, as in the
case with defective rolling bearings and
gears, better results are achieved with an
acceleration signal. Extensive real-world
measurements and laboratory experiments
show that replacing acceleration with jerk
and snap improves results even further. In
several cases, snap has been shown to be
better than jerk. Crackle and pop signals can also be used to detect
impact-causing faults.
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Figure 1. Snap signals from a pulp washer’s rolling bearings:
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460 Insight • Vol 63 • No 8 • August 2021
Insight • Vol 63 • No 8 • August 2021 461
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