CRITICAL SPEED AND NATURAL FREQUENCY ANALYSIS OF CRACKED ROTOR SHAFT

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 06, June 2019, pp. 232–240, Article ID: IJMET_10_06_018
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=1
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
CRITICAL SPEED AND NATURAL FREQUENCY
ANALYSIS OF CRACKED ROTOR SHAFT
Abhijeet H. Kekan
Department of Mechanical engineering, K. L. deemed to be university, Green Fields,
Vaddeswaram, Guntur District, and Andhra Pradesh, India
B. Raghu Kumar
Department of Mechanical engineering, P.V.P. Siddhartha Institute of Technology Kanuru,
Vijayawada, Andhra Pradesh, India
ABSTRACT
Excessive Vibrations in machine shaft cause catastrophic failure of machine parts.
Vibration in the machine part increases due to crack generation and propagation in
machine shaft. Dynamic properties of rotor shaft such as natural frequency of
vibration and critical speed get altered due to change in crack depth and crack
location. In this paper, an experimental approach was developed to observe the effect
of variation in a crack depth and location. In the present experiment single crack was
generated in a rotary shaft at 4 different locations (100 mm, 200 mm, 300 mm, and
400 mm) from bearing support. Further, three different crack depths (17%, 33% and
50% of shaft diameter) was provided at each crack location. Critical speed and
natural frequency of rotary shaft were obtained experimentally using Fast Fourier
Transform. Critical speed varies in the range of 5% to 8.5% due to change in crack
depth and crack location. Variation of natural frequency was observed 6% to 12% due
to change in crack depth and crack location. Frequency vs. amplitude analysis of
rotary shaft carried out using Fast Fourier Transform analyzer.
Keywords: Crack location, Crack depth, Critical speed, Fast Fourier Transform,
Natural frequency.
Cite this Article: Abhijeet H. Kekan and B. Raghu Kumar, Critical Speed and Natural
Frequency Analysis of Cracked Rotor Shaft, International Journal of Mechanical
Engineering and Technology, 10(06), 2019, pp. 232–240.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=6

Critical Speed and Natural Frequency Analysis of Cracked Rotor Shaft
1. INTRODUCTION
Vibrations in the main shaft of machines are undesirable. This vibration causes failure of
other machine parts such as bearings and gearbox. This leads to unplanned maintenance and
loss of production. To avoid these consequences monitoring of dynamic properties of rotor
shaft should be done frequently. Anuj Kumar Jain et al. [1] observed stiffness with respect to
crack depth and found that there is variation in stiffness of shaft as crack depth increases. M.J.
Gómez, C. Castejón and J.C. García-Prada [2] made a comparison of various signal
processing technique. Artificial Neural Network used for classification of signals. M.J.
Gómez, C. Castejón and J.C. García-Prada [3] used Wavelet Packet Transform energy
difference to identify crack depth. It was found that crack with depth of more than 12% of
Shaft diameter can be detected reliably using Wavelet Packet Transform energy difference
technique. Mohamed. R. et al. [4] generated a crack at depth 0%, 40%, 50%, 60% of shaft
diameter and amplitude Vs. Frequency graph was drawn. It was found that amplitude of
vibration changes with respect to a change of depth of the crack. Crack in shaft identified with
help of change of frequency respect to change in crack depth. V. Sudheer Kumara et al. [5]
performed experiments on a shaft with different crack depth at a different location.
Displacement versus frequency graph drawn using Fast Fourier Transform (FFT) analyser.
Increase in displacement curve observed as crack depth increases. K. Vigneshwaran and R.K.
Behera [6] developed Finite element model for a beam with breathing crack. It was observed
that the natural frequency of vibration decreases as crack depth increases. Alok Ranjan Biswal
et al. [7] developed Finite element model for a cracked tapered beam. Reduction in natural
frequency of vibration was observed as crack depth increases. Sandeep Das et al. [8]
developed adaptive Neuro-Fuzzy interface system (ANFIS) to access crack depth and
location.M.J. Gómez, et al. [9] carried out crack depth and location identification using
wavelet packets transform energy. T.Sunil Kumar et al. [10] made a crack of varying depth at
a different location on the cantilever beam. Reduction in natural frequency of vibration was
observed as crack depth increases. It was also observed that there was a reduction of natural
frequency as crack location distance increases from the fixed end. K.M. Saridakis et al. [11]
proposed an analytical model for cracks in a shaft and used Artificial Neural Network and
fuzzy logic to determine the crack location and crack depth. Sachin K. Singh and Rajiv Tiwari
[12] developed multi crack detection and localization algorithm. It was found that the
algorithm successfully detects a crack in most of the cases. José Fernández-Sáez et al. [13]
used the first two natural frequency of vibration of a shaft for identification of single crack in
the shaft. Hamid Khorrami et al. [14] observed the effect of crack location, crack depth and
crack orientation variation on vibration properties as critical speed and harmonic response.
Chaozhong Guo et al. [15] carried out crack detection in rotor using Empirical mode
decomposition technique and compared results with FFT results. It was observed that the
Empirical mode decomposition technique can be practically used for crack detection.
Debabrata Gayen, et al. [16] carried out finite element analysis of cracked shaft. It was
observed that natural whirl frequencies and critical speeds of shaft decrease as crack depth
increases. D.Koteswara Rao and Tarapada Roy [17] developed a mathematical model of
functionally graded rotary shaft. Gayen D, Chakraborty D. and Tiwari R. carried [18] out
finite element analysis of cracked functionally graded shaft. Ghanbari Mardasi et al. [19] used
wavelet transform to localize small depth crack. A. P. Stawiarski [20] proposed algorithm for
crack detection and localization using wavelet propagation. Tejas Aher et al. [21] carried out
FEA analysis of cracked beam and found that natural frequency of transverse beam can be
used to detect the crack. D.I.Sampio and R.Nicoletti [22] developed approximated entropy
algorithm to detect the crack. A.P.Bovsunovsky [23] found that efficiency of vibration based
crack detection depends on the compliance of structure. A. P. Stawiarski [24] found the
effectiveness of vibration based crack detection depends on a comparison of relative change

Abhijeet H. Kekan and B. Raghu Kumar
of shaft compliance. J. Xiang et al. [25] developed a method for crack detection using wavelet
transform and genetic algorithm. Victor Girondin et al. [26] developed a mathematical model
for the healthy and cracked shaft. X Li et al. [27] developed a maintenance policy for gear
shaft assembly and modelled deterioration of shaft and gear. Severity of crack can be
identified from the natural frequency and critical speed of rotary shaft. In the present study an
experimental approach developed to avoid catastrophic failure of machine parts. To avoid
intense vibration critical speed range should be avoided. Rotor shaft vibration frequency
should be other than natural frequency of shaft to avoid resonance. Therefore by observing
natural frequency and critical speed of rotary shaft crack location and crack depth can be
predicted. This will avoid the catastrophic failure of the rotary shaft system.
2. EXPERIMENTAL SET UP
Figure 1 shows an experimental setup to measure critical speed and Natural frequency of
vibration. A shaft was supported between two bearings. To the overhanging portion of shaft
crack was generated at a different location. A shaft was rotated with the electric motor and
with help of FFT readings were noted.
Figure 1 Experimental setup for FFT readings.
2.1. Shaft
Set up consist of 12mm diameter and 880 mm long shaft held between two roller element
bearings. Shafts made up of steel 304 materials.
2.2. Crack locations and crack depth
Figure 2 shows shaft was supported between two roller element bearings. A single Hacksaw
cut was made to free end of the shaft at 4 different locations i.e. 100, 200, 300 and 400mm
distance from bearing 2 supports.

Figure 2 Crack Locations distance from bearing support
Table 1 show four crack locations given 3 different crack depths 17%, 33% and 50% of
shaft diameter. Figure 3 shows crack generated on a free end of the shaft.
Table 1 Different crack location and depth
Crack location from bearing 2 (mm) Crack depth % of shaft diameter
100 17 33 50
200 17 33 50
300 17 33 50
400 17 33 50
Figure 3 Crack on shaft
2.3. Electric Motor
Shaft rotated with 3 phase electric motor. Pulley mounted on the shaft. Electric motor and
pulley were connected with help of v belt. An electric motor is 1/3 HP and 2800 RPM speed
limit.
2.4. Variable voltage drive
Variable voltage drive apparatus vary rotary speed of shaft by varying applied voltage to the
electric motor. Variable voltage drive operates from 3-phase power at 208 V to240 V. The
speed of the electric motor can be adjusted by variable voltage drive.
2.5. Fast Fourier Transform (FFT) Analyser
Vibrations of the shaft can be measured with help of FFT analyzer. The accelerometer of FFT
mounted on bearing mounting to measure vibrations of the shaft. Accelerometer senses the

vibration on bearing mounting and converts it to the electric signal which was given as input
to the FFT device. FFT convert time domain signal to a frequency domain and draw
frequency vs. amplitude graph.
3. EXPERIMENTAL PROCEDURE
The main objective of conducting an experiment to get critical speed and natural frequency
values of the rotary shaft. Each type shaft supported between two roller elements bearings and
rotated with help of the electric motor. A speed of electric motor altered by the variable
voltage drive. Speed of motor gradually increased from zero rpm to motors maximum speed
capacity 2800 rpm.
The first shaft with crack location 100 mm and crack depth 17% of shaft diameter
supported between two bearings and rotated with help of the electric motor. As the speed of
motor reaches near to critical speed rotating shaft, shaft start to vibrate. At Critical speed
natural frequency of vibration of shaft and frequency of forced vibration becomes equal. At
critical speed rotary shaft vibrate with maximum amplitude this phenomenon is called as
resonance. At resonance speed of shaft was measured with help of digital tachometer. The
frequency of vibration of shaft measured with accelerometer of FFT analyzer.
After FFT reading and tachometer readings motor was stopped. With help of hacksaw
crack depth increased from 17% to 33% of shaft diameter. Crack depth increased in a loaded
condition to avoid misalignment of the shaft. Above procedure was repeated to draw
frequency Vs amplitude graph and to take tachometer reading. Again crack depth increased
from 33% to 50% of shaft diameter. FFT reading and tachometer readings were noted. After
completion of 3 crack depth readings for a shaft of crack location 100mm shaft was removed
from set up and shaft with crack location 200mm mounted to set up. Again by varying crack
depth from 17% to 33% and from 33% to 50% of shaft diameter FFT reading and tachometer
readings were noted. Same procedure repeated for a shaft with crack location 300mm and
400mm. Graph of frequency vs. Amplitude was drawn with help of FFT analyzer for each
crack depth.
4. RESULTS AND DISCUSSION
Section 4.1 describes the frequency amplitude graph and its variation with respect to crack
depth. Section 4.2 addresses the variation of critical speed with respect to variation in crack
depth and crack location. Section 4.3 addresses the variation in natural frequency of shaft with
respect to variation in crack depth and crack location.
4.1. Frequency versus amplitude graph
As shown in figures 4 (a,b,c,d) frequency Vs amplitude graph for healthy shaft and shaft with
crack depth 17%, 33% and 50% of shaft diameter are drawn. The horizontal axis of a graph
represents a frequency of vibration. The vertical axis of a graph represents the amplitude of
vibration. This graph was plotted using data acquired from FFT analyzer. Conversion of time
domain graph to frequency domain graph carried out by using Fast Fourier Transform (FFT).
FFT convert time domain signal to frequency domain signal. After studying figures 4
(a,b,c,d), Increase in amplitude of vibration was observed as crack depth increases. It was
observed that as crack depth increases the frequency of vibration decreases.

(a) Frequency versus amplitude graph for shaft
with no crack.
(b) Frequency versus amplitude graph for crack
depth 17% of shaft diameter.
(c) Frequency versus amplitude graph for crack
(d) Frequency versus amplitude graph for crack
Figure 4 Frequency versus amplitude graphs
4.2. Critical Speed of shaft
Graph of critical speed at the various crack location for various crack depth was drawn. The
horizontal axis represents crack location at various distances from bearing support. The
vertical axis represents a critical speed of the shaft. Figure 5 shows the effect of variation of
crack depth on a critical speed of the shaft. It was observed that as crack depth increases the
critical speed of shaft decreases. It was also observed that as the location of crack away from
bearing support critical speed decreases.
Amplitudeofvibration(m/S2)
Frequency of vibration(Hz)

Figure 5 Critical speed of shaft
4.3. Frequency of vibration
Graph of a frequency of vibration at the various crack location for various crack depth was
drawn. The horizontal axis represents crack location at various distances from bearing
support. The vertical axis represents a frequency of vibration of the shaft. Figure 6 shows an
effect of variation of crack depth on the frequency of vibration of a shaft. Reduction in
frequency of vibration was observed as crack depth increases. It was also observed that as the
location of crack away from bearing support frequency of vibration decreases.
Figure 6 Frequency of vibration of shaft

5. CONCLUSIONS
An experimental set up created to observe dynamic behaviour of a shaft by varying crack
depth and crack location. In this experiment critical speed and natural frequency of vibration
of shaft obtained for different crack depth and crack location. Critical speed and natural
frequency of vibration of the shaft were obtained experimentally. Frequency vs. amplitude
graph obtained using FFT.
Critical speed varies in the range of 5% to 7% due to an increase in crack depth from 17%
to 50% of shaft diameter. Variation of natural frequency was observed 6% to 8% due to an
increase in crack depth from 17% to 50% of shaft diameter. Critical speed varies in the range
of 6.5% to 8.5% for change in crack location distance from 100 mm to 400 mm. Variation of
natural frequency was observed 10% to 12% for change in crack location distance from 100
mm to 400 mm.
It was observed that as a depth of crack increases a critical speed of shaft and frequency of
vibration of the shaft decreases. As the crack distance from bearing support increases a critical
speed of shaft and frequency of vibration decreases. It was observed from the Frequency
versus amplitude graph that as crack depth increases amplitude of vibration increases. It was
also observed from Frequency versus amplitude graph that as crack depth increases the
frequency of vibration of shaft decreases.
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