

<4D6963726F736F667420576F7264202D20DFD1C7D120C8C7E5D120E6DEC7D3E320E6DAE3C7CF32392D203432>


This is an open access article under the CC BY license : 

 
Al-Khwarizmi 

Engineering 

Journal 
 

Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, June, (2022) 

P. P. 29- 42 

 
Comparative study of vibration analysis in rotary shafts between 

Rayleigh's and Dunkerley's methods 
 

Karrar Baher *                        Qasim A. Atiyah **  

Imad A. Abdulsahib*** 
*, **, *** Mechanical Engineering Department / University of Technology / Baghdad / Iraq 

*Email: me.19.20@grad.uotechnology.edu.iq 
**Email: 20044@uotechnology.edu.iq 

***Email: 20018@uotechnology.edu.iq 

 
(Receive 2 Marh 2022; Accepted 18 May 2022) 

https://doi.org/10.22153/kej.2022.05.001 

 
Abstract 
 

The importance of vibrations in rotating rotors in engineering applications has been examined, as has the best 

approach to interpreting vibration data. The most extensively used analytical approaches for rotating shaft vibration 

analysis have been investigated. In this research, a detailed study was made of the Rayleigh and Dunkerley methods 

due to their importance in the special calculations to find the amplitude of vibrations in the rotation system. The 

multi-node method was used to calculate both Dunkerley's and Rayleigh's methods. An experimental platform was 

built to study the vibrations that occur in the rotating shafts, and the results were compared with theoretical 

calculations and with different distances of the bearings. It proved that there is very little error between the 

experimental and theoretical results. The vibration signal from the sensors was analyzed using the LABVIEW 

program. Rayleigh's method was compared to the exact method, and it was considered the most accurate method. It 

was found that it made very little difference, up to about 0.06%. As for the Dunkerley method, the difference between 

it and the proper method is about 4%, which is acceptable. Then a comparison was made between Rayleigh's and 

Dunkerley's methods, and it was found that Dunkerley's method is the most appropriate in the calculations. 

 
Keywords: Vibrations, Rotating-Bearing System, Rayleigh, Dunkerley, LABVIEW. 

 
1. Introduction 
 

Rotordynamics is the study of the dynamics 

of rotating machines. Rotordynamics varies from 

structural vibration research because of 

gyroscopic moments, cross-coupled forces, and 

the possibility of whirling instability [1]. Many 

industrial applications, including onboard space 

vehicles, revolving machinery in electrical power 

plants, and power transmission gear trains, utilize 

rigid rotor systems aided by linear or nonlinear 

elastic bearings. In rotating systems vibration can 

cause inefficiency, malfunction, and even 

catastrophic failure. As a result, modeling and 

understanding their complex behavior has 

become a prominent study topic [2-4]. 

A rotor-bearing mechanism can show 

undesirable subcritical super-harmonic 

resonances when the rotor's spinning speed is a 

part of its natural frequency [5]. Vibration is a 

natural occurrence in rotating machinery, but it 

has the potential to reduce productivity [6, 7]. 

Thus, while investigating the work of 

machines in general (and rotating machines in 

particular), vibrations are a serious concern for 

designers, engineers, and researchers. As a 

consequence, the focus will be on studying 

vibrations, determining the most appropriate 

method of analysis, and   

determining the values of critical frequencies 

arising from an imbalance in machines. 

Mass imbalance is the most common cause of 

harmonic excitation in rotating machinery. An 


Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
30 

imbalance can arise during the assembling of 

machine components or during the 

manufacturing process of machine components. 

Even if a rotor is adequately balanced when it 

first starts up, its stability will diminish over 

time. Another vibration response, such as 

vibration from a nearby unbalanced rotating 

machine, could be excited at the base of a 

spinning machine with its own mass imbalance 

oscillation [2]. One of the most common causes 

of machine vibration is the inertia of the 

machine's moving elements. In a reciprocating 

motion, several components move back and 

forth. Newton's laws require that a force is 
applied to accelerate the mass, as well as a 

response from the force to the machine's 

structure. Periodic deflections are perceived as 

vibrations because the forces are generally 

periodic [1]. 

Tiwari [8] proposed a well-conditioned 

recognition technique for simultaneous 

calculation of residual imbalances, bearing 

stiffness, and damping coefficients based on the 

rotor's clockwise and counter-clockwise 

reactions. Reddy and Srinivas [9] investigated 

the dynamic analysis of a rotor with base 

excitation. With time histories, phase diagrams, 

and frequency responses, the effect of base 

excitation frequency and amplitude on rotor 

dynamics is demonstrated. Wang et al. [10] 

focused on the imbalance reaction, and they 

proposed an algorithm for detecting residual 

imbalances in the rotor and bearings at the same 

time. The rotor was represented as a homogenous 

and continuous Rayleigh beam. Yang and 

colleagues [11] used a new sort of TVRBSE 

based on the formulation of absolute node 

coordinates and Rayleigh beam theory under an 

arbitrary Lagrange-Euler description to develop 

a dynamic model of a moving and axially rotating 

Rayleigh beam. Zhu and J. Chung [12] used the 

proposed dynamical model to investigate the 

vibration and stability of a rotating Rayleigh 

beam with axial motion. To completely consider 

the terms of rotating inertia, they used the 

Rayleigh ray theory. Farshbaf Zinati, R., et al. 

[13] analyzed the stabilization and nonlinear 

vibration of a simply supported axially moving 

Rayleigh viscoelastic beam fitted with 

intermediate nonlinear support. Aouadi, M. A., & 

Lakrad, F. [14] discussed the three-dimensional 
bending linear free vibrations of rotating 

Rayleigh beams. The destabilization of free 

vibrations was found to be dependent on the 

linearization method that was used. Faraji 

Muhairi, M, et al. [15] studied the effect of 

angular velocity on the balance and vibration of 

a simply supported Rayleigh shaft. The 

distinctions between the Rayleigh and Euler-

Bernoulli models are presented. The influence of 

the slenderness ratio on the instability threshold 

and natural frequencies is seen. Zhu and Chung 

[16] addressed the beam's dynamic behaviors and 

properties, as well as a novel rotating beam 

model that is currently being implemented. The 

spinning beam's dynamic behavior and vibration 

frequency were compared in action. While 

examining the spinning beam, the Rayleigh beam 

model was found to be more accurate than the 

Euler-Bernoulli beam model.  
Tamrakar and Mittal [17] used an impact 

hammer test to determine the system's 

fundamental frequencies, then determined the 

speed at which whirling occurs in the system. 

They used Dunkerley's natural frequency 

approach to verify the experimental results. Levy 

[18] has created an iterative technology based on 

Dunkerley's method for delivering natural 

vibration frequencies to discrete systems at the 

same time. Low [19] validated a Dunkerley 

expression referring to a uniform beam holding 

several masses. When compared to the result 

associated with the original property equation, it 

is discovered that Dunkerley's expression can 

yield a good approximation in general. Due to the 

impossibility of conserving computational time, 

Low strongly advised the Dunkerley approach 

for beams transporting more than two masses at 

separate locations.  

Rayleigh and Dunkerley's approaches for 

analyzing vibrations have been utilized in prior 

studies, but without specifying which is better in 

the study or expressing a clear comparison 

between them. For this reason, it is necessary to 

clarify the two ways and choose which is the 

most appropriate in mathematical calculations 

for studying spinning machines and determining 

their frequency values. 

The amplitude of vibrations in rotational 

systems will be determined using the Rayleigh 

and Dunkerley methods in this paper. It will also 

rely on the multi-node approach to calculate each 

of the Dunkerley and Rayleigh methods, 

compare them to one another, then compare both 

ways to the precise method to determine which is 

more accurate and recommended to utilize. 
 

2. Mathematical Analysis 
 

The stator, which supports the bearing, is 

frequently assumed to be stiff when modeling a 


Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
31 

rotor device. In real-world applications, 

however, the rotor is frequently included in a 

more flexible structure, such as the framework of 

an aero-engine, which adds more compliance to 

the system and theoretically affects the influence 

of bearing nonlinearities [3]. 
 

2.1 Rayleigh’s Method 

 
The Rayleigh method for estimating the 

system's natural frequencies will be presented in 

this study. The system will be continuous, and 

you will be able to utilize this method to estimate 

the fundamental natural frequency of continuous 

systems [20]. It is critical to conduct a modal 

analysis of the rotors to avoid resonance during 

operation. A dynamic study of the rotating-shaft 

system under operating conditions is also 
necessary to assess the dynamic properties of the 

rotating system. By seeing the shaft as a spinning 

beam model, the system was mathematically 

modeled [21, 22]. 

 
Fig. 1. Simply Supported Beam with finite nodes 

 
The uniform beam for the number of nodes 

is shown in figure 1, and the masses for the 

nodes are computed using the following 

equations [23]: ��� � ��/2 � �	��
��2                           … 1�  
�� � �	� 
�2 � �	���
���2                         … 2�  
where � � 1, 2, 3, . . . , � ��� � �	��
��2                                             … 3�  

The moment of inertia for the uniform solid 

beam is: �� � �64 ∗ �                                                 … 4�  
     It is possible to represent both the 

stiffness and the flexibility of a system's 

elastic behavior. The equations of motion for 

normal mode vibration in terms of stiffness K 

[23]:  !"#$%& � $'&�()* � 0                         … 5�  
     In the stiffness conception, the force is 

expressed as a displacement: (-* � $'&()*                                               … 6� 

     Stiffness is the polar opposite of flexibility. 

The displacement is given in units of force in 

this case: ()* � $'&.�(-*                                          … 7�  ()* � $0&(-*                                          … 8�  
 

The 0�,2 coefficients of the flexibility matrix 

are 

3)�)#⋮)5
6 � 70�� 0�# ⋯ 0�50#� 0## ⋯ 0#5⋮ ⋮05� 05# ⋯ 055

9 3:�:#⋮:5
6   … 9�  

 
The flexibility influence coefficient 0�,2� 

is the bending at i due to a unit load exerted at 

j with all other forces equal to zero. 

Deflections associated with :� � 1 and :# �:< � 0  are represented in the preceding 

matrix's first column. In the second column, 

you'll find deflections for :# � 1 and :� �:< � 0, and so on [23]. 

     By multiplying Eq. 5 by $'&.� � $0& , it 
is simple to determine the equation of motion 

in terms of flexibility:  !"#$0&$%& � $�&�()* � 0                  … 10�  
Where,  ()* is deflection vector matrix  $�& is mass matrix $0& influence coefficient matrix equal $=&.� $=& is stiffness matrix $�& is unit matrix, '.� ∗ ' 

 >! 1"# $�& � $0&$�&>                                 … 11� 

     When a lumped-mass system has a 

diagonal mass matrix, Eq. 11 becomes. 

 
Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
32 

?! 1"# 71 0 ⋯ 00 1 ⋯ 0⋮ ⋮0 0 ⋯ 19 � 70�� 0�# ⋯ 0�50#� 0## ⋯ 0#5⋮ ⋮05� 05# ⋯ 055
9 7�� 0 ⋯ 00 �# ⋯ 0⋮ ⋮0 0 ⋯ �5

9? � 0                       … 12�  
 

Fig. 2. Load exerted between the two bearings on a simply supported beam. 

 
Figure 2 shows a simply supported beam 

with an applied load, from which the 

deflection equations for beams can be derived. 

The deflection will be as shown in Figure 2: 02� � @bcL# ! b# ! C#� 6EIL⁄            … 13� 0�� � @a#I# 3EIL⁄                   … 14� 0J� � @adL# ! a# ! d#� 6EIL⁄            … 15� 

 
  Now the fundamental natural frequency of 

the beam is calculated based on equations 9 

and 13 to 15. )� � -� ∗ 0�� � -# ∗ 0�#� -< ∗ 0�<�. . . �-5 ∗ 0�5 )# � -� ∗ 0#� � -# ∗ 0##� -< ∗ 0#<�. . . �-5 ∗ 0#5 )5 � -� ∗ 05� � -# ∗ 05#� -< ∗ 05<�. . . �-5 ∗ 055  

                                                          …16� 

Where        - � � ∗ L. 

The fundamental natural frequencies are 

obtained using equation 16 for a deflection 

given to several nodes in the simply supported 

beam. 

Rayleigh's methods could be used to 

estimate the fundamental frequency of a beam 
or shaft defined by a series of lumped masses. 

The resulting constant deviation curve will be 

taken into account for weights %�L, %#L, %<L, ... which correspond to )�, )#, )<, ... deviations. The stored energy in the 

beam due to strain is [21] MNOP � 12 L%�)� � %#)# � %<)< � ⋯ �                                                                    … 17� 

'. Q � 12 "#%�)�# � %#)## � %<)<# � ⋯ �                                                                           … 18� MNOP � '. Q                                          … 19� "# � L∑%�)�∑%�)�#                                        … 20� 

 
2.2 Dunkerley’s Method 
 

Dunkerley's method uses the natural 

frequencies of the constituent elements to 

derive the fundamental frequency of 

composite construction. It is acquired by 

leveraging the fact that the higher natural 

frequency bands of practically all vibrating 

systems are large in comparison to their 
fundamental frequencies. The frequency 

equation can be used to determine 

eigenvalues for a large system of degrees of 

freedom [20]: 

The simplification procedures are done on 

equation 12, and then the formula is 

expanded to get the following formula: 

 1"�# � 1"## �. . . � 1"5#� 0����� 0##�#� . . . �055�5  

                                                         …21� 

Because the intermediate frequencies "#, "<, …, "5  are often significantly larger than 

the fundamental frequency "�, and therefore 
 

Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
33 

1"�# ≪ 1"�#             � � 2,3, . . . , � 1"�# ≈ 0���� � 0##�#� . . . �055�5                                                                         … 22� 
This equation is referred to as Dunkerley's 

formula. Eq. 22 produces a fundamental 

frequency that is never exactly the same as the 

exact number. In some cases, it would be easier 

to rewrite eq. 22 as: 1"�# ≈ 1"�5# � 1"#5# �. . . � 1"55#                   … 23� 

Where "�5 � 1 ∕ 0������/# � =�� ∕����/# is the natural frequency of a single-

degree-of-freedom system with mass ��  and a 

stiffness spring =�� , � = 1, 2, …, n. The lowest 

frequency of elasticity systems is calculated 

using Dunkerley's method. 

 
3. Experimental Work 

 
For this project, an experimental test 

platform was built to measure natural 

frequencies and take vibration readings. The 

main iron platform, which is a metal plate 

constructed of iron with dimensions of 120 cm 

x 40 cm x 5 mm that serves to carry the other 

parts was employed. A shaft is a spinning 

member that transmits rotational motion in the 

system. It is made of AISI 1045 steel and 

measures 80 cm in length and 20 mm in 

diameter. Two metal iron pieces, 40 cm long, 

10 cm wide, and 10 mm thick were also used. 

The major role of these components is to 

support the pillow block bearing. In addition to 

being placed on the linear ball bearing block to 
modify the distances between the bearings, 

these pieces have a longitudinal groove 

targeted at balancing devices and removing 

misalignments. There's also a linear sliding rail 

that holds the linear ball bearing block. Its job 

is to vary the distance between the bearings 

depending on where the measurements are to 

be made.  

The system also has a ball bearing, (model 

UCP 204) made of cast iron with an inner 

diameter of 20 mm, a length of 124 mm, a 

thickness of 31 mm, a center height of 33.3 

mm, and an overall height of 65 mm. A three-

phase electric motor with a capacity of 0.37 

kW and a maximum speed of 2825 rpm was 

also used. A coupling’s main functions are to 
connect the shaft to the motor and reduce 

misalignment. It's also composed of iron, with 

a 43-mm outside diameter and a 98-mm length. 

The system also includes a power source for 

varying the motor's rotational speed, an 

Arduino UNO, and accelerometers mounted 

on a ball bearing to monitor the intensity of 

shaft vibrations, with measurements taken 

solely for the vertical axis. The details are 

shown in figures 3 and 4. 

The vibration signal from the two 

accelerometers (mpu6050 and adxl335), with a 

variable rotational speed starting at 500 rpm 

and ending at 3000 rpm, was analyzed using 

the LABVIEW program. As indicated in figure 

5, the sensors (mpu6050 and adxl335) were 

also employed for the calibration process 

between them. The readings from the 

LABVIEW program are in Hertz, and the 

system's frequency readings are taken on the Z 

vertical axis. 

  
Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
34 

 
Fig. 4. The main components of the system. 

 
The main components of the system in this work as shown in Figure 4: 

1- Motor 

2- Coupling 

3- Ball bearing 

4- Rotating shaft 

5- Sensors 

6- Main base 

7- Linear Ball Bearing Block 

8- Linear Sliding Rail 

9- Iron pieces 

10- Arduino UNO 
 

Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
35 

 
Fig. 5. Calibration between MPU6050 and ADXL335. 

 
4. Results and Discussion 
 

The natural frequencies are derived using 

Dunkerley's approach from the critical 

frequency values that were retrieved using 

equation 22 and the results are compared with 

Rayleigh's uniform beam values, yielding the 

difference ratio between the two ways, as 

shown in Table 1. Given that Rayleigh's 

technique is more precise, it's worth noting that 

as the number of nodes employed for the 

uniform beam grows, the difference between  

the values reduces and the results converge. 

This suggests that the two procedures are in 


Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
36 

agreement and that the results are accurate. 

In practice, LabVIEW software will be used 

to find the frequency domain using mpu6050 

and adxl335 sensors, with the two sensors 

being used to calibrate the results and note 

their reliability. The investigation was carried 

out in six distinct bearing positions. 
The amplitude of rotating shaft vibrations at 

six distinct places along the vertical Z-axis was 

studied using the LabVIEW software in figures 

7 to 12, and these positions are depicted in the 

simple schematic figure of Figure 6. When the 

distance between the bearings is lowered, it has 

been noticed that the values of the natural 

frequencies begin to fall. At around 65 Hz, the 

natural frequency reaches its maximum value, 

which is the first critical frequency of a 

spinning shaft in the first position. In the sixth 

position, it reaches its lowest critical frequency 

of 7 Hz. The values of the natural frequencies 

usually decrease as the distance between the 

bearings decreases, indicating that when the 

shaft becomes a hanging shaft, the value of the 

stresses is lower which reduces the size of the 

whirling along the shaft. 
 
 
Table 1, 

The frequencies of the uniform beam are 

calculated by using Rayleigh-Dunkerley's 
approach for a range of different nodes: 

 
The analytical approaches were 

investigated and compared to the exact method 

[20], with the Rayleigh method showing a 

difference of 0.0585 percent when compared 

to the exact method, and with the Dunkerley 

method showing a difference of 3.8788 percent 

when compared to the exact method. 
 

Fig. 6. Sketch between the bearings. (a) first position, (b) second position, (c) third position, (d) forth 

position, (e) fifth position, (f) sixth position. 

 
No.  

Nodes 

Natural Frequency 
(Hz) Difference 

% 
Rayleigh Dunkerley 

5 - Nodes 62.52421 60.22316 3.68025 

7 - Nodes 62.77290 60.50608 3.61116 

9 - Nodes 62.56431 60.11276 3.91845 

12 - Nodes 62.56956 60.10748 3.93496 


Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
37 

 
Fig. 7. Natural frequency of first position. 

 
Fig. 8. Natural frequency of second position. 

 
Fig. 9. Natural frequency of third position. 

 
Fig. 10. Natural frequency of fourth position. 

 
Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
38 

 
Fig. 11. Natural frequency of fifth position. 

 
Fig. 12. Natural frequency of sixth position. 

 
In addition, a comparison was made 

between the experimental and numerical sides, 

and a percentage difference was discovered, 

which was close, as shown in Table 2. 

Because real values exist on the 

experimental side and are more dependable in 

calculations and comparisons, there are values 

detected in the numerical and empirical aspects 

that are not found in the theoretical aspect that 

has been properly calculated. 

 
Table 2,  

Comparing the natural frequencies of the shaft numerically and experimentally. 

No. of Position 1 2 3 4 5 6 

1st Natural 

Frequency 

(Hz) 

 
Experimental 65.013 59.548 50.835 20.134 5.536 5.235 

Numerical 60.762 67.235 60.933 22.651 6.6278 4.889 

Difference % 6.996 11.433 16.572 11.112 16.473 7.077 

2nd Natural 

Frequency 

(Hz) 

 
Experimental 125.845 155.435 105.735 93.648 100.153 35.686 

Numerical 141.88 141.88 99.612 111.11 118.38 35.656 

Difference % 11.301 9.553 6.146 15.715 15.397 12.325 

3rd Natural 

Frequency 
(Hz) 

Experimental 200.146 230.233 155.084 120.875 130.478 120.235 

Numerical 220.82 237.23 141.88 141.88 141.88 141.88 

Difference % 9.362 3.047 9.306 14.804 8.036 15.256 

 
The first five natural frequencies were 

compared in table 3 between the numerical and 

experimental sides using ANSYS. The values 

of the initial natural frequencies turn out to be 

close to a substantial extent, implying 

increased reliability. 

 
Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
39 

Table 3,  

Comparison of natural frequencies experimentally and numerically by ANSYS. 

No. of Natural 

Frequency 
Experimental (HZ) Numerical (HZ) Difference % 

1st 65.013 60.762 6.99615 
2nd 125.845 141.88 11.3018 

3rd 200.246 220.82 9.31709 

4th 380.856 389.94 2.32959 

5th 440.218 420.77 4.622003 

 
When the Rayleigh and Dunkerley methods 

were compared, it was discovered that the 

difference between them was relatively minor. 

For example, when using 7 nodes in the 

calculations for the two methods, the 

difference reached 3.6 percent, taking into 

account that the Rayleigh method is lengthy in 

the solution because it requires 25 flexibility 

values, whereas the Dunkerley method 

requires only 5 flexibility values. Bear in mind 

that the time at once is much less than the 

Rayleigh method. This saves effort in the 

calculations and is important in factories if 

productivity is important. 

 
5. Conclusions 

 
The Rayleigh and Dunkerley procedures for 

four states of nodes of a uniform shaft, which 

are used to determine the vibrations of rotating 

shafts, have all been thoroughly researched.  

Through this study, the following 

conclusions were obtained: 

1- By using a number of nodes to calculate the 

values of the natural frequencies of 

Rayleigh’s and Dunkerley's methods, it was 

found that there is convergence in the 

results between the two methods, as the 

greater the number of nodes, the greater the 

accuracy and the more convergence in the 

results between the two methods.  

2- The most suitable application method was 

determined to be the Dunkerley approach, 

which is close in results to the accurate 

method, with a little variation from the 

Rayleigh method of less than 4%.  

3- The value of the natural frequencies reduces 

as the distance between the supporting 

bearings of the rotating shafts decreases. 
This result was tested and confirmed in six 

different positions for bearings.  

4- When comparing experimental and 

numerical data, it was discovered that the 

difference in findings did not surpass 20 %. 

This indicates that the values were 

converging and the results were accurate. 

5- The amplitude of the vibrations generated 

in the system increases as the spinning 

shaft's speed increases. It can be shown that 

if the rotational speed is kept low, the 

system becomes more stable because the 

critical speeds are avoided. 

 
List of Symbols   

 VW Cross sectional area at node � �#� VXY Cross sectional area of first element �#� VZY Cross sectional area of last element �#� [ Diameter (m) \ Young’s modulus (Pa) ], X Force (^) (]* Force vector matrix (^) _ Gravitational acceleration   � ∕ `#� aW Area moment of inertia for the uniform 

solid beam (� ) $a& Unit matrix b Stiffness coefficient (N/m) $b& Stiffness matrix (N/m) b. c Maximum kinetic energy (J) dW Length of element at node i (m) dXY Length of first element (m) dZY Length of last element (m) d Length of shaft (m) eW Ith mass of nodes (Kg) eXY Mass of first node (Kg) eZY Mass of last node (Kg) f, e Mass (Kg) $f&, $e& Mass matrix (Kg)  g Volume (�<) heij Maximum potential energy (J) k Weight force applied to the beam (^) l Natural frequency (rad/s) m Deflection of Beams (m) (m* Deflection of Beams vector matrix (m) n Flexibility influence coefficient (m/N) $n& Influence coefficient matrix (m/N) o Mass density (=L ∕ �<) 


Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
40 

6. References 
 

[1] J. M. Vance, F. Y. Zeidan, and B. G. 

Murphy, Machinery vibration and 

rotordynamics. John Wiley & Sons, 2010. 

[2] F. M. A. El-Saeidy and F. Sticher, 

"Dynamics of a Rigid Rotor 

Linear/Nonlinear Bearings System Subject 

to Rotating Unbalance and Base 

Excitations," Journal of Vibration and 

Control, vol. 16, no. 3, pp. 403-438, 2009, 

doi: 10.1177/1077546309103565. 

[3] A. H. Haslam, C. W. Schwingshackl, and A. 

I. J. Rix, "A parametric study of an 

unbalanced Jeffcott rotor supported by a 

rolling-element bearing," Nonlinear 

Dynamics, vol. 99, no. 4, pp. 2571-2604, 

2020, doi: 10.1007/s11071-020-05470-4. 

[4] A. Khadersab and S. Shivakumar, 

"Vibration analysis techniques for rotating 

machinery and its effect on bearing faults," 

Procedia Manufacturing, vol. 20, pp. 247-

252, 2018. 

[5] J. Heikkinen, B. Ghalamchi, J. Sopanen, and 

A. Mikkola, "Twice-Running-Speed 

Resonances of a Paper Machine Tube Roll 

Supported by Spherical Roller Bearings: 

Analysis and Comparison With 

Experiments," in ASME 2014 International 

Design Engineering Technical Conferences 

and Computers and Information in 

Engineering Conference, 2014: American 

Society of Mechanical Engineers Digital 

Collection.  

[6] R. V. Daniel, S. A. Siddhappa, S. B. 

Gajanan, S. V. Philip, and P. S. Paul, "Effect 

of Bearings on Vibration in Rotating 
Machinery," presented at the IOP 

Conference Series: Materials Science and 

Engineering, Karunya University, 

Coimbatore 641114, Tamil Nadu, India., 

2017. 

[7] G. Chandrashekar, W. Raj, C. Godwin, and 

P. S. Paul, "Study On The Influence Of 

Shaft Material On Vibration In Rotating 

Machinery," Materials Today: Proceedings, 

vol. 5, no. 5, pp. 12071-12076, 2018, doi: 

10.1016/j.matpr.2018.02.182. 

[8] R. Tiwari, "Conditioning of regression 

matrices for simultaneous estimation of the 

residual unbalance and bearing dynamic 

parameters," Mechanical Systems and 

Signal Processing, vol. 19, no. 5, pp. 1082-

1095, 2005/09/01/ 2005, doi: 

https://doi.org/10.1016/j.ymssp.2004.09.00

5 

[9] M. R. Reddy and J. Srinivas, "Vibration 

Analysis of a Support Excited Rotor System 

with Hydrodynamic Journal Bearings," 

Procedia Engineering, vol. 144, pp. 825-

832, 2016, doi: 

10.1016/j.proeng.2016.05.093. 

[10] A. Wang, W. Yao, K. He, G. Meng, X. 

Cheng, and J. Yang, "Analytical 

modelling and numerical experiment for 

simultaneous identification of unbalance 

and rolling-bearing coefficients of the 

continuous single-disc and single-span 

rotor-bearing system with Rayleigh beam 

model," Mechanical Systems and Signal 
Processing, vol. 116, pp. 322-346, 2019, 

doi: 10.1016/j.ymssp.2018.06.039. 

[11] S. Yang et al., "Dynamic modeling and 

analysis of an axially moving and spinning 

Rayleigh beam based on a time-varying 

element," Applied Mathematical 

Modelling, vol. 95, pp. 409-434, 2021. 

[12] K. Zhu and J. Chung, "Vibration and 

stability analysis of a simply-supported 

Rayleigh beam with spinning and axial 

motions," Applied Mathematical 

Modelling, vol. 66, pp. 362-382, 2019. 

[13] R. Farshbaf Zinati, M. Rezaee, and S. 

Lotfan, "Nonlinear vibration and stability 

analysis of viscoelastic rayleigh beams 

axially moving on a flexible intermediate 

support," Iranian Journal of Science and 

Technology, Transactions of Mechanical 

Engineering, vol. 44, no. 4, pp. 865-879, 

2020. 

[14] M. A. Aouadi and F. Lakrad, "Linear 

flexural natural frequencies and stability 

analysis of spinning Rayleigh beams: 

application to clamped-clamped beams," 

in MATEC Web of Conferences, 2018, 

vol. 241: EDP Sciences, p. 01002.  

[15] M. Faraji Mahyari, K. Faraji Mahyari, and 

S. Fardpour, "New Approach to Instability 

Threshold of a Simply Supported Rayleigh 

Shaft," Journal of Solid Mechanics, vol. 6, 

no. 2, pp. 150-157, 2014. 

[16] K. Zhu and J. Chung, "Dynamic modeling 

and analysis of a spinning Rayleigh beam 

under deployment," International Journal 

of Mechanical Sciences, vol. 115-116, pp. 

392-405, 2016, doi: 

10.1016/j.ijmecsci.2016.07.029. 
[17] R. Tamrakar and N. Mittal, "Experimental 

Investigation of Shaft Whirl Carrying 3 

Rotors," Trends in Mechanical 

Engineering & Technology, vol. 5, no. 2, 

pp. 17-21, 2015. 


Karrar Baher                                  Al-Khwarizmi Engineering Journal, Vol. 18, No. 2, P.P. 29- 42 (2022) 

 
41 

[18] C. Levy, "An iterative technique based on 

the Dunkerley method for determining the 

natural frequencies of vibrating systems," 

Journal of sound and vibration, vol. 150, 

no. 1, pp. 111-118, 1991. 

[19] K. Low, "A modified Dunkerley formula 

for eigenfrequencies of beams carrying 

concentrated masses," International 

Journal of Mechanical Sciences, vol. 42, 

no. 7, pp. 1287-1305, 2000. 

[20] S. S. Rao, Mechanical Vibrations in SI 

Units, Global Edition, Sixth ed. Pearson, 

2017. 

[21] J. Metsebo, N. Upadhyay, P. Kankar, 
and B. N. Nbendjo, "Modelling of a 

rotor-ball bearings system using 

Timoshenko beam and effects of 

rotating shaft on their dynamics," 

Journal of Mechanical Science and 

Technology, vol. 30, no. 12, pp. 5339-

5350, 2016. 

[22] N. Wang, D. Jiang, and H. Xu, 

"Dynamic characteristics analysis of a 

dual-rotor system with inter-shaft 

bearing," Proceedings of the Institution 

of Mechanical Engineers, Part G: 

Journal of Aerospace Engineering, vol. 

233, no. 3, pp. 1147-1158, 2019. 

[23] W. T. Thomson, Theory of vibration 

with applications. CrC Press, 2018. 

 
 ) 2022( 29- 42، صفحة  2العدد، 18مجلة الخوارزمي الهندسية المجلد                                              كرار باهر تعيب       

42 

 
 ودونكرلي رايلي طريقتين  باستخدام  الدوارة األعمدة  في  االهتزازات  تحليل

  
  قاسم عباس عطية**                   كرار باهر تعيب*   

 عماد عبد الحسين عبد الصاحب***         
 *،**،***قسم هندسة الميكانيك/ الجامعة التكنولوجية/ بغداد/ العراق
me.19.20@grad.uotechnology.edu.iq  :البريد   االلكتروني* 

20044@uotechnology.edu.iq  :البريد  االلكتروني** 
20018@uotechnology.edu.iq  :البريد  االلكتروني*** 

 
 الخالصة
 

 االهتزاز لتحليل استخداًما األكثر التحليلية الطرق على دراسة اجراء تم. الهندسية التطبيقات في الدوارة لألعمدة االهتزازات أهمية توضيح تم
 الرياضية  الحسابات  في  ألهميتهما نظراً  Rayleighو Dunkerley لطريقتي  تفصيلية  دراسة  إجراء  تم البحث هذا  في أنه  حيث. الدوران  عمود في

 منصة  إنشاء  تم .Rayleighو Dunkerley طريقة من  كل  في للحساب  المتعددة  العقد  طريقة استخدام  تم.  الدوران  نظام  في  االهتزازات  سعة إليجاد
 هناك أن ثبت. المحامل بين المختلفة ولمسافات النظرية الحسابات مع النتائج ومقارنة الدوارة األعمدة في تحدث التي االهتزازات لدراسة تجريبية

 طريقة مقارنة تمت. المستشعرات من القادمة االهتزاز إشارة تحليل تم LABVIEW برنامج باستخدام. والنظرية التجريبية النتائج بين جيد توافق
ا، بسيطًا فرًقا تحدث أنها وجد وقد. دقة األكثر الطريقة اعتبارها وتم الدقيقة، بالطريقة رايلي  دنكرلي طريقة أما٪. ٠٫٠٦ حوالي إلى يصل جدً

 طريقة أن ووجد ،Dunkerleyو Rayleigh طريقتين بين مقارنة إجراء تم ثم. مقبولة وهي٪ ٤ حوالي يبلغ الدقيقة طريقة وبين بينها فالفرق

Dunkerley الحسابات في األنسب هي.