https://orcid.org/0000-0001-6852-1521
Figures
Abstract
This study introduces an enhanced numerical approach for analyzing the dynamic behavior of a rotor-bearing system subjected to unbalanced excitation from a gearbox drive shaft. The Newmark-β method with the integration of a variable time-step algorithm was used, allowing the system to be solved rapidly and accurately without compromising stability. This technique enables a precise computation of displacement and torsional deformation of the rotating shaft during its operational cycle. The proposed computational model is validated against experimental data, showing deviations of displacement in normal operation below the critical speed of about 6%. A comprehensive parametric analysis is conducted to evaluate the influence of rotational speed, trial mass, and initial phase angle on the system dynamics. The findings confirm that our enhanced numerical approach yields rapid convergence and reliable predictions, making it a valuable tool for dynamic analysis of rotating systems.
Citation: Tran TL, Nguyen VP, Le CC, Dang TN (2025) Investigation on the influence of unbalanced shaft component in gearbox on displacement using the Newmark-β method. PLoS One 20(11): e0335632. https://doi.org/10.1371/journal.pone.0335632
Editor: Ha Quang Thinh Ngo, VNUHCM HCMUT: VNUHCM-Ho Chi Minh City University of Technology, VIET NAM
Received: January 9, 2025; Accepted: October 14, 2025; Published: November 5, 2025
Copyright: © 2025 Tran et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting information files.
Funding: This work belongs to the project in 2025 funded by Ho Chi Minh City University of Technology and Education, Vietnam.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
The drive shafts in gearboxes are common components with variety of applications in industries, operating at various speeds and load types. Among many factors causing failure of shafts, vibration due to unbalance is a major cause. Single and coupled vibration modes, including torsional, longitudinal and transverse vibrations induce fatigue, fracture and tribological issues on the rotating shaft components [1–3]. These vibrations result in displacement, performance of gear transmission, wear and cracks [4–6].
Many studies have been carried out on dynamic aspects of the of rotor systems. The modelling of the rotor systems often uses the Jeffcott rotor model having a massless axis and a mass disk placed in the middle of the shaft. Modern analytical methods have been utilized to provide a foundational understanding of rotor dynamics and to conduct simple model experiments [7,8]. In addition, [9] and [10] studied a rotor system with a dynamic model, in which a disc is placed in the middle of a massless elastic rotating shaft. The equations of motion are obtained to Lagrangian dynamics for transverse – torsional vibrations. [11] derived the equation of motion by assuming that the diesel engine drive system can be approached as a simple rotor model such as the Jeffcott rotor. A modified version of this rotor model was also used for analysing the coupled torsional–transverse vibrations of a propeller shaft resulting from misalignment induced by shaft rotation [12]. Besides, [13] modelled a flexible rotating shaft system subject to bending and torsion coupled with the shaft and disk moving away from the center point of the shaft. In these studies, it is evident that shaft displacement caused by vibration is a complex problem. Therefore, it is necessary to investigate the shaft’s displacement in detail due to unbalance, using a new approach and model in the study of rotor dynamics.
The Newmark-β method, that is a widely used numerical integration technique in finite element analysis, is particularly effective for simulating dynamic systems. A study by [14] used this method to analyze the dynamic characteristics of spur and helical gear systems, while a subsequent study focused on the influence of housing flexibility on gear transmission dynamics [15]. An improved Newmark-β method was applied in [16] to determine nonlinear dynamics and reduce crankshaft torsional vibration. An enhanced version of the method was proposed for long-term simulations [17], offering improved convergence in evaluating the effects of nonlinearities on engine crankshaft torsional behavior. Collectively, these studies confirm that the Newmark-β method is a suitable and reliable tool for addressing time-varying oscillation problems.
In general, the current studies lack experimental evidence to support the theoretical computation. This study represents a computational dynamic model for a rotor-bearing system under unbalanced excitation, using the Newmark-β method with a variable time-step algorithm to determine the unbalance amount and shaft displacement. Various key experimental parameters, including rotational speed, trial mass, and initial phase angle, are put into account. The displacement and the amount of unbalance, determined from experiment, are compared with the numerically computational values to verify the validity of the computational model and thus to evaluate their effects on system dynamics. The motional orbit, representing shaft displacement, is also determined to evaluate the influence of the unbalance on the deflection of the shaft. This will allow to predict the fatigue strength, thus to accurately predict performance during operation.
2. Dynamic model of the rotor system
2.1 The proposed model
This issue arises from the eccentricity and imbalance of rotating components. If the gearbox operates under long-term conditions, it may lead to wear and eventual failure due to fatigue fracture. Fig 1 shows a shaft assembly in the gearbox of a lathing machine, comprising the drive shaft 3, bearings 1 and 5, and gears 2 and 4. Fig 2 represents the schematic diagram of a generalized model comprising a shaft and various assembled disks.
2.2 Parameters of the model
The proposed model of transmission shaft with two disks is shown in Fig 3. The specimens for evaluating the fatigue strength of metallic materials were used, according to ISO 1143:2010. This specimen type is also well-suited for evaluating location to failure [18]. Seven nodes, from node 1 to node 7, on the rotating shaft are selected to be investigated. Nodes 1 and 7 are the bearing positions. Nodes 2 and 6 are the disk placements and nodes 3 and 5 correspond to the positions with the maximum cross-section on the shaft. Node 4 has the smallest cross-section, where the fracture occurs.
The displacement vector qi for the shaft nodes i (i = 1–7) is:
where the displacements xi and yi along the X- and Y-axes and torsional angle αi at the investigated positions are
Since the loads are applied onto the gears at nodes 2 and 6, the rotation angle on the shaft segment between nodes 2 and 6 are thoroughly investigated. The torsional angles at nodes 3 and 5 are assumed to be small and negligible (α3 = α5 = 0). Thus, the torsional angles at nodes 2, 4, and 6 are where i = 2, 4, 6. The governing equation for the rotation angle is given as:
where (t) is the total rotation angle at node i as time t;
is the initial phase angle at node i;
is the torsional angle at node i;
is the angular velocity at node i (i = 2, 4, 6). The kinetic energy of the system is represented as:
where T t is translational kinetic energy; Tr is rotational kinetic energy; ,
are the vectors of translational velocities in the X and Y directions;
is the vector of angular velocities of the components;
is the mass matrix corresponding to the system from node 1 to node 7; [
] is the static moments of inertia concerning the study nodes 2, 4, 6. The elastic potential energy of the system is:
where [Kx} and [Ky] are respectively the stiffness matrices corresponding to translational stiffness in the X- and Y-axes, [Kt} is the torsional stiffness matrix related to angular displacements. According to Fig 3, an examination of the shaft segment from node 2 to node 6 represented by l3 to l6 was performed. Fig 4 shows the spring modeling of the shaft segment.
In the torsional shaft problem, x2, x4, and x6 are equated to the torsional angles α2, α4 and α6 of the shaft at nodes 2, 4 and 6, respectively. From the modelling of the shaft segment in Fig 4, the elastic potential energy is written as:
The dissipated energy of the system is given by:
where D is the total energy dissipated due to damping in the system; [Cx], [Cy}, and [Ct} are respectively the damping matrices corresponding to translational damping in the X, Y directions and axial torsional damping. The dissipated energy due to damping is written as:
where are the torsional damping at nodes 2, 4, and 6. The Lagrange’s equations of the second kind can be written in terms of the system as follows [19]:
where T is the total kinetic energy, V is the total potential energy of the system; is force due to the mass of the node;
is centrifugal force due to unbalance. The differential equation of motion for the rotor at this time is:
where is the mass matrix of the system;
is damping matrix;
is a stiffness matrix. The displacement vector
includes the displacements in the X-direction, Y-direction, as well as angular rotation,
is excitation force vector:
By solving equation (10) for i = 2, 4, 6; we obtain:
2.3 Solution method
Using the Newmark time-stepping analysis method to solve Equation (9), the following variables and matrices are involved:
Take the derivation of Equation (9), we have:
where
Transforming Eq. (12), we obtain:
Equation (13) can be generalized as:
where is the effective stiffness matrix, determined as:
is the effective force vector, determined as:
In this study, the implicit Newmark–β method was employed, since it does not require adherence to the time step condition as . The parameters
were adjusted during the time-stepping loop and set to
A constant time step of Δt = 0.01s was selected for the simulations to ensure stability and computational efficiency [20]. Fig 5 shows the algorithm flowchart, used for computing the displacement
, the velocity
, and the acceleration
.
3. Experimental procedure
An AISI 1045 steel transmission shaft as shown in Fig 3 with the unbalances m2 and m6 was prepared to investigate the behavior of nodes 1–7 during rotational operation. Table 1 shows the geometry dimensions and mechanical properties of the shaft [12,21,22].
Fig 6 shows the vibration testing machine, designed and fabricated for study the vibration behavior of the rotating shaft. The shaft was driven by an AC motor and an encoder was attached to the motor spindle to record the actual speed of the rotating shaft. A laser sensor measured the displacement of the rotor. The accelerometer and processor recorded the amount and positions of unbalance on the rotor. The operating tests were performed in controlled modes.
4. Results and discussions
4.1 The effect of rotational speed on shaft displacement
In Fig 7, the rotor oscillates when operated arround the first critical speed (mode 1). Under the effect of centrifugal force, the shaft is deflected and the shaft center will oscillate with displacement at position 4 with amplitude ymax.
Neglecting the damping coefficient C, the first critical speed of the system can be determined according to [23] using the following expression:
In this case, the critical speed is:
To ensure the stability of the rotor in the model and corresponding to the actual working speeds of the equipment, the experimental speed ranges of the rotor were selected as follows: n1 = 800 rpm, n2 = 1500 rpm and n3 = 2000 rpm, in which n3 is selected to be less than 70% of critical speed of 2132 rpm [24]. Fig 8 shows the displacement of node 4 in the X direction at speed of n3 = 2000 rpm, determined from the experiment and the Newmark–β numerical simulation. Generally, the amplitude measured from the experiment is larger than those using Newmark–β numerical computation because various factors in the manufacturing process may affect the unbalanced conditions.
Table 2 compares the X and Y displacement of node 4, determined from the Newmark-β method and from the experiment. The displacement determined experimentally is higher than those determined from the theoretical simulation because the actual stiffness of shaft and bearing is not uniform and due to the inaccuracy and clearance between shaft and bearing. For the eperation below the first critical speed, the vibrations remain stable, with displacement amplitudes of X = ± 0.046 and Y = ± 0.081. The relative error of displacement in the X- and Y-axes for speeds of 2000 rpm is 6% and 12%, respectively. As the rotor reaches the first critical speed of 3000 rpm, the vibrational amplitude significantly increases, with displacements of X= ± 0.101 and Y= ± 0.119 mm because of the resonance effects of the rotor components. In general, the displacement in the direction Y is higher than in the X direction because of the gravity force of the components, acting downward, resulting in the higher vertical vibration. The relative errors in displacements are higher by 12% and 18%.
A comparison between the numerical results obtained using the Newmark–β method and the experimental data indicated that the displacement deviation of approximately 6%, which is acceptable for engineering analysis and validates the accuracy of the computational model.
4.2 The effect of unbalance on shaft displacement
In engineering, for some design circumstance or unavoidable eccentricity in the manufacturing process, the unbalance exists and should be taken into account. The unbalance, eccentricity, and angular position (the angle between the planes of trial masses m2 and m6) will be analyzed hereafter. The masses m2 and m6 were mounted at three positions 00–1800, 00–900, and 00–00 such that the direction of m6 respectively makes angular positions of 1800, 900 and 00 to the direction of m2, as shown in Fig 9.
4.2.1 Symmetrical trial mass (angular position: 00–00 on both disks).
The structure of the unbalanced rotor is shown in Fig 9, where the trial masses m2 and m6 are in the same direction. According to the unbalance standard, the corresponding test masses considered in the range follows as G1, G2.5, G6.3 and G16 [25]. For G1, the allowable eccentricity e (n3 = 2000 rpm) is:
At the minimum speed n1, the allowable test mass for G16 is calculated as follows:
Three kinds of trial masses 10 g, 20 g, and 30 g and rotational speeds of 800, 1500 and 2000 rpm were selected to determine the unbalanced, using the commercial Erbessd – Instruments interface, as shown in Fig 10. For the highest trial mass mtrial of 30 grams, the unbalanced amounts are respectively 41.18 mm/s and 39.06 mm/s for two nodes 2 and 6.
Table 3 represents the measurement results of rotor unbalance at angular position 00–00 with corresponding to load values and speeds. The corresponding unbalance significantly increases with the increase of trial mass mtrial and the rotational speed n respectively. The unbalanced amount G was highest of approximately 41.1 mm/s at n of 2000 rpm with mtrial of 30 g and got lowest of 0.58 mm/s at 800 rpm and mtrial of 0 g.
Fig 11 and Table 4 represent the variation in horizontal X-axis displacement at node 4 under different operating speeds. The results clearly show the displacement amplitude increase with the rotational speed, particularly as the rotor approaches its first critical speed. This behavior reflects a corresponding rise in dynamic excitation, which is characteristic of resonance phenomena in rotor-dynamic systems.
4.2.2 Asymmetrically placed trial mass (angular positions 00–900 and 00–1800).
By changing the initial phase angle, the results of rotor unbalance at positions 00–900 and 00–1800 are shown in Table 5. At a constant rotational speed of 2000 rpm, variations in the load angular position 00–00, 00–900, and 00–1800 and trial mass mₜᵣᵢₐₗ of 10, 20, and 30g result in corresponding changes in the component unbalance. The experimental results indicate that the lowest unbalance of 2.6 mm/s occurs at a trial mass of 10 grams and angular positioned at 00–1800. This is because the centrifugal forces of m2 and m6 canceled out. On the contrary, the unbalance G obtained the highest value of 42.3 mm/s at the mass 30g and angular positioned at 00–00 because of the resonance of centrifugal forces of m2 and m6.
In addition, Fig 12 and Table 6 show a decreasing trend of displacement amplitudes by ±0.098 mm, ± 0.079 mm, and ± 0.054 mm as the load position shifts from 00–00 to 00–900 and then to 00–1800, respectively.
4.3 Motional orbit
To evaluate the influence of the unbalanced mass position on the rotor’s motion trajectory, a series of experiments was conducted at a rotational speed of 2000 rpm. A trial mass 30g was mounted on the two discs of the rotating shaft, with various combinations of phase angular positions between the loads. Three unbalanced mass distribution cases were investigated for three angular positions 00–00, 00–900 and 00–1800. The motion orbit of the shaft center at the speed n3 = 2000 rpm is represented in Fig 13. The orbits for load-free rotation and the unbalanced masses in the opposite angular directions 00–1800 obtains the smallest value, while the orbit for angular position 00–00 is most unstable. This is because of the cancelation or resonance of centrifugal forces of the masses, well agreeing with the experimental displacements in Table 5. Furthermore, the data indicate that the oscillation observed with a trial mass of 30g for the angular position 00–00 closely resembles the shaft behavior at its first critical speed.
It can be observed that changes in the angular positions of the trial mass 0°–0°, 0°–90°, and 0°–180° strongly affect the vibration amplitude, thus the shape of the shaft center orbit and well agrees to the expected behavior predicted based on the influence parameters. The greater amount of unbalance and their positions concentration will generate stress accumulation and micro-cracks, resulting in shaft failure.
5. Conclusions
The following conclusions are made:
- a. The effects of such key parameters as rotation speed, eccentricity due to unbalanced and initial phase angle during operation were simulated. The experimental equipment was built and experimental results were compared with the simulated data.
- b. The vibration becomes severely unstable when rotation speed approaches the critical speed, as shown in Tables 3 and 4. The displacement amplitude significantly increases with the amount of unbalance, as shown in Table 5.
- c. The initial phase angle of the trial mass position has great influence on the vibration of the rotating shaft, thus the fatigue bending strength and performance of the component, as shown in Tables 5 and 6 and Fig 13. By determining the vibration and unbalance conditions of the shaft component, the fatigue limitation can be predicted to ensure the safety and long-term stability of the rotor system.
- d. For operation under the critical speed, the unbalance analysis using the Newmark-β method well agrees with the experimental result, as represented in Table 2, showing that the Newmark-β method is a reliable and stable approach for analyzing the behavior of rotor-bearing systems. This allows rapid and high computational efficiency to predict the shaft operation with fast convergence time.
Further research may be proceeded on the following issues:
- i. Unbalance analysis for double-phase and composite materials.
- ii. Analysis for materials with thin films, coating layer or the surface layer with residual stress.
- iii. Unbalance analysis in high temperature conditions.
Acknowledgments
The authors would like to express their appreciation to the staff of the Metallurgy Laboratory at Ho Chi Minh City University of Technology and Education for their assistance.
References
- 1. Cohen R, Porat I. Coupled torsional and transverse vibration of unbalanced rotor. J Appl Mech. 1985;52:701–5.
- 2. Ananda Rao M, Srinivas J, Rama Raju VBV, Kumar KVSS. Coupled torsional–lateral vibration analysis of geared shaft systems using mode synthesis. J Sound Vib. 2003;261(2):359–64.
- 3. Chen X, Liao M. Transient characteristics of a dual-rotor system with intershaft bearing subjected to mass unbalance and base motions during start-up. ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition; Oslo, Norway; 2018.
- 4. Chalabi I. Comparative service life analysis for gears according to different failure criteria. J Fail Anal Prev. 2020;20:2137–44.
- 5. Wang C, Li H, Zhang K, Hu S, Sun B. Intelligent fault diagnosis of planetary gearbox based on adaptive normalized CNN under complex variable working conditions and data imbalance. Measurement. 2021;180:109565.
- 6.
Muszynska A. Rotordynamics. 1st ed. New York: Taylor & Francis; 2005.
- 7.
Friswell M, Penny J, Garvey S, Lees A. Dynamics of rotating machines. 1st ed. Cambridge University Press; 2010.
- 8.
Tiwari R. Rotor systems: analysis and identification. CRC Press, Taylor & Francis Group; 2018.
- 9. Yuan Z, Chu F, Lin Y. External and internal coupling effects of rotor’s bending and torsional vibrations under unbalances. J Sound Vib. 2007;299(1–2):339–47.
- 10. Hong J, Yu P, Ma Y, Zhang D. Investigation on nonlinear lateral-torsional coupled vibration of a rotor system with substantial unbalance. CJA. 2020;33(6):1642–60.
- 11. Han HS, Lee KH, Jeon SH, Park SH. Lateral-torsional coupled vibration of a propulsion shaft with a diesel engine supported by a resilient mount. J Mech Sci Technol. 2017;31:3727–35.
- 12. Halilbeşe AN, Zhang C, Özsoysal OA. Effect of coupled torsional and transverse vibrations of the marine propulsion shaft system. J Mar Sci Appl. 2021;20:201–12.
- 13. Das AS, Dutt JK, Ray K. Active control of coupled flexural-torsional vibration in a flexible rotor–bearing system using electromagnetic actuator. Int J Nonlin Mech. 2011;46:1093–109.
- 14. Kong X, Hu Z, Tang J, Chen S, Wang Z. Effects of gear flexibility on the dynamic characteristics of spur and helical gear system. Mech Syst Signal Process. 2023;184:109691.
- 15. Kong X, Tang J, Siyu C, Hu Z. Effects of gearbox housing flexibility on dynamic characteristics of gear transmission system. J Vib Control. 2020;27(17–18):2097–108.
- 16. Ma K, Du J, Liu Y, Chen X. Torsional vibration attenuation of a closed-loop engine crankshaft system via the tuned mass damper and nonlinear energy sink under multiple operating conditions. Mech Syst Signal Process. 2024;207:110941.
- 17. Ma K, Du J, Liu Y. Nonlinear dynamic behavior analysis of closed-loop self-excited crankshaft model using improved Newmark-β method. Nonlinear Dyn. 2022;111(6):5107–24.
- 18.
ISO 1143:2010. Metallic materials - rotating bar bending fatigue testing.
- 19.
Sussman GJ, Wisdon J. Structure and interpretation of classical mechanics. MIT Press; 2015.
- 20. Fan Y, Zhao C, Yu H, Wen B. Dynamic load identification algorithm based on Newmark-β and self-filtering. Proc Inst Mech Eng C: J Mech Eng Sci. 2020;234:96–107.
- 21. Yanhong MA, Yongfeng W, Cun W, Jie H. Interval analysis of rotor dynamic response based on Chebyshev polynomials. CJA. 2020;33(9):2342–56.
- 22. Tian G, Shuqian C, Tiancheng Z. Theoretical and experimental study on the transient time-frequency characteristics of the bending-torsional coupling motions of a rub-impact dual-rotor system. Shock Vib. 2021;2021:8817228.
- 23.
Chen WJ, Gunter EJ. Introduction to dynamics of rotor-bearing systems. Trafford Publishing; 2007.
- 24.
Darlow MS. Balancing of high-speed machinery. NY: Springer Verlag; 2012.
- 25.
ISO 1940/1:2013. Balance quality requirements of rigid rotors. International Organization for Standardization.