Dynamic Response of a Rotating Assembly under the Coupled Effects of Misalignment and Imbalance

In rotating machinery, the second most common fault after imbalance is misalignment. Misalignment can have a severe impact on equipment and may reduce the machine’s lifetime considerably. In this paper, the simultaneous eﬀect of imbalance and misalignment (parallel or angular) on the vibration spectra of rotating machinery will be discussed. A numerical model is developed and used to obtain the time and frequency responses of the rotor-coupling-bearing system to the simultaneous eﬀect of these faults. The numerical model shows that the imbalance was mainly related to the peak located around 1X, whereas misalignment was linked to the peak around 2X. In addition, the parallel misalignment fault magniﬁes the 2X amplitude of the displacement response, whereas the response of angular misalignment is captured at the 2X and 4X amplitudes. This study also examines the eﬀects of changing the model’s rotational speed, misalignment level, and coupling type for angular and parallel misalignments.


Introduction
Rotating machinery is an essential part of many industrial sectors, and thus monitoring the condition of such machinery is continuously attracting researchers. e coupling is a vital part of any rotating machine, but it is often neglected because of its low cost against the total cost of the equipment. Flexible couplings are essential mechanical elements used in rotating machinery to allow power transmission between a driver and a driven shaft. Couplings are also used because they can compensate for inevitable misalignments between linked shafts. In the research literature, a strong focus on explaining the misalignment behavior in rotor dynamics has led to the experimental approach being used less frequently than numerical simulations. Suitable experimental methods include vibration analysis, torque, temperature capturing, and motor current consumption, whereas simulation methods include structural analysis, finite element (FE) analysis, and mathematical derivations. Out of all these methods used to capture misalignment faults, vibration spectrum analysis is the most widely accepted one. Here, we propose a numerical model that can evaluate the vibration response of misaligned shafts in rotating systems.

Misalignment in Rotating
Shafts. Vibration in rotating machinery can sometimes be so dangerous that it destroys critical parts of the machine. e two primary sources of vibration within rotating mechanisms are rotor imbalance and shaft misalignment. Rotor imbalance is recognized as one of the most common origins of machine vibration, and it is present to some degree in nearly all rotating equipment. Physically, imbalance occurs when the center of mass (axis of inertia) of a rotating assembly does not coincide with the center of rotation (the geometric axis). Misalignment occurs when a rotating driveshaft, and the driven shafts coupled to it, does not rotate around the same central axis. Despite efforts to aligning interconnected shafts accurately, perfect alignment between shafts is challenging to maintain and achieve. Consequently, rotating machines often operate under less than optimum alignment conditions. is vexing problem has challenged and intrigued maintenance professionals for decades because it can arise from many sources, including manufacturing and mounting tolerances, operational deflections caused by thermal expansion and distortion, foundation accommodation, working forces, and worn bearings [1]. e many different causes of misalignment are why this malfunction is known to be the second most frequent source of faults in rotating machinery, with only mass imbalance being more prevalent [2]. By itself, this problem represents more than 60% of failures reported in the industry [3,4]. In the last decade, misalignment detection techniques have developed rapidly. However, there remains a need for a mechanism that describes misalignment phenomena more scientifically. ere are many experimental methods for predicting misalignment faults, such as monitoring the motor's current, torque, acoustic, and vibration signals. e vibration-based diagnostics are most often used because vibration signals provide an abundance of mechanical information and are easier to collect [5]. Nevertheless, the vibration response of a system including a rotor, a coupling, and a bearing subjected to the coupled effects of misalignment and imbalance faults has not yet been fully explored.

Parallel and Angular Misalignments.
In rotor dynamics, shafts have either parallel or angular misalignment, or both, as shown in Figure 1.
As portrayed in Figure 1, parallel misalignment is the condition when the axes of rotation are not collinear and do not intersect with each other. However, when the two axes of rotation are not collinear, but their centerlines intersect, this is known as angular misalignment. Parallel and angular misalignments can occur vertically and horizontally. Combined misalignment is the case where both parallel and angular misalignments exist simultaneously.

Experimental and Numerical Investigation of the Rotor-
Coupling-Bearing System. In general, misalignment in the coupling leads to vibration throughout the mechanical system. e characteristic used to identify shaft misalignment includes a high level of axial vibrations, a 180°p hase shift between axial vibrations on the shaft tips, and the manifest presence of a harmonic component in the signal spectrum at double the rotation speed; these are all widely accepted signs for fault diagnosis [6,7], based on system vibrations. Sudhakar and Sekhar [8] reviewed the various methods that have been used for modeling the coupling, the effects of misalignment, and the condition monitoring techniques. Expressions for the forces and moments generated by parallel misalignment were defined by Gibbons [9]. Xu and Marangoni [10] explored misaligned rotor systems with imbalance and misalignment faults numerically and experimentally with flexible and helical couplings. Typically, peaks at the rotation speed (1X) and double the rotation speed (2X) were predominant in the vibration spectra. Sekhar and Prabhu [11] modeled the effects of coupling misalignment on rotor vibrations with eight degrees of freedom (DOFs) per node and developed expressions for the forces and moments involved in angular misalignment.
Despite the fact that 2X components on signal spectra are frequently reported as a sign of misalignment [12][13][14], varying results have been published, as shown in the theoretical work of Al-Hussain and Redmond [15], who showed that parallel misalignment manifests as synchronous vibrations. Moreover, the experimental results of different researchers revealed distinct spectral content for particular couplings under the same conditions of misalignment [3].
A simple linear mathematical model with flexible coupling was developed by Redmond [16] to analyze the forces in the system. He claimed that parallel misalignment by itself produces multiharmonic (i.e., 1X, 2X, and 3X) static and dynamic system responses. Lees [17], on the contrary, investigated rigid rotors and developed an analytical model for a linear system that included parallel misalignment but no damping. is linear model generated responses at multiple harmonics of the shaft's rotation speed. Jalan and Mohanty [18] used the residual generation method to develop a model-based fault detection method for a simple system involving a rotor and a bearing.
Similarly, Hariharan and Srinivasan [19,20] developed an FE model of a simple rotor-bearing system with flexible coupling and correlated the FE model with experimental results for parallel misalignment only and found that in this case, 2X component was dominant. Xu and Marangoni [12] conducted experiments which showed that the vibration responses caused by misalignment occurred at even multiples of the rotational speed (2X, 4X, etc.). Patel et al. [21] developed a coupled rotor model that applied Timoshenko's beam theory as well as the effects of parallel and angular misalignments. Patel et al. used an experimental setup to discover diagnostic features in the 1X to 3X range. Sekhar and Prabhu [11] numerically estimated the effects of parallel misalignment on the vibration response at 2X within the rotating system. Jun-Lin and Yu-Chih Liu [22] used the concept of multiscale entropy alongside wavelet denoising to detect shaft misalignment. Dewell and Mitchell [13] showed experimentally that the vibration components at 2X and 4X mainly depend on misalignment in the coupling.
More recently, Hujare and Karnik [4] carried out experimental and numerical analysis (using FE models) for an aluminum shaft-rotor-bearing system with parallel misalignment. Misalignment effects at the coupling location were simulated via a nodal force vector. ey observed that the impact of parallel misalignment was dominant at the characteristic 2X frequency rather than the 1X and 3X frequencies.
e natural frequency has a crucial role in determining the speeds at which the characteristic 1X, 2X, and 3X frequencies reach their maximum amplitudes.
More recently, Li et al. [23] used an asymmetrical generator rotor system supported on journal bearings to examine the nonlinear dynamic behavior induced by parallel 2 Shock and Vibration misalignment. By using numerical techniques (e.g., rotor orbits and their frequency spectra, Poincaré maps, and the greatest Lyapunov exponent), Li et al. [23] demonstrated that the supersynchronous component in the frequency spectrum at 2X was the key to identifying and diagnosing rotor misalignment. Wand and Jiang [24] established a dynamic model for researching imbalance-misalignment coupling faults in a dual rotor system with intershaft bearing. Numerically and experimentally, they investigated the effect of the rotational speed ratio, mass eccentricity, misalignment angle, and parallel misalignment on the vibration characteristics of outer and inner rotors. e proposed model was verified by cascade plots, time waveforms, and frequency spectra. Recently, Wang and Gong [25] developed a comprehensive model to study misalignment and imbalance faults for a rotor system, of six degrees of freedom, via a FE approach. Misalignment effects were considered at the coupling location through the application of nodal force and moment vectors. Wang and Gong [25] managed to highlight some exciting features of parallel and angular misalignments in horizontal and vertical displacement forces and moments. Srinivas et al. [26] analyzed various system faults induced by angular misalignment within coupled rotor-train systems integrated with auxiliary active magnetic bearing (AMB) support. e method quantified the effect of misalignment by estimating additive coupling stiffness. e use of additive coupling stiffness to assess the severity of angular misalignment was a novel concept presented in this work. Sawalhi et al. [27] showed a detailed FE and dynamic simulation model of a vibration test rig. e result of the simulation was compared with experimental results. Both the simulated and experimental results showed an increase in the lower and higher harmonics of the shaft's rotational speed when the acceleration vibration signal with smearing was considered. Recently, the work presented in [28] investigated the deformation in a hexangular flexible coupling that joined a pair of rigid rotors that were misaligned (angular and parallel misalignments). e relationships among the vibration responses of the 1X, 2X, and 3X components and the moving orbits of the coupled rotors were simulated numerically at different speeds of rotation with and without misalignment. e results provided theoretical support for diagnosing and detecting faults within rotating machinery that includes hexangular flexible coupling.
e majority of the cited articles mainly focused on experimental and/or numerical investigations of angular and parallel misalignments. However, very few studies have analyzed a realistic case where misalignment and imbalance coexist simultaneously. Until now, these conditions have not been systematically investigated, although perfect balance in a rotating system is hard to achieve in practice, as some degrees of imbalance will always be present. Many of the previous studies had noticeable limitations since their systems were very complicated, or their theoretical developments were based on unrealistic assumptions that did not describe the combined effect of imbalance and misalignment, which were unable to capture real physical phenomena or were not applied in extreme cases of loading and deformation. erefore, these systems are not easily available and/or often practically not applicable. Accordingly, this article aims to discuss the vibration features, nonlinear dynamics, and parameter properties of the combined effect of imbalance and misalignment in rotating machinery.
is paper is structured as follows: Section 2 presents the model used in this paper and provides a mathematical derivation of the forces and moments impacting the system as a result of misalignment and imbalance. Section 3 presents the results of various numerical experiments that were conducted and presents discussion of the results. Finally, Section 4 presents the conclusions of this research.  shafts is partially supported, from the coupling side, by a ball bearing. To make the analysis nonspecific for a particular type of coupling and to ease comparisons with other investigations, the coupling was simulated as two disks, interconnected by stiffness and damping elements. A diagram of the assembly model, including the main components and geometric parameters, is presented in Figure 2. e coordinate system was chosen in this model so that one of the axes (the z-axis) lay along the axial (longitudinal) direction of both shafts, whereas the x-axis and the y-axis lay along the radial directions. In the rest of the model description, the right-hand side of the coupling is indicated by the index 1, and the left-hand side is indicated by 2. Each of the coupling disks is able to move independently in three directions and to rotate freely around three axes. erefore, with respect to Node 1, located at the center of gravity of the outer surface of disk 1, and to Node 2, located at the center of gravity of the outer surface of disk 2, the system's motion is described in total by 12 DOFs, in which each node has six DOFs, as indicated by the following equation:

Modeling the
where x is the displacement in the radial horizontal direction, y is the displacement in the radial vertical direction, z is the displacement in the axial/longitudinal direction, θ is the rotation around the z-axis, β is the rotation around the xaxis, and c is the rotation around the y-axis. A second-order differential equation determines the motions of the rotor-coupling-bearing system shown in Figure 2: where m 1 is the mass of Subsystem 1 (the right-hand side of the coupling), m 2 is the mass of Subsystem 2 (the left-hand side of the coupling), I x1 is the mass moment of inertia of Subsystem 1 in the x direction, I x2 is the mass moment of inertia of Subsystem 2 in the x direction, I y1 is the mass moment of inertia of Subsystem 1 in the y direction, I y2 is the mass moment of inertia of Subsystem 2 in the y direction, I z1 is the mass moment of inertia of Subsystem 1 in the z direction, and I z2 is the mass moment of inertia of Subsystem 2 in the z direction. e coefficients of the mass matrix are obtained by the following derivative rule:

Determination of the Stiffness Matrix.
Based on the previously defined degrees of freedom, the equation for potential energy, in the shafts and coupling, is given by where the index B refers to the bearing, S refers to the shaft, C refers to the coupling, and T refers to torsion. In the previous expressions, since the two springs KB and KS are connected in parallel, their equivalent values will be Similarly, the coefficients of the stiffness matrix are obtained by the following derivative rule: Consequently, the stiffness matrix [K] is given as Shock and Vibration

Determination of the Damping Matrix.
ere are numerous paths to damping, and in a complex structure, several means of damping may take place simultaneously at different locations throughout the structure. e damping considered in this work accounts for the interconnecting parts, at the endpoints of the shafts, namely, the bearings and the coupling, and the structural damping of the shafts themselves.
Viscous damping is a formulation of the damping phenomena, in which the force of damping is proportional to the velocity. Most often, viscous damping refers to dashpot, a simple technique used to model the energy dissipation in mechanical systems and thus represents several dissipative phenomena such as heat, friction, and plastic yielding. For those reasons, several dashpots were introduced in the system to account for viscous damping behavior. For the proposed model, the dissipation energy equation because of viscous damping is detailed as follows: e coefficients of the viscous damping matrix are obtained by using the following derivative rule: e produced viscous damping matrix [C v ] is detailed below: It is quite common to describe the structural damping matrix [C s ] of the system by a matrix that is proportional to the mass and the stiffness matrices, which can be expressed as where a and b are the constants; according to [25], they have the following values: a � 5 and b � 1.35 × 10 − 5 , respectively. e following expression describes the total damping effect: where [C st ] is the structural damping matrix, [C v ] is the viscous damping matrix, and [C] is the total damping matrix e model of the rotor-coupling-bearing system has several assumptions that apply throughout this paper: (i) Both shafts were connected to the coupling by an interference fit to avoid introducing keys to the system (ii) e bearings did not impose any longitudinal stiffness or damping (iii) e discs of the system are responsible for unbalance (introduced into the system as eccentricity in the excitation force equation) (iv) e coupling's stiffness is independent of the angle of rotation (v) e stiffness and damping coefficients of the coupling are fully described in three directions (one axial direction and two radial directions) (vi) Imbalance and misalignment are the only excitation forces in the system (vii) e gyroscopic effect is neglected since the rotation speed is low (since less than 1000 rpm) 6 Shock and Vibration

Modeling the Forces of Imbalance and Misalignment.
e general arrangement of the coupled shafts with parallel and angular misalignments is shown in Figure 3. e total misalignment ∆E is defined as the sum of the parallel misalignment Δy and the angular misalignment α, where O 1 and O 2 are the center of articulation for Subsystems 1 and 2, respectively: e excitation force that applies within the coupling is induced by two phenomena, which are the imbalance (denoted IB) and the misalignment (denoted MA). erefore, the vector force has two components: 2.2.1. Forces due to Imbalance. e imbalance force is related to the dynamic eccentricity of the system that appears in the two parts of the coupling. e radial imbalance force can be defined as follows: where e 1 is the mass eccentricity of Subsystem 1, e 2 is the mass eccentricity of Subsystem 2, ω is the rotational speed of the shafts, t is the time, r is the radius of eccentricity in the coupling, and ϕ is the phase between the imbalance and the misalignment forces.

Forces due to Parallel Misalignment.
Misalignment is not easy to detect on machinery that is running. e radial forces transmitted from one shaft to another, through a coupling, are typically combination of static (i.e., unidirectional) and dynamic forces that are difficult to measure externally. Unfortunately, outside evaluation of how much force is being applied to the couplings is practically unavailable.
In the literature, the attempts to theoretically describe and quantify the internal forces inside a coupling are relatively limited. Because of the difference in the internal geometry, a theoretical model developed for a particular type of coupling is generally not suitable for other types. Of particular interest is the theoretical model developed by Wang and Jiang [24], for a disc coupling, and refined by Wang and Gong [25].
As developed in [25], in the case of a parallel misalignment, the radial forces F x1 and F y1 on the right-hand side of the coupling are expressed as where K x and K y are the coupling stiffness in the x and y directions, respectively, and ΔE is the total misalignment. In the previous equations, one can see that for the developed expressions, as expected beforehand, the force in the x direction is harmonic. At the same time, the one in the y direction, because of the supporting condition in that direction, is a combination of static and harmonic components. Moreover, in both directions, the forces are pulsating with double the rotation frequency, i.e., the radial forces change two times their directions for one rotation cycle of the coupling. e radial forces on the left-hand side of the coupling are simply the opposite of the opposite side: Shock and Vibration 7

Forces due to Angular Misalignment.
e angular misalignment forces were derived by following the methodology of Xu and Marangoni [12] and Wang and Gong [25]. Figure 4 illustrates the torque decomposition of the coupling subsystems in which an angular misalignment fault is present. e driving torque T, from the motor, splits into two components, T Z and T R , when it passes through the misaligned coupling, as shown below: e torque component T z is along the rotor axis, while the component T R is perpendicular, causing lateral shaft bending deflection. e bending moment T R can be decomposed into two components, in the x and y directions, which can be expressed as follows: As mentioned by Xu and Marangoni [12] and Wang and Gong [25], the ratio of the relative velocity between the misaligned shafts is calculated as where C 1 � 4 cos α 3 + cos 2α , By differentiating equation (25) and making minor rearrangements, we can obtain an expression for the acceleration at Node 2 as follows: Next, the torque caused by angular misalignment is calculated via the following equation: Consequently, the torques in the x and y directions are expressed as 1 + C 2 cos 2ωt 2 cos ωt, e misalignment force vector for parallel and angular misalignments is defined as follows:

Determination of Flexible Coupling
Stiffness. e two couplings used in this study are aluminum spiral couplings ( Figure 5), with an outer diameter of 24/16 inches (38.1 mm) and an inner diameter of 10/16 inches (15.875 mm). ese are named the "white" and "black" couplings due to the color of their coatings. e two couplings are similar in all aspects apart from the length and degree of their spiral grooves. e white coupling has 380°g rooves, while the black coupling has 525°grooves. e coupling flexibility influences the dynamics of the rotatory system, especially under misalignment conditions. e ability of a flexible coupling to accommodate misalignment is a vital feature to reach desired performance in terms of vibrational behavior. erefore, accurate estimation or measurement of the coupling stiffness is always a fundamental step in any theoretical or numerical investigation.
e stiffness values of the actual couplings were estimated by FE simulations, as portrayed in Figure 6. e stiffness of the coupling in a specific direction is calculated by correlating 8 Shock and Vibration the load applied in that direction with the resulting deflection in the same direction. e torsional stiffness is obtained in a similar way, except for the load, which is replaced by a moment. e reliability of any modeling process depends not only on the development of the model itself but also on the numerical values of its physical parameters. erefore, it is necessary to obtain accurate and realistic values obtained via FE analysis. As a first check, some of the simulation results for the white coupling were compared with the available data provided by the manufacturer [29]. e comparison displayed in Table 1 shows a reasonable degree of matching between both sets of results. e linear and torsional stiffness values of the white coupling resulting from the FE simulation are summarized in Table 2.

Determination of the Coupling's Damping Coefficients.
e damping values of the coupling in three directions were obtained experimentally by conducting an impact test with a PCB PIZOTRONICS impact hammer (Model PCB-086, 0-500 lb, 10 mV/lb), an ICP accelerometer (Model No. 352C33, 100 mV/g), and a BETAVIB data acquisition unit, as displayed in Figure 7. e results are summarized in Table 3.

Stiffness and Damping of the Bearings and Shaft.
e bearing's stiffness and damping values were estimated in accordance with the previous work of Badri [30]. e longitudinal, lateral, and torsional stiffness values of the shafts were obtained by applying the laws of solid mechanics.

Numerical Solution of the Equations of Motion for the Rotor-Coupling-Bearing System.
To solve the equations of motion numerically, a Matlab code was created to compute the model's states by applying an explicit Runge-Kutta (4,5) formula for numerical integration. e parameters of the rotor-coupling-bearing system used in the numerical solution for the white coupling are listed in Tables 4-6. Table 4 presents the parameters of the rotor-bearing system; the specific parameters of the white and black couplings are shown in Tables 5 and 6, respectively.

Results and Discussion
is section presents the simulation results for the rotorcoupling-bearing system and discusses the findings. In the simulation that examined the model's response to imbalance and misalignment forces, the white coupling was used. e simulation results are presented and analyzed in the time and frequency domains. In the time domain, the results are given up to the limit of 0.5 seconds, which allows visualizing ten cycles of vibration. e results are displayed up to 100 Hz in the frequency domain, as this is the range that encloses 1X, 2X, 3X, and 4X.

Effect of Pure Imbalance on the Vibration Response.
e effect of imbalanced forces on the vibration response of the rotor-coupling-bearing system was investigated in both the radial and angular directions.

Response to an Imbalance in the Radial Directions.
e response of the system to an imbalance in the radial directions (x 1 , x 2 , y 1 , y 2 ) was investigated at a rotation speed of 1200 rpm (1X � 20 Hz). An eccentricity of 1 mm was introduced to the system in vertical and horizontal directions. Figure 8 shows the time-and frequency-domain responses in the x direction, and Figure 9 shows the responses in the y direction.

Shock and Vibration 9
As expected, the imbalanced excitation makes the system oscillate in a harmonic way that manifests itself as a pure sine wave in the time domain and a single peak in the frequency domain.
e peak amplitudes of the displacement time responses appeared at 5.214 × 10 −3 µm for the x 1 direction and 5.265 × 10 −3 µm for the y 1 direction. e peak amplitude of the displacement time responses were 3.597 × 10 −3 µm for the x 2 direction and 3.633 × 10 −3 µm for the y 2 direction. Moreover, the vibrations at Node 1 are higher than those at Node 2 as expected because Subsystem 1 had a greater mass than Subsystem 2 (equations (17)-(23)). A slight change appeared between the amplitudes in the time and frequency domains, mainly caused by an error in the calculation of the fast Fourier transform (FFT) function in Matlab. Furthermore, the response of the time and frequency domains to displacement in the radial direction at Node 1 was almost constant, suggesting that the displacement responses of x 1 and y 1 directions are similar. e same was true for the displacement response at Node 2. is was expected because the stiffness and damping of the coupling, bearings, and shafts in the model were independent of both the vertical and horizontal directions.

e Unbalanced Response of the System in the Angular
Direction. For the same conditions of speed (1200 rpm) and eccentricity (1 mm), the response of the system to an imbalance in the two angular directions (θ 1 , θ 2 ) is displayed in Figure 10.         Once more, the response is harmonic. It manifests as a sinusoidal curve in the time domain and a single peak centered at 1X in the frequency domain.

Vibration Response to the Combined Effects of Imbalance and Parallel Misalignment.
Parallel misalignment has a strong influence on the vibration response of all rotating assembly systems. In this section, a particular case of a 1 mm imbalance, combined to the simultaneous effect of a 1 mm gap of parallel misalignment, was simulated and analyzed.
e system was supposed to rotate at the same speed (1200 rpm). Figures 11 and 12 present the responses in the time and frequency domains to faults in the radial directions x and y.
In Figures 11 and 12 , the time-domain plots are no longer harmonic, but rather periodic. In the frequency domain, instead of a single peak, two peaks are now dominating the spectrum. e first one is the fundamental frequency of the rotating assembly (1X), and the second one is its second harmonic (2X).
Although the time domain's waveform changed considerably in both radial directions (x and y), one can see that the amplitudes of x 1 and y 1 at Node 1 and x 2 and y 2 at Node 2 are comparable. is can be explained by the same arguments as for pure imbalance, previously described in Section 3.1.1. Table 7 shows the peak amplitudes in the radial direction obtained from the spectrum.
If we admit that the first peak (1X) is related to imbalance and the second peak (2X) is related to misalignment, the results reported in Table 7 clearly show that the amplitude of 2X is much more sensitive to parallel misalignment. is finding confirms the results already known to and widely accepted by many practitioners and researchers (Section 1.3).
Moreover, the amplitude of 2X is almost constant in all directions. is is probably a result of the model properties being similar in those directions.

Effect of Varying the Amount of Parallel Misalignment.
e next step in our investigation of misalignment was to alter the degree of parallel misalignment and analyze the effects on the dynamic behavior of the rotor-couplingbearing system. To pursue this inquiry, a set of parallel misalignment values, ranging from 0.2 mm to 1.2 mm with a step of 0.2 mm, were included in the simulation model. During these simulations, the rotational speed was maintained at 1200 rpm. e results of this investigation are presented in Figure 13.
e results displayed in Figure 14(a) show that the amplitudes at Node 1 (for the right-hand side of the coupling) are higher than those at Node 2 (the left-hand side). Moreover, one can easily see that varying the parallel misalignment has practically no influence on the amplitude of the 1X peaks in all radial directions. However, the amplitude of the 2X peaks was found to increase linearly as the degree of parallel misalignment increased.

Vibration Response to the Combined Effect of Imbalance and Angular Misalignment.
e goal of this section is to investigate the vibrations patterns emerging from a faulty   system when angular misalignment is present alongside imbalance. Once again, the rotational speed and the eccentricity value were kept identical to the previous cases (1200 rpm and 1 mm of eccentricity). However, in this case, an angular misalignment of 1°was introduced. Figures 15-17 display the new simulation results.
In conclusion, an angular misalignment of 1°was found to generate two comparable peaks at 20 Hz (1X) and 40 Hz (2X). e peak amplitudes in the radial direction obtained from the spectra are shown in Table 8.
Once again, the combined effect of imbalance and angular misalignment had the same effect on the amplitude of the fundamental peak (1X) as it did when the imbalance was acting alone. However, when a small angle of 1°was added to the mechanical system, this had a significant effect on the amplitude of the second harmonic of speed 2X. e response in the angular directions θ 1 and θ 2 , under the combined effects of imbalance and angular misalignment, is displayed in Figure 16.
At first glance, several comments could be made about the results. First, the noticeable difference in the time waveform shapes between radial and angular directions. Second, compared with the case of pure imbalance, one can see that the response of the time domain in the angular direction increased considerably because of one-degree angular misalignment. Remarkable 1X, 2X, and 4X frequency components can be observed in the amplitudes of the spectrum, with the 2X component being the strongest. Table 9 summarizes the peak amplitudes in this scenario.
In conclusion, a comparison between the case of a single defect (pure imbalance) and the case of combined defects (imbalance and angular misalignment) shows that the angular misalignment did not have a noticeable  Figure 17: Effects of different degrees of angular misalignment on the vibration response. Table 9: Angular spectrum of the system including imbalance and misalignment.
Amplitudes (in degrees) effect on the amplitude of 1X. In contrast, the increase in the amplitude of 2X and 4X, noted in many previous studies (Section 1), was clearly visible. Unlike the case of parallel misalignment, the amplitudes of 2X and 4X were higher at Node 1 because of a more significant moment of inertia.

Effect of Varying the Angular Misalignment.
Our next step in investigating misalignment was to change the degree of angular misalignment and examine its effects on the dynamic behavior of the rotor-coupling-bearing system. A range of angular misalignment values was considered, from 0.2 to 1.2 degrees in steps of 0.2 degrees. For all simulations, the rotational speed remained at 1200 rpm. e simulation results are displayed in Figure 17.
Varying the angular misalignment had almost no effect on the amplitude of 1X. is peak is clearly related to imbalance. However, the angular misalignment could be recognized by the harmonics of the fundamental frequency (1X). e amplitudes of 2X and 4X increased as the degree of angular misalignment increased. In particular, the 2X component was the most sensitive, which confirms the findings of previous researches. Moreover, the amplitudes of 2X and 4X under angular misalignment were higher at Node 1 than at Node 2 because of the more significant moment of inertia in Subsystem 1 (equation (28)).

Effects of Changing the Shift of Phase between Imbalance and Misalignment.
e phase relationship between the imbalance and the misalignment forces is investigated in this section. To pursue this inquiry, a set of phase values, ranging from 0 to 180°, were included in the simulation model. During these simulations, all other running conditions were maintained constant (the speed at 1200 rpm, imbalance eccentricity at 1 mm, angular misalignment at 1°, and no parallel misalignment). e results of this investigation are displayed in Figures 18-20. Figure 18 displays the simulation response in the radial direction x 1 , computed both in the time and the frequency domains. A stepped variation of the phase shift, introduced between the imbalance and the misalignment forces, is found to alter the shape of the time waveform significantly. Such an effect is not inflected to one particular direction, but rather to all of them. e phase diagram between x 1 and x 2 is a typical example of such effect. e orientation of the loop, displayed in Figure 19, is directly related to the amount of phase shift between the imbalance and misalignment. Moreover, a close look to x 1 and x 2 graphs reveals an obvious variation of their amplitudes. e simulation displayed in Figure 20 shows that, on a cycle of 180°variation of the phase shift, there is around 15% variation on x 1 and x 2 amplitudes.

Effects of Changing the Rotational Speed.
To make our investigations into misalignment more thorough, we changed the rotational speed and examined the effects on the       rotor-coupling-bearing system's dynamic behavior in the presence of parallel and angular misalignments.

Effects of Altering Rotational Speed in the Presence of Imbalance and Parallel Misalignment.
In the established rotor-coupling-bearing system with a constant parallel misalignment fault of 1 mm, the rotational speed was changed from 1200 rpm to 2700 rpm in steps of 300 rpm. e results are shown in Figure 21.
As portrayed in Figure 19, the response in the radial directions (x 1 , y 1 , x 2 , y 2 ) at 1X increased significantly with increased rotational speed. is was mostly caused by the unbalanced force that acted on the system (Section 3.2). Furthermore, the increase in the 1X response was more rapid at Node 1 (x 1 and y 1 ) than at Node 2 (x 2 and y 2 ). is is probably because at Node 1, the system was more flexible than it was at Node 2. Moreover, the amplitude of 2X was constant despite the change in rotational speed. is was expected because the model of parallel misalignment force depends on the system's stiffness rather than rotational speed.

Effect of Altering the Rotational Speed in the Presence of Imbalance and Angular Misalignment.
e effects of changing the rotational speed from 1200 rpm to 2700 rpm in steps of 300 rpm on a rotor-coupling-bearing system with both imbalance and angular misalignment faults were examined. Angular misalignment of 1°was maintained throughout the process. e results are shown in Figure 22. Figure 22 shows that all peaks for 1X and its harmonics are sensitive to speed variation but to different degrees. In particular, the 2X and 4X amplitudes were found to be the most responsive to changes in speed variation, and they were more sensitive at Node 1(θ 1 ) than at Node 2(θ 2 ). is can be explained by the system being more flexible at Node 1 than at Node 2.

Effects of Changing the Coupling Type.
e effects of changing the flexible coupling type were investigated for systems with parallel and angular misalignments.

Effects of Coupling Type on a System with Imbalance and Parallel Misalignment.
e effects of changing the coupling type on the model response for combined imbalance and parallel misalignment of 1 mm were examined at a rotational speed of 1200 rpm. e results in the radial directions x 1 and x 2 are shown in Figure 23, and the results in the radial directions y 1 and y 2 are shown in Figure 24.
Based on the time response graphs, one can see that the amplitudes were higher for the white coupling, which was stiffer and was therefore unable to accommodate the deformations as in the case of the black one. In the frequency domain, the peaks located at 1X did not show a perceptible variation (a difference of only 2.4%). However, for the peaks located at 2X, the amplitudes for white coupling were 60% higher because of the difference in the stiffness.

Effects of Coupling Type on a System with Imbalance and Angular Misalignment.
e effects of swapping the white coupling for the black one were analyzed for a system with imbalance and 1°of angular misalignment, at a  rotational speed of 1200 rpm. Figure 25 presents the simulation results for this particular case. Figure 25 shows that changing the coupling type did not produce a noticeable difference in the vibration spectrum recorded in the θ direction. We expected the black coupling to absorb more vibrations, which would result in reduced amplitudes. However, the moments of inertia in the two couplings were small, and the difference between them was only minor, meaning that no effect of changing the coupling could be seen.

Validation of the Numerical Model.
To verify our proposed model, the simulation presented here was compared with a similar study by Wang and Gong [25] that provided a numerical simulation of a system that included parallel misalignment. is particular model was chosen because the authors provided enough data to allow a comparison of the two models. Wang and Gong simulated the vertical and horizontal displacement response at the coupling location. Some of the parameters had to be assumed, as the authors had not supplied them, and we drew on our prior expertise to make these assumptions. Figure 26 illustrates the simulated model of Wang and Gong [25].
Before we can compare the models, we needed to clarify several assumptions so we could implement the model with the available data. Each system was divided into two subsystems (with a specific mass, stiffness, and damping values) and analyzed with 12 DOF, as was the case for our model. e stiffness in the axial direction was assumed to be 10 times higher than the bending stiffness; the angular stiffness was assumed to be 1/1000 of the radial stiffness. ese were the ratios obtained here for a flexible coupling (Section 2.3). Table 10 presents the data provided by Wang and Gong.
Wang and Gong [25] provided data on the vertical and horizontal radial vibrations of the rotating system at Node 1.
eir system was tested with 1 mm of parallel misalignment at a rotation speed of 3800 rpm. We used our model with the parameters provided by Wang and Gong to obtain the response, displayed in Figure 27, in the time and frequency domains.
Similarly, for both studies, the amplitudes in the horizontal direction were higher than those in the vertical direction.
is aligned with our expectations because the bearings were fixed to the ground, preventing vertical movement. Table 11 compares the amplitudes of our model against that of Wang and Gong.