#### Abstract

To extract any information from a complex data vibration, the time-frequency technique represents a suitable tool and a better indicator of the overall asset health of a rotating machine. Fluid influences on a cracked driveshaft and dynamic analysis of a rotating Cardan shaft are discussed in this paper. The study of the forced vibration signals of the rotating Cardan shaft focused uniquely on the impact of the hydrodynamic forces near the cracked driveshaft and its transmission of power to a driven shaft. By merging each subsystem consisting of the breathing crack mechanism in a rotating shaft, the perturbation function between misaligned shafts and transient hydrodynamic forces results in a strongly nonlinear system of partial differential equations, which becomes complicated when analyzing the dynamic response of the Cardan system. In view of the complexity of the Cardan system, the instantaneous frequency identification based on synchrosqueezing wavelet transform filters is numerically used to provide a reliable indication of the rotating system equipment’s health. A conclusive dynamic analysis through the wavelet synchrosqueezing demonstrated that the Cardan joint and the hydrodynamic resistance forces have a high impact on the system’s vibration characteristics and significantly influence the transmission motion with a continuous increase of the driven shaft vibration amplitudes.

#### 1. Introduction

Flexible Cardan shaft systems, which facilitate the transmission motion of parallel or misaligned shafts, are frequently used in various kinds of industrial drivelines such as the automobile industry (heavy-duty vehicles), aircraft, or the global maritime industry. As a part of the rotating machinery, it is one of the critical components of a mechanical system and has been studied using different mathematical models. A comprehensive historical review of the model is illustrated in [1]. However, these models constituted of a single or double universal joint (Hooke’s joint, U-joint, or Cardan joint) are complex even in the linear case. Hence, discretization methods such as Taylor–McLaurin series, or a reliable numerical method called monodromy matrix [2, 3], and Lyapunov exponent or averaging method [4–6] have been proposed to estimate the self-excited vibration and transient responses of the drive system under parametric fluctuation as well. In the early years, the focus was mainly on the parametric instability in a multibody shaft system excited by universal joints for different complicated rotor-bearing systems [3, 7, 8]. The above study concentrated and worked well for a linear system. Various studies have been conducted to distinguish the parametric excitation in a Cardan shaft by constructing as much as possible a mathematical model which may indeed represent the actual physical system by considering the lumped mass model [9, 10]. It was demonstrated that, when considering such a simple rotating shaft-disc model, the dynamic vibration and the natural frequencies of the coupled shaft system are underestimated, leading to incorrect analysis and making the Cardan shaft oscillation one of the most commonly misunderstood issues [11].

Several theoretical and numerical works that illustrate various nonlinear phenomena and parametric excitation were developed [12–14], emphasising nonlinear parameters. However, nonlinear effects such as breathing cracks on connected shafts by a Hooke’s joint are crucial in Cardan shaft dynamic analysis and have been analysed in [14–16]. These effects were mainly due to the nonlinear characteristics of a time-varying stiffness, the disturbance of the fluctuated Hooke’s joint, and the mass unbalance of the rotor system. Because of their compactness, noise, and high load capacity, many researchers have conducted the Cardan shaft analysis since Porter [9, 17] to derive a differential equation for predicting the synchronous critical resonance range associated with such a coupled rotor system. These defective phenomena mostly occur on the driveshaft rather than the driven shaft, which instead manifests high vibration [18]. As the driveshaft is aligned with the motor, the driveshaft becomes more sensitive to periodic excitation due to rotational speed, and Hooke’s joint disturbance may further lead to early failure. Failure of such a system can result in a sudden disruption of the power supply between its source and the consuming device. Therefore, research should be conducted to analyse and anticipate multiple faults which can be generated due to the defectiveness mechanism of the Cardan shaft. Research in fluid-rotor interaction is going on for more than 60 years. Despite a comprehensive list of references provided by Ibrahim [19], there are hardly any study reports explicitly regarding the failure of the Cardan shaft, especially when the system is partially submerged in a fluid medium. To the authors’ knowledge, the dynamic analysis of a transmission model using a Cardan joint partially submerged by a viscous fluid has not been analysed yet. For the viscous fluid-driveshaft, the hydrodynamic behaviour is much more complicated. It is found in the literature that due to the complexity of the mechanism of the Cardan shaft, various sources of excitation such as the hydrodynamic forces and unbalanced shaft that can be encountered in practice have been neglected. Thus far, research on the Cardan shaft did not as well consider all necessary degrees of freedom as discussed in [14, 16]. As demonstrated by research in practice, Cardan shaft failures are often manifested as being composite (various elements or parts instead of single, but extracting the features of composite defects is much more difficult than single ones) [20, 21]. It is, therefore, logical to analyse the dynamic mechanism of the shafts coupled by a single or double Hooke’s joint, which considers the effect of the lateral and torsional deflection of the shafts, the fluid-driveshaft interaction, the breathing crack mechanism inside the fluid, and the unbalanced effect on the shafts.

In the current work, a model for analyzing the high vibrations of a Cardan shaft focusing on coupling malfunctions and extracting the features of a cracked driveshaft system coupled to a single Hooke’s joint in a viscous fluid is developed. Earlier works [16, 20–23] have shown that resistance fluid has a significant impact on rotor system behaviour, and the wavelet technique is an effective signal processing tool for crack detection, which will be used here for crack feature extraction in a fluid medium. The use of the rotating reference frame will assist in a better understanding of the nature of the shaft whirl and facilitate the shaft modelling and the fluid-driveshaft coupling. Although more degrees of freedom are adopted for the whole twin-rotor system using the Lagrangian approach, some reduction procedures, as mentioned in the preliminary investigation [22], will be implemented for the fluid-driveshaft interaction without any loss of accuracy. The first part of the work deals with the formulation of the problem, which includes crack effects and fluid resistance forces. Gyroscopic effects have been ignored because of the system parameters under consideration. Once an adequately discretized model is established, the numerical procedure is examined, presenting the relevant results. Various nonlinear effects, such as breathing and crack-fluid interactions, are investigated. As the impact of noise, the interaction between misaligned components and the transmission of vibration energy from the drive to the driven shaft or driveshaft to the viscous fluid contributes to the fact that the faults of the Cardan shaft tend to be complicated. Because of this, the synchrosqueezing wavelet algorithm is associated with the classical frequency technique to conduct the research on extracting the vibration characteristic of an unbalanced and cracked primary shaft under different modes of compound faults.

#### 2. Modelling Approach and the Governing Equation

Consider the system representing the schematic Cardan shaft system which includes two unbalanced shafts connected by single Hooke’s-joints, as illustrated in Figure 1. The model of the coupled rotor system considers the nonlinear variation of the stiffness of the primary shaft and the influence of viscosity on the breathing mechanism, as shown in Figure 1. The primary and secondary shafts were considered flexible and had torsional stiffness. The driveshaft system is spinning with an average constant angular velocity of . The angular misalignment between the shafts is defined by . The Cardan shaft vibration model is theoretically established under the assumption that the gyroscopic effects due to the rotating disc and the anisotropic bearing effect may be ignored without losing a good understanding of the fundamental behaviour of the system. Only the lateral and torsional vibration of a two-shaft should be considered. Thus, the interconnected rotor system via the single Hooke’s joint can be modelled as a De Laval shaft system. The two rotors are simplified into two identical shaft discs with an eccentric mass attached to each disc. In Figure 1, *O*_{1} is the motor-driveshaft center, and *O*_{2} is the driven shaft disc center. It is considered that the two shaft-discs modal masses are which carries an eccentric mass and with unbalanced mass, respectively, at the midspan of each shaft.

The coordinate diagram used to establish the governing equation of the model is shown in Figure 2, where at time *t*, the centers of the shaft discs are at points and which are the rotating coordinates system attached to the respective discs. The origins and are the intersections of the disc plane and the line between the bearing centers; *e* = *m*_{u} is the eccentricity of the disc, and *m*_{u} is the lumped mass of the rotor system.

##### 2.1. Coupling Model and Lagrange’s Formalism

The theoretical model of the coupled rotor system consists of seven degrees of freedom constituted of four lateral displacements of the two shafts at the location of the discs (, ) and (, ), correspondingly, two rigid-body rotations , , and two torsional deflection angles, and . Two stationary reference systems, (, , ) of the input shaft and (, , ) of the output shaft as illustrated in Figure 2, represent the total lumped mass system. Coordinate is attached to the motor while is fixed to the center of the secondary shaft such that is coincident, respectively, with the motor input shaft and the central axis of the output shaft bearing, as shown in Figure 1. The two global position vectors are defined for the eccentric mass *m*_{u}. The pairs of vectors , characterize the centers of the shaft masses and , respectively. The rotating kinetic energy of the rotor system is expressed aswhere the specific inertias and their actual displacements are as follows: (mass moment of inertia of the motor) experiences rigid-body rotation only; represents elastic deformation of front shaft spring , that is, superposed on the elastic rotation of ; represents elastic deformation of output shaft spring , that is, superposed on the flexible rotating output shaft, that is, .

The kinetic energy of the primary shaft is given as [21]

The kinetic energy of the secondary shaft is constituted as

The total kinetic energy of the Cardan shaft system combines the partial input shaft1 kinetic energy and the partial output shaft 2 kinetic energy given by

##### 2.2. Global Cardan Shaft Elastic Strain Energy

The potential energy includes shafts 1 and 2 bending strain and torsional strain energy aswhere are the respective stiffness coefficients connected to the system generalized coordinates, and and are the shaft’s torsional stiffness coefficients.

##### 2.3. The Total Cardan Shaft Rayleigh Dissipation Expression

The Rayleigh dissipation function was derived using the viscous fluid’s modal damping as follows:where, are, respectively, the degree-of-freedom dashpots of damping coefficients, and and are the torsional vibration damping of the first and second shaft.

##### 2.4. Expression of the Governing Equations of the Cracked Cardan Shaft

By substituting the total energy function defined in equations (4)–(6), and performing all relevant derivations using Lagrangian formalism through each generalized coordinate, the elements of the respective matrices and vectors are then given as follows.(i)For the generalized coordinate , the variable mass elements were expressed as(ii)For the generalized coordinate , the variable mass elements were expressed as(iii)For the generalized coordinate , the variable mass elements were expressed as(iv)For the generalized coordinate , the variable mass elements were expressed as(v)For the generalized coordinate , the variable mass elements were expressed as(vi)For the generalized coordinate , the variable mass elements were expressed as(vii)For the generalized coordinate , the variable mass elements were expressed as where that depends on and is a small perturbation parameter modelled on the basis of Hooke’s joint kinematics and expressed in detail in [23] and

Applying the same principle of Lagrange’s equation, the stiffness elements of the twin-rotor system are expressed based on equation (5) by

The derivation of the damping expression of the cracked rotor system is obtained from equation (6), and the whole damping components of the Cardan shaft system can be expressed aswhere and are, respectively, the stiffness and damping ratio of the rotor system. To identify the excitation characteristics of the breathing crack (deflection and misalignments) in a fluid medium, the system is operated with the help of an electric motor with an oscillatory input torque. The applied torque treated as an exciting external force is a major component of the machine thrust and depends on the angular velocity; hence, the output torque will undergo a cyclic change similar to that of the driven shaft velocity [17].

The system response for the intersection angle () and the input motor torque acting on the system at maximum input rotational speed = 1500 rpm are presented in Figure3(a).

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##### 2.5. The Breathing Function of a Transverse Crack Introduced in the Model

The alternating mechanism of crack opening and closing in static shaft deflection is synchronous with the rotational speed, as illustrated in Figure 4. The change of the neutral axis is taken into account in calculating the stiffness matrix of the cracked shaft. The recently published breathing mechanism introduced in [24, 25] is used in this article, and its complete derivation will not be repeated.

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Referring to Figure 4(a), the whole area moments of inertia of *A*_{1} about the fixed and axes are denoted by and , respectively, which are constant during the rotation and are given bywhere is the moment of inertia of the cross-sectional area during the fully closing crack. The moment of inertia of the cracked cross-section are and about the fixed and axes defined in [25] for are expressed as:where . The cross-sectional moments of inertia of *A*_{1} about the centroidal and axes are time-varying quantities expressed in terms of and as

The transverse crack’s breathing functions and can be summarized as follows [25]:where the positive even numbers controlling the breathing function deflections are denoted by and , . The instantaneous area moment of inertia values around the principal centroidal directions, and , are approximated using a closed Fourier series expansion by

As a result, the whole time-varying stiffness matrix part of the response of the driveshaft is approximated aswhere *L* characterizes the length of the rotor, denotes the shaft elastic modulus , and are the estimated moment of inertia related to the fixed and axis, respectively. Since the area moment of inertia of the cross-coupling cracked driveshaft is symmetric about the axis, and from the results presented in [25] where the product of the moments of inertia of the cracked shaft around the *X*_{1}*Y*_{1} axis is much smaller compared to the moments of inertia in the *X*_{1} and *Y*_{1} directions, therefore, . The dynamic response of the cracked driveshaft is displayed in Figure 3(a).

#### 3. Introduction of Hydrodynamic Forces into the Model

Contrary to the previous study published in [22], the behaviour of the fluid around the disc-related eccentric mass is considered in this investigation. Fluid equations that reflect the viscous fluid’s resistance to system motion consider the driveshaft-disc elasticity and viscous damping. Assuming a laminar flow around the driveshaft (low Reynolds number), the fluid equation could be used to establish the pressure distribution and hydrodynamic force influencing the driveshaft excitation. The fluid in contact with the driveshaft surface spins at a uniform angular speed . The laminar flow around the rotating driveshaft-disc rotating in a polar coordinates system is reduced aswhere symbolizes the fluid pressure, is the Laplacian operator, the dynamic fluid viscosity, and represents the density of the viscous fluid density. The velocity component and expressed in the polar coordinate and , respectively, can be defined in terms of the stream function as

The analysis includes the acceleration potential , in which gradient gives the acceleration vector:

If equations (26) and (27) are introduced into equations (24) and (25), along the radial and circumferential coordinates and , then the linearized expression of equation (28) gives

The velocity potential function can be represented as the sum of two components using two stream functions, such as

The partial stream function is, therefore, written as

The constants A and B are determined from the boundary conditions, and *K*_{1} is a Bessel function of the second kind. Because the fluid influences the eccentric mass , is assumed to be a weak time-varying function. In other words, the simplest approximation of the distance from the center of a reservoir filled with a viscous fluid iswhere is a perturbation function of the fluid around the eccentric mass, and and are the vector component projected in *x*_{1,}*y*_{1} frame. The fluid velocity contact and of the fluid at the driveshaft parallel to the direction of shaft oscillation with amplitude is given in the following form:

The total stream function is then expressed in the following form:

The boundary parameters between the surface of the driveshaft and the fluid are equivalent to those of the shaft at , resulting in hydrodynamic pressures of

The hydrodynamic pressure distribution is evaluated by integrating the fluid pressure distribution over the intended submerged input shaft length *L* as follows:

By integrating by part, the first term of equation (36) and applying relations (34) and (35),

Using an asymptotic Bessel function expansion with significant input, *K*_{0}(*kR*) and *K*_{1}(*kR*) are defined as Kelvin’s functions. Over the predicted area the and direction, the resulting resistive hydrodynamic force acting on the driveshaft can be lowered as follows:

The fluid centrifugal forces and arising at the disc section with eccentric mass, which acts outwards in the radial direction, are represented as

Fluid mass expressions and viscous damping coefficients per unit of fluid depth are as follows:where represents the dynamic effects of the displaced mass of the fluid along the driveshaft length *L;*; is the Euler constant; and is a dimensionless quantity representing the average amplitude of the driveshaft vibration and can be found in [22]. The change in minimum hydrodynamic thickness is shown versus time for high input shaft speed and low Reynolds in Figure 5(b). The analysis involving hydrodynamic forces is only valid for laminar flows for which the Reynolds number does not exceed 2000. As expected, the oscillation frequency of the Cardan shaft and the fluid kinematic viscosity bring the response of the fluid to values bounded at the resonance.

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The dynamic equation of the Cardan shaft model is obtained by incorporating the fluid mass and nonlinear viscous damping factors into the Cardan system equations of motion. The dynamic equation of the unbalanced and cracked Cardan shaft system in a viscous fluid was reduced to a system with seven degrees of freedom by considering the nonlinear disturbance caused by the Cardan joint, the discrete representation of the coupled rotor model, and the time-varying stiffness induced by a transverse crack.

It is observed that the first part of equation (41) obtained through derivation of the governing equations by applying the Lagrangian equations represents, respectively, the Carioles couple vector corresponding to the quadratic velocity of the exciting torque system (), the velocity vector of the nonlinear self-excited disturbance through the Hooke’s joint () associate with the fluid forces () fully expressed in equation (39).

When the viscous hydrodynamic force is combined with the nonlinear breathing crack, the dynamic equation of the coupled twin-rotor system becomes highly nonlinear and more complicated. The nonlinear wavelet synchrosqueezing transform (NWSST) algorithm detects the oscillation feature of instantaneous frequency (IF), which can qualitatively describe the transient stiffness’s oscillation feature. It extracts the characteristics usually hidden in weak components and searches for the strongly oscillated transient stiffness phenomenon. NWSST to the problem of nonlinear oscillations evoked in [26, 27] is used to detect a crack in a viscous medium. A generic description of NWSST is also provided, along with an improved peak detection approach for estimating the IF of the Cardan shaft system.

#### 4. Feature Extraction of Cracked Rotor System Using Synchrosqueezing Techniques

##### 4.1. Nonlinear Wavelet Synchrosqueezing Transform (NWSST)

A brief algorithm of the IF extraction based on NWSST for a given time-varying signal *x*(*t*) is presented. To extract the IF, consider a pure harmonic signal wave . Since the IF of the vibration signals generated by the cracked drive rotor system offers a fast and highly oscillatory feature, NWSST is developed to extract this feature hidden in the vibration signals. The short-time Fourier transform (STFT), which naturally applies Fourier transform to a signal and its inversion, can be defined aswhere is a parametric form of window function with a window size of , and this window function facilitates the division of the nonstationary signal into small parts which are supposedly stationary. For the time-shift derivative of , the following derivation is obtained:where is the STFT of computed with the window function , which is the first differential of the initial window . A larger window produces a coarse representation in time but fine in frequency. The density of an energy spectrogram is defined as

Similarly, a description of is expressed as the ratio of the two time-frequency representations and , that is,

For a multicomponent periodic signal , based on the theorem in [27], giveswhere and are corrective terms, is the signal amplitude, is the instantaneous phase of the signal. If the window size is sufficiently small, then the ratio can be approximated as

For a mono-component signal , according to Parseval’s theorem, it can be noted that is equal to which is a constant value. Hence, becomes

As seen in equations (47) and (48), it should be noted that the TF representation is only relevant for the phase but is independent of the signal amplitude . Moreover, the value of its module will rapidly increase when , and the value of the ratio at frequency is as follows.

Additionally, when , the value of its module rises substantially, and the value of the ratio at frequency becomes

Thus, can be used to detect the hidden signal even with a weak amplitude.

The signal , which uses the summation of sinusoidal functions to model multicomponent frequency modulation signals, can be characterized by the expressed model:where, respectively, the amplitude and phase *A*_{k}(*t*), *ϕ*_{k} (*t*) are defined according to the analytical signal . The instantaneous pulsation of the component is the first derivative of its phase:

Thus, the IF is . Since the changes of and are usually much slower than the change of , the component can be considered locally as a harmonic signal with amplitude and frequency . In this concept, if *K* *=* *1*, the signal is called a single signal, and if *K* *>* *2*, the signal is called a multicomponent signal. The model defined by Equation (50) applies for multicomponent FM signals and allows the modelling of *K* time-varying IF structures.

In order to more intuitively describe the effects of dynamic parameters on IF and quantify the relationship between oscillated IF and dynamic parameters, the average range of oscillated IF is defined in the time domain as follows. Set the maximum and minimum amplitude of the IF in the *j*^{th} cycle for the oscillated IF are *IF*_{max,j} and *IF*_{min,j,} respectively, and then, the peak-peak value of the oscillated IF in the *j*^{th} cycle is obtained by [26]

Hence, the mean range of the oscillating IF becomeswhere *n* is the number of cycles for the oscillating IF. When combining the hydrodynamic force with the nonlinear perturbation function for a specific crack depth, the oscillated feature of the IF is characterized by using the NWSST. The periodically oscillated feature of the IF taken as an index to qualitatively describe the strongly oscillated transient stiffness of the rotor system and extract the crack’s symptoms is applied to the dynamic equation of the coupled oscillation.

##### 4.2. System Parameters Used for Numerical Simulations

In order to simulate the operating conditions of the Cardan shaft system in a viscous fluid and extract the feature of the time-varying stiffness induced by the transverse crack, the physical and geometric properties listed in Tables 1 and 2 are used here. The equivalent moment of inertia, *J*_{D1} and *J*_{D2}, of the propeller shafts was 0.62 kgm^{2}, the engine equivalent inertia, *J*_{M}, of the system was 12.091 kgm^{2}, the equivalent torsional rigidity, *K*_{T1} and *K*_{T2}, was 5.8 kNm^{−1}/rad. The torsional attenuation, *C*_{T1} and *C*_{T2}, was 15.77 Nms/rad. The simulation is conducted through two scenarios: firstly, without viscous fluid around the cracked and misaligned shaft, and secondly, with hydrodynamic force surrounding the cracked driveshaft.

##### 4.3. Simulation and Dynamic Analysis of the Unbalanced Cardan Shaft System

The Runge–Kutta–Fehlberg method, generally suitable for numerical solutions of ordinary differential equations, is used to solve the system equations using the fourth-order adaptive step and create a new state space. This method applied in Eq. (48) will be used to calculate the nonlinear dynamic responses of the rotor system. The speed of rotation is among the key parameters influencing the dynamic characteristics of the Cardan shaft system so that the dense fluid oscillates precisely at the same vibration frequency . The resulting rotor velocity profile and motor torque are shown in Figures 3(a) and 3(b), respectively, to obtain the system response over a wider range of vibration frequencies. The simulation analysis is performed at a chosen maximum rotational speed of 1500 rpm to mimic the real operation of the motor torque. The dynamic response of a cracked driveshaft system, which fluctuates periodically at the rotating frequency, as shown in Figure 5(a), may offer an indicative index to monitor the “breathing” behaviour of the cracked system under the influence of fluid forces. The cracked driveshaft depth is chosen such that .

In this section, the oscillation feature of an unbalanced vibrating rotor system with a transverse crack was obtained at the rotating speed of 1447.74 rpm. Due to the strong nonlinearity of the proposed system, the numerical analysis is carried out using the fourth-order Runge–Kutta method with a set of 0.001 s time steps used to refine the response.

To validate the model proposed in this study, the numerical simulation parameters were chosen to be the same as those of the experiment. The responses were displayed using the shaft displacement, orbits, and frequency spectra. Vibration readings were performed in lateral displacement, frequency, and orbits with a maximum driveshaft vibration. The overall selected plots are displayed within this work for ease of reading and effective comparison. Figures 6(a) and 7(a) show the influence of the fluid on the motion of the center of the driveshaft. The displayed cracked driveshaft orbit combines some swirling loops with a maximum amplitude of 3 × 10^{−5} mm, which are more complex and disorganized when observing the driven shafts. Moreover, fluid around the transverse crack significantly reduced the distorted loops to a quasicircular circle at a low amplitude of 2 × 10^{−7} mm. This implicates that Hooke’s joint disturbance highly contributes to the distorted elliptical-shaped movement of both uncracked driven shafts with a slightly amortized vibration and disturbed loops of the immersed cracked driveshaft, as shown in Figure 7(b). In the orbit analysis of Figures 6(a), 6(b), 7(a), and 7(b), there are big differences between an *X* and Y displacement scale. For example, the maximum peak-to-peak amplitude in the first chart in the *X*-direction is about 1.5 × 10^{−8} m, and the maximum peak-to-peak amplitude in the *Y*-direction is about 6 × 10^{−5} m. The vibration in one direction is 400 times higher than in the other. The differences may be related to the mass, stiffness, or damping of the supporting structure in different directions.

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When the crack parameters are considered, the lateral deflection, their spectrum, and applied fluid forces have not changed the observations mentioned previously. Analysis of Figures 6(c) and 6(e) shows that the lateral deflection amplitude and its corresponding frequency spectrum amplitude become more prominent with the presence of multiple proportional subharmonics. It is known and confirmed in [22] that a transverse crack introduces components with frequencies of 2*X*, 3*X*, 4*X*, and more, which are considered to have lower amplitudes. However, in the present case, the amplitudes of the subharmonics are disproportionate with a slight unbalance magnitude in the case where the cracked driveshaft is coupled to the Hooke’s joint. These higher harmonic amplitudes become less intense as fluid damping amortizes the driveshaft’s vibration, which occurs as the system approaches resonance. The driveshaft and coupling phenomenon provoked additional components with or without hydrodynamic forces and generated some occurrences of weak subharmonics near the 1*X* resonance peak. These feature results can be verified in the driveshaft signals of Figure 7(e) and the driven shaft signals of Figure 7(f), where the highest amplitude is observed at the high spot synchronous frequency 1*X* is a resonance speed at approximately 100 Hz. At the same time, the other components (1/*X*, 2*X*, and 3*X*) emerge in higher harmonics with smaller amplitude. A partial conclusion derived from the spectra given in Figures 6 and 7 is that since the Hooke’s joint is primarily considered, the period of onset of the subharmonics and the development of the 2*X* and 3*X* harmonics are the primary result of crack breathing regardless of the influence or not of the viscous fluid around the cracked driveshaft. However, the amplitude of superharmonic growth is faster in the motor shaft than the amplitude of the driven shaft, which may indicate the existence of a crack in the rotor system, and the transmission of motion to the secondary shaft is done with crack fault symptoms. Observing both results in Figures 6 and 7, nothing significant can be seen in terms of amplitude or frequency defaults; the spectrum features of both shafts look almost alike and can be misinterpreted in terms of fault discretization and Hooke’s joint mixed mode. In order to isolate the response signals at the primary natural frequency of the system and discretize the crack features in these signals, a synchrosqueezing technique is applied to either a low-pass or a band-pass filter to the twin-rotor signals with specified cut-off frequencies. The program detrends the filtered signals to remove residual linear trends and computes the synchrosqueezing wavelet transform of these denoised signals.

In the case of the present analysis, the signal evolution through the time-frequency representation was used to obtain a higher resolution TF analysis. The SWT technique extracted the maximum energy TF ridge cycles per sample. A series of improved IF methods were adapted to extract the spectral distortion of the frequency through the rotor signal, as shown in Figures 8(a), 8(c), and 8(e). It is observed that there is a significant change in terms of weak frequency extraction and energy concentration in the extraction of the crack feature using the SWT. The normalised power spectrum of the cracked driveshaft increases drastically compared to the driven shaft response, where the features of the breathing crack are significant and distinctively perceptible. The synchrosqueezing power spectrum of the driven shaft is shown in Figures 8(b), 8(d), and Figure 8(f) and includes both unbalance and cracking effects. The higher first harmonic peak corresponds mainly to the disequilibrium effect and the associated *N* × superharmonic speed peaks are precisely due to the combination of misalignment and the breathing crack. These figures also show that the energy concentration of the system fluctuates strongly in the vibration start-up period at different frequencies [260 Hz, 400 Hz, and 490 Hz]. A stable vibration remains after a transient vibration process (Figure 8(f)). The fluids attached to the cracked driveshaft because self-excited oscillations and significant disturbances different from those observed in nonsubmerged rotor models. It is noticed that the amplitude of variations of the driven shaft decreases considerably, and the peak amplitudes are only about 1/4 of those in the case when a driveshaft without fluid force was used. This means that the driveshaft with more substantial damping will minimize the amplitude of variation of both shafts. As a result, it is observed that the differences in dissipated energy and instantaneous frequency are very symmetrical and very close to being periodic. If the evolution through time of the peak at 0.45 Hz is followed, it can be seen that despite the effect of the fluid, there remains the primary frequency and some peaks of subharmonics, characteristic, respectively, of the unbalance and the presence of cracks around a defined frequency.

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Moreover, the curve in Figure 9(c) shows that the driveshaft sudden jumps as if the nature of the system’s behaviour was changing from one mode to another, which corresponds to chaotic behaviour. Figure 9(e) displays an attenuated signal in the presence of viscous fluid forces. Some high-frequency components appear in the NSTFT scalograms between 140 Hz and 400 Hz and above 50 Hz. The overall response of the vibrating system obtained using the SWT shows that the differences in the concentrated energy of the system with or without viscous fluid are significantly different. These results should agree reasonably well with the expected physical behaviour of the actuator. Therefore, an experimental setup should be conceived for the present model validation in various twin-rotor transient motion operations.

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Figure 10 shows the TFR obtained by the NWSST, in which the distance of frequency shift is 500 Hz. In Figure 10(a), various high-amplitude components (bands) with frequencies ranging from 0 to 150 Hz are depicted. It can be seen that the amplitude of IF is about peak rad/s, and its fluctuating frequency is around 55 Hz, which is consistent with the set parameters. As a result, the RPM range was separated into three band segments to observe the peak evolution and identify crack features in system. A defect (crack) in a dense medium is indicated by the existence of a small band of concentrated vertical peaks in Figure 10(b). Their expansion is more pronounced and more robust along the aperiodic axis. This finding indicates that any change in fluid characteristics that reduces crack breathing throughout the rotor system reduces the likelihood of crack shaft failure. Synchronization techniques are found to be very sensitive to the dynamic characteristics of the cracked rotor and the properties of the fluid, especially for low Reynolds numbers. Considering the increasing interest in fluid-structure interaction (FSI) problems, this work can be extended to instability analysis and FSI-induced forced-excited oscillation of the flexible rotor in uniform flows at high Reynolds numbers. The latest research study clearly indicates that control of sound transmission through structures is important in the noise analysis of vibrating structures [27]. Therefore, the direct relationship between sound transmission from the flow rate fluctuation and the natural frequencies of the oscillating driveshaft can be investigated in future work using a robust controller, namely a self-adjusting boundary layer published in [28]. This technique is against uncertainties in sensors, and actuators control the noise radiated by the flow and prevent the occurrence of vibration phenomena from the rotating transmission shaft.

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#### 5. Conclusions

In conclusion, from the twin-rotor analysis in a viscous fluid using the classical fast Fourier transform technique, nothing major except the common subharmonics 1*X*, 2*X,*…, 5 *X* as defaults were detected at different locations associated with the particular crack phenomena. The overall vibration amplitude of the submerged cracked driveshaft was so low (*μ*m) and appeared to be at an acceptable level of vibration and frequency. However, for any change arising in the system that might be linked with a crack vibration and a Hooke’s joint disturbance in the system, the detection of the different phenomena has been assessed qualitatively with three different time-frequency techniques. Subsequently, the nonstationary response of the system was characterized by an instantaneous frequency (IF) and the nonlinear wavelet synchrosqueezing transform (NWSST) at the first critical speed. The synchrosqueezing program has proven to be able to visualize the phenomenon captured, showing at which point and frequency band it was detected. In addition, the features of the breathing crack phenomena seen from the 3D trend graphs with a low-frequency response were able to quantify each phenomenon and evaluate the percentage of motion transmitted to the second shaft. It is concluded that the vibration generated by the coupling joint and the high fluctuation of the connected rotors is directly proportional to the transient stiffness of the flexible driveshaft. However, the hydrodynamic forces around the cracked driveshaft attenuated the excitation of the driveshaft by decreasing its vibration amplitude. The vibration amplitude will gradually decrease if the fluid viscosity increases with decreasing fluid density; that is, more fluid density means less vibration. The results recommended that although numerical techniques can predict the influence of the submerged driveshaft crack, the measured data could show much better results. This issue convinced the authors to carry out practical tests in the laboratory and present their results according to the experimental setup.

#### Appendix

The governing equations were derived using the Lagrangian formalism in Equation (41). The coupling elastic deformations across Hooke’s joint are represented by a system of seventh-order inertia equations with nonlinear excitation as follows.(1)Derivation with respect to is given in their final forms as follows::(2)The velocity vector representing the nonlinear self-excited couple related to is(3)Derivation with respect to *;*(4)Derivation with respect to ;