Vibration Characteristics of Rotor System with Loose Disc Caused by the Insufficient Interference Force

The rotating parts looseness is one of the common failures in rotating machinery. The current researches of looseness fault mainly focus on non-rotating components. However, the looseness fault of disc-shaft system, which is the main work part in the rotor system, is almost ignored. Here, a dynamic model of the rotor system with loose disc caused by the insufficient interference force is proposed based on the contact model of disc-shaft system with the microscopic surface topography, the vibration characteristics of the system are analyzed and discussed by the number simulation, and verified by the experiment. The results show that the speed of the shaft, the contact stiffness, the clearance between the disc and shaft, the damping of the disc and the rotational damping have an influence on the rotation state of the disc. When the rotation speed of the disc and the shaft are same, the collision frequency is mainly composed of one frequency multiplication component and very weak high frequency multiplication components. When the rotation speed of the disc and the shaft is close, the vibration of the disc occurs a beat vibration phenomenon in the horizontal direction. Simultaneously, a periodical similar beat vibration phenomenon also occurs in the waveform of the disc-shaft displacement difference. The collision frequency is mainly composed of a low frequency and a weak high frequency component. When the rotation speed of the disc and the shaft has great difference, the collision frequency is mainly composed of one frequency multiplication, a few weak high frequency multiplication components and a few low frequency multiplication component. With the reduction of the relative speed of the disc, the trajectory of the disc changes from circle-shape to inner eight-shape, and then to circle-shape. In the inner eight-shape, the inner ring first gradually becomes smaller and then gradually becomes larger, and the outer ring is still getting smaller. The obtained research results in this paper has important theoretical value for the diagnosis of the rotor system with the loose disc.

comparing critical speed and vibration mode. Zhang et al. [3] proposed a rotor model with nonlinear oil film force, and solved this mode by the numerical method of Runge-Kutta, and obtained frequency characteristics of the rotor system. Lu et al. [4] proposed a coupling model of the rotor system with looseness and rub-impact faults, and discussed the influence of the looseness stiffness and the clearance between rotor and stator on the dynamic characteristics of the rotor system based on the nonlinear finite element method and the contact theory. Based on nonlinearity measure, Mian et al. [5] proposed an evaluation method of the pedestal looseness in bearing-rotor system with piecewise-linear stiffness, damping and nonlinear elastic force under constant rotational speed at constant speed. In Ref. [6][7][8][9], the corresponding model was proposed, and the dynamic characteristics of the proposed model were analyzed. Liu [10] proposed a dynamic model of looseness-rubbing coupling fault in the rotor-bearing system with dual-disk and threesupport. Wang [11] proposed a model of rotor-supportcasing in the rotor system with support looseness fault. Yu [12] proposed a dynamic model of rotor-bearing system with loose support. Chen [13] proposed a dynamics model in a rotor-ball bearing-stator coupling system with rotor unbalance-loose coupling fault. For the complex movement of a single loose bearing pedestal rotor-bearing system, Zhang [14] proposed a nonlinear non-steady oil-film force model with short bearings. Cao et al. [15] proposed a nonlinear dynamic equation of rotor-bearing system with looseness fault between the pedestal and the casing. Xu et al. [16] proposed a dynamics model of dual rotor system with loose bearing pedestal based on the dual rotor structure of experiment rig. Yang et al. [17] analyzed the vibration features of the rotor system with respect to the effects of geometrical nonlinearity, rotor-stator rub and pedestal looseness, and revealed the change rules of resonant characteristic and rub region under different loose stiffness. Cao et al. [18] proposed a vibration differential equation with piecewise-linear stiffness by considering the system with one-side support looseness and the rotor with unbalanced extraneous exciting force. Cao et al. [19] proposed a dynamic model of rotor-bearing-pedestal system to investigate the vibration characteristics and stability due to fit clearance. However, the above researches ignore the looseness fault in the disc-shaft system. Because the working environment of the disc in the rotor system is complex, and the working time is also long, the looseness of the disc-shaft easily occurs. When the interference between the disc and shaft is insufficient, the relative sliding of disc-shaft will be easily caused. This relative sliding has an influence on the dynamic characteristics of the rotor system with a loose disc. At present, some progress has been made in the looseness of rotating parts of rotor system [20][21][22][23][24][25]. Behzad [20,21] proposed the model of the looseness of disc-shaft system with large clearance based on the hypothesis that the rotating speed of the disc is constant, and the disc and the shaft are always in contact. However this model ignored the collision and friction of disc-shaft, and assumed that the disc and the shaft rotated at the same speed. Obviously, the engineering practice does not meet these requirements. In practice, the disc-shaft looseness in the rotor system generally also occurs when the clearance is small, even no clearance [22]. In our works [23,24], a dynamic model of rotor system with the clearance between the shaft and the disc is proposed in Ref. [23],and a rotor system model with disc-shaft looseness with non-steady-state oil film force is proposed in Ref. [24]. Wei [25,26] studied the disc-shaft looseness fault in rotor system caused by the excessive speed of the shaft. However these research results in disc-shaft system do not consider the influence of the discshaft looseness fault on the motion state of the rotor system with the insufficient interference force.
Therefore, the motion differential equations of the rotor system with loose disc caused by the insufficient interference force is proposed based on the contact model of the disc-shaft system with the microscopic surface morphology, and the numerical simulation and experiment test have been completed. The influence factors of the motion state of the disc, the vibration characteristics of the disc, the variation of the displacement difference of disc-shaft, and the trajectory of the disc. These obtained outcomes provide theoretical support for the fault diagnosis of the disc-shaft system with slight loose disc in engineering practice. Figure 1 is a schematic diagram of disc-shaft contact [27]. In Figure 1, the outer ring represents the reference plane of the inner diameter of the disc and the inner ring Then, at any angle φ , the clearance between the disc and shaft is as follows When the height of micro-convex, which is represented by z, is higher than the clearance between the disc and shaft, which is represented by �d(φ) , The normal force generated by the micro-convex can be obtained [24] where γ 1 is the radius of the disc, γ 2 is the radius of the shaft, γ is the radius of the micro-convex, E 1 is the elastic modulus of the disc, E 2 is the elastic modulus of the shaft, and E is the elastic modulus of the micro-convex. Suppose that the normal distribution function of z on the contact surface is φ(z) , the mean of normal force generated by a single micro-convex at any angle ϕ is expressed as

Contact Model of Disc-Shaft System with the Microscopic Surface Morphology
(2) (3) Let η is the density of the micro-convex, r 1 d(ϕ) is the length of the micro-convex, then the number of microconvex is ηlr 1 dϕ on the micro-area segment with a width of l. The total force on the entire contact surface can be obtained by integrating Eq. (7) Since the contact force is symmetric with respect to o 1 o 2 , Eq. (8) can be simplified as follows The radial force generated by a single micro-convex at any angle φ is �F r = �F cos ϕ , then the mean of radial force is �F r (ϕ) = �F (ϕ) cos ϕ . Therefore, the total radial force generated over the entire circumference can be expressed as follows Suppose that the contact of each micro-convex meets to the conditions of friction law of Coulomb in the process of contact of disc-shaft, let µ is the friction coefficient at the contact point, and �ω = ω 1 − ω 2 , where ω 1 is the rotation speed of the disc, ω 2 is the speed of the shaft. Then the tangential force generated by a single micro-convex at any angle ϕ is �F t = sgn(�ω)�F cos ϕ , then the mean of the tangential force is �F t (ϕ) = sgn(�ω)�F (ϕ) cos ϕ . the total tangential force generated over the entire circumference can be expressed as where sgn(�ω) is the step function, which is expressed as The frictional force F f generated by the entire contact surface of disc-shaft can be expressed as follows: The components of the force from the disc on the shaft in two coordinate directions are respectively expressed as follows. Figure 2 shows a rotor system with concentrated mass and rigid support. This system is composed of a rotor and a loose disc. From Ref. [20], the influence of gyroscopic effect on the dynamic response of the system is very little, for simplicity, the gyroscopic effect is ignored. The dynamic response of the rotor system can be obtained by integrating the equation of motion.

Motion Equations of the Rotor System with Loose Disc
The vibration equation of the shaft is expressed as follows where m 1 , c 1 and k are the mass, damping coefficient and stiffness coefficient of the shaft respectively, e 1 and ω are the eccentric distance and the rotation speed of the shaft respectively. f x and f y are the components of the force from the disc on the shaft in the x-direction and the y-direction respectively.
The vibration equations of the disc are is expressed as follows where m 2 is the mass of the disc , c 2 is the vibration damping coefficient of the disc, e 2 is the eccentric distance of the disc.
The rotational vibration equation of the disc is expressed as follows where j is the moment of inertia of the disc, c 3 is the rotational damping coefficient of the disc.
For the convenience of calculation and analysis, dimensionless to the equations [15][16][17], The dimensionless equation of motion can be written as Due to the nonlinear of Eq. (18), its analytical solution is hardly obtained. Here the fourth-order Runge-Kutta [28][29][30] method is used to solve Eq. (18). In order to ensure the convergence of the solution and reduce the calculation error, the time step of the solution is set to π 640.

Analysis of Motion State of the Disc
When the disc-shaft looseness fault occurs, the speed of the disc is no longer the same as speed of the shaft, and the motion state will inevitably change, this is also the unique characteristic of this fault. From Eqs. (10), (11), (13) and (18), the speed of the shaft Ω, contact stiffness C, clearance H, damping of the disc ξ 2 and rotational damping ξ 3 have an influence on the motion characteristics of the disc-shaft system. Therefore, the motion state of the disc is discussed mainly from the above several factors.

Influence of Shaft Speed
The change of relative speed of the disc θ ′ 1 at different rotation speed of the shaft Ω is shown in Figure 3. The change of relative speed of the disc θ ′ 1 with speed of the shaft Ω is shown in Figure 4. From Figure 3a, when Ω≤1, the shaft speed of the disc is basically the same as that of the shaft. From Figure 3b, when 1 < Ω ≤ 2.6 , The disc is no longer at the same speed as the shaft, θ ′ 1 decreases with the increase of Ω . From Figure 3c, when 2.6 < Ω ≤ 5.8 , the rotation speed of the disc is no longer stable and in a surge state. From Figure 3d, when 5.8 < Ω , the rotation speed of the disc returns to a stable state again, and the relative speed of the disc θ ′ 1 continues to decrease with the increase of Ω . From Figure 4, when Ω is low, the disc and the shaft basically rotate at the same speed. With the increasing of Ω , θ ′ 1 decreases gradually.

Influence of the Contact Stiffness of Disc-Shaft
Due to the contact stiffness of disc-shaft C = 4 3lηr 1 E ′ σ 2σ γ π , i.e., there is a linear relationship between C and the contact stiffness of disc-shaft. Therefore, the influence of the elastic modulus on the motion state of the disc is known by discussing the influence of C on the motion state of the disc. Figure 5 is the change of relative speed of the disc θ ′ 1 under the different C.
When C ≤ 4 × 10 6 N/m , θ ′ 1 gradually increased with the increase of C, which is seen in Figure 5a. When 4 × 10 6 N/m < C ≤ 6 × 10 7 N/m , the rotational speed of the disc is in the state of surge, and θ ′ 1 gradually increases with the increases of C, which is seen in Figure 5b. When 6 × 10 7 N/m < C ≤ 3 × 10 8 N/m , the rotation speed of the disc recovers to the stable state again, and θ ′ 1 continues to increase with the increases of C, which is seen in Figure 5c. When C > 3 × 10 8 N/m , the rotation speed of the disc is the same as that of the shaft. With the increase of C, the rotation state of the disc is more stable. The increase of C is the increase of the contact stiffness, also is the increase of the interference force and the reduction of the looseness fault of the disc-shaft, therefore the rotation speed of the disc is getting more stable.

Influence of the Clearance of the Disc-Shaft
Let H is the clearance of the disc-shaft. Figure 6 shows the change of relative speed of the disc θ ′ 1 under the different clearance of the disc-shaft H. From Figure 6, θ ′  Figure 7 is the change of θ ′ 1 at ξ 1 = 1/3 . Firstly, the influence of ξ 1 is analyzed. From Figure 7, under the condition of ξ 2 = 2 and ξ 3 = 1 3 × 10 9 , With the change of ξ 1 , the simulation result demonstrates that the rotation state of the disc is almost constant, that is, the motion state of the disc is not sensitive to the change of ξ 1 .

Influence of Damping
Next, the influence of ξ 2 is analyzed, under the condition of ξ 1 = 1 3 and ξ 3 = 1 3 × 10 9 , With the change of ξ 2 , the simulation result shows that when ξ 2 ≥ 12 , θ ′ 1 is always stable at 0.61. However, when ξ 2 ≤ 12 , the vibration amplitude of the disc becomes larger and larger, the rub-impact of the disc-shaft also gets more and more intense. The rotation speed of the disc is sharply increased from a certain value, which is lower than the rotational speed of the shaft, to fluctuation at the rotational speed of the shaft, and the sharp increase of rotating speed and serious vibration of the disc occur simultaneously. The serious change of the disc-shaft rubbing causes the frictional force to change drastically, thus causes θ ′ 1 to fluctuate at θ ′ 1 =1. Figure 8 shows that the change of motion state of the disc at ξ 2 = 0.1 . From Figure 8, the amplitude of the disc is particularly large at this time, and is still increasing, this is obviously abnormal.
Finally, the influence of ξ 3 is analyzed under the conditions of ξ 1 = 1 3 and ξ 2 = 2 . Figure 9 shows the change of θ ′ 1 with ξ 3 . The simulation result shows that the relative speed of the disc θ ′ 1 gradually decreases with the increase of ξ 3 . When ξ 3 is smaller, θ ′ 1 decreased rapidly, and when ξ 3 is larger, θ ′ 1 decreased slowly. The bigger the rotational damping is, the lower the rotating speed of the disc is.  Figure 10 shows that the time-frequency characteristics of the disc-shaft when the rotation speed of the disc is the same as that of the shaft. From Figure 10b, the spectrum of X 2 , i.e., in the horizon direction, is mainly composed of one frequency component. However, there is a process of collision between the disc and the shaft, and the frequency components of the disc-shaft collision should occur in the spectrum. The disc-shaft displacement difference R is a parameter that directly describes the process of the disc-shaft collision. From Figure 10d, one frequency multiplication is its main component in the collision frequency, and the amplitude of other frequency components are extremely weak compared with that in the X 2 spectrum, therefore the collision frequency of the disc-shaft is not found in the spectrum of X 2 . Figure 11 shows that the time-frequency characteristics of the disc in the x-direction and the time-frequency characteristics of the disc-shaft displacement difference R when θ ′ 1 = 0.86 . From Figure 11b, the rotating frequency component of the disc and the rotating frequency component of the shaft the spectrum are both in the spectrum of X 2 . The collision frequency component also doesn't occur in the spectrum of X 2 . Since the rotating frequency component of the disc and the rotating frequency component of the shaft are close, the beat vibration of the disc occurs in the waveform of X 2 , which is seen in Figure 11a. The periodic similar beat vibration also occurs in the waveform of the displacement difference of the disc-shaft. The result shows that the displacement difference of the disc-shaft has a causal relationship with the vibration of X 2 . From Figure 11d, the spectrum of R is composed of a few low frequency multiplication components and a few weak high frequency components. Similarly, because the amplitude of the collision frequency component is very small, the collision frequency component is also not found in the X 2 spectrum. Figure 12 shows that the time-frequency characteristics of the disc in the x-direction and the time-frequency characteristics of the disc-shaft displacement difference R when θ ′ 1 = 0.42 . From Figure 12b, the rotating frequency component of the disc and the rotating frequency component of the shaft the spectrum are both in the spectrum of X 2 . The collision frequency component also doesn't occur in the spectrum of X 2 . From Figure 12d, the spectrum of R is composed of one multiplication component. A few weak low frequency multiplication components and a few weak high frequency components also occur in the spectrum of R. Similarly, because the amplitude of the collision frequency component is very small, the collision frequency component is also not found in Figure 12b.  Figure 13 shows the trajectory of the disc under the different θ ′ 1 . From Figure 13, With the decreases of θ ′ 1 , the trajectory of the disc changes from circle-shape to inner eight-shape, and then return to circle-shape. In the inner eight-shape, the inner ring first gradually becomes smaller and then gradually becomes larger, the outer ring is still getting smaller. The appearance of the inner eightshape is caused by the difference of θ ′ 1 . The reason why the outer ring has been getting smaller is that the centrifugal force is getting smaller due to the smaller and smaller relative speed of the disc-shaft θ ′ 1 .

Experimental Research
The experimental system is composed of ZT-3 rotor test bench, Bentley displacement sensor and MULLER-BBM sampling system. Figure 14 is the ZT-3 rotor test bench. The ZT-3 rotor test bench consists of the power output system and the rotor system with loose disc. The flexible coupling is used as the demarcation point, the right half part of demarcation point is the power output system, which consists of motor, joint coupling, shaft, bearing seat, flexible coupling and phase bonder. The right side of the flexible coupling is connected to a shaft with a length of 320 mm, and the left side of the flexible coupling is connected to a shaft with a length of 500 mm. Both ends of the shaft are supported by the sliding bearings. In order to ensure the accuracy of the experimental results of the rotor system with loose disc, the flexible coupling is used to make the power output system only output torque, not horizontal or vertical vibration. The phase bonder is used to measure the rotational frequency of the shaft by an eddy current sensor. The motor in the experiment is a DC motor with the output power of 250 W. The speed controller is used to realize stepless speed in the range of 0−10000 r / min. The left half part of the demarcation point is a rotor system with loose disc, which consists of a rotating shaft, two bearing seats, and a disc. The diameter of the shaft is 9.5 mm, the mass of the disc is 0.612 kg, the outer diameter of the disc is 76.2 mm, and the distance between the centers of the two bearing seats is 422 mm. This experiment is used to analyze the vibration characteristics of the rotor system with loose disc. Figure 15 is the structure of the loose disc, which consists of the  inner ring and the outer ring. These two parts are assembled together by a cone surface, and the pressure is provided by the screw thread. When the screw thread rotates clockwise, the inner ring and the outer ring are extruded by the cone surface contact, so that the diameter of the inner ring reduces. Therefore, the screw thread is only turned clockwise to fix the disc on the shaft, then turned counterclockwise, thus the clearance can be generated between the disc and the shaft. Figure 16 shows that the waveform and spectrum in the x-direction when the disc and the shaft is at the same speed. Figure 17 shows that the waveform and spectrum in the x-direction when the speed of the disc and the speed of the shaft is close. Figure 18 shows that the waveform and spectrum in the x-direction when the disc and the shaft have a large difference in rotation speed. From Figure 16, when the disc and the shaft is at the same speed, the collision frequency is mainly composed of one frequency multiplication component. From Figure 18, when the disc and the shaft have a large difference in rotation speed, compared with Figure 16, except for one frequency multiplication component in the collision frequency, a few weak high frequency multiplication components and a few weak low frequency multiplication components also occur in the collision frequency. From Figure 17, when the speed of the disc and the speed of the shaft is close, the vibration of the disc occurs a beat vibration, the collision frequency is mainly composed of a few low frequency components. The experimental results are consistent with the simulation results. Figure 19 shows that the trajectory of the disc under the different θ ′ 1 . When the disc and the shaft is at the same speed, the trajectory of the disc is basically a circle-shape. With the decreases of θ ′ 1 , the trajectory changes from circle-shape to inner eight-shape, and then return to circle-shape. In the inner eight-shape, the inner ring first gradually becomes smaller and then gradually becomes larger, and the outer ring is still getting smaller. the experimental results are also consistent with the simulation results. The experiment results show that the simulation model is very reasonable.

Conclusions
A dynamic contact model of the rotor system with loose disc is proposed based on the contact model of the discshaft system with the microscopic surface topography. The influences of Ω, C, H, ξ 2 and ξ 3 on the motion state of the disc are discussed. The time-frequency characteristics of the disc and the motion trajectory of the disc are discussed. The experiment results verified the effectiveness of the proposed model. The obtained conclusions are as follows.
(1) The rotation speed of the shaft, the contact stiffness, the clearance, the damping of the disc, and the rotational damping all have an influence on the motion state of the disc. Compared with the rotor system with a loose disc caused by the clearance fit, The variation of vibration characteristics of the proposed model is smaller and more stable. The rotor system with slight fault is more stable and conforms to objective laws.  are same, the collision frequency is mainly composed of one frequency multiplication component. When the rotational speed of the disc and the rotation speed of the shaft is close, the beat vibration occurs in the x direction of the disc. Simultaneously, the periodical similar beat vibration phenomenon also occurs in the waveform of the disc-shaft displacement difference. The collision frequency is mainly composed of low frequency multiplication components and weak high frequency multiplication components. When the disc and the shaft have a large difference in rotation speed, the collision frequency is mainly composed of one frequency multiplication component, a few weak high frequency multiplication components and a few weak low frequency multiplication components.
(3) With the decrease of the relative rotation speed of the disc, the trajectory of the disc changes from changes from circle-shape to inner eight-shape, and then return to circle-shape. In the inner eight-shape, the inner ring first gradually becomes smaller and then gradually becomes larger, and the outer ring is still getting smaller. The appearance of the inner eight-shape is caused by difference in the speed between the disc and the shaft. With the rotating speed of the disc is getting lower and lower, the centrifugal force provided by the disc is getting smaller and smaller, and the outer ring of inner eight-shape is getting smaller.