Vibration Response Analysis of a Time-varying Stiffness Wheel with Contrate Gear under Unbalanced Excitation

The vibration and response of multi-contact interface micro-motion time-varying stiffness wheel are analyzed, and the nonlinear vibration response characteristics caused by the unbalance force of multi-contact interface contrate gear face of a gas turbine rotor are studied. Firstly, the local sliding model of dry friction-damped contact is extended to establish a holistic and local unified sliding model. The equivalent stiffness and damping of the damping device are calculated by means of the equivalent linearization method and the first harmonic balance method. Finally, the finite element model of multi-contact contrate gear rotor wheel is established to analyze the influence of different unbalanced masses on vibration response under nonlinear frictional contact. The results show that the response caused by the unbalance mass of multi-contact interface wheel with contrate gear is mainly radial response, and the peak value of resonance response increases with the increase of unbalance force.


Introduction
Heavy duty gas turbine widely used drum rod combination rotor structure, rod rotor because of its excellent lightweight, high strength, easy to assemble and disassemble characteristics, the structure mainly has two forms of center rod and circumferential rod.[1] According to the torsional mode, the connection structure of the tie rod rotor can be divided into plane friction torsional and end tooth torsional.The end teeth of the wheel have high stiffness and moderate self-alignment performance, which play an important role in the connection and torsional transmission of the tie rod rotor of heavy gas turbines.Different from the whole rotor, the tie rod rotor is not a continuous whole, with many contact surfaces and complex contact conditions.The failure rate of the gas turbine rotor system is as high as 45%, and the operation and maintenance cost accounts for more than 60% of the whole power plant [2].Understanding the mechanism of interface contact state and dynamic response is of great significance to the prediction of dynamic performance and reliability analysis of the whole machine.There is no uniform design criterion for the study of natural frequency and dynamic characteristics of bolted tie rod rotors.The nonlinear behavior of the multi-contact interface is affected by the excited force and the operating state, and the structure parameters are many and the boundary conditions are complex.It is necessary to use the extension of finite element linear method and nonlinear method to analyze.
The contact stiffness of the disc contact interface is different from that of the continuous matrix structure, which shows that the frequency response function changes with the preload force and the exciting force.Contact stiffness analysis is the basis of accurate modeling and dynamic response calculation of tie rod rotors.The calculation of contact stiffness between disks can be divided into two methods: micro calculation and macro simulation.For the contact of rough surfaces, theoretical analysis models can be mainly divided into statistical model and fractal model [3].Hertz contact gives the expression of the contact pressure, permeability and contact radius between a rigid sphere and an elastic plane, which is the basis of the contact mechanics model [4].Based on statistical theory, Greenwood Williamson proposed the hypothesis that rough surface micro-convex bodies follow Gaussian function, established the two-plane elastic contact model (GW model), and extended the Hertz contact model to planar contact [5].The contact interface of mechanical structure generally has external interference force under working condition, and the contact interface under external excitation produces macroscopic vibration response, and the vibration response characteristics of the rough consolidated interface directly affect the dynamic service performance of the mechanical structure.Studying the nonlinear vibration response characteristics of the rough consolidated interface is helpful for improving the service life of mechanical parts.It is of great theoretical and practical significance to improve the dynamic service characteristics of mechanical equipment.In order to consider the influence of connection structure when calculating rotor vibration characteristics, a lot of research work has been done by domestic and foreign scholars in recent years.At present, there are three equivalent methods, such as equivalent spring method, virtual material method [6] and modified stiffness matrix method.As for the study of the nonlinear vibration response characteristics of the rough interface, it is difficult to accurately reveal the nonlinear vibration response mechanism of the contact interface with rough morphology because of the lack of the relationship between the rough morphology and the contact load and the influence law that has not been clarified.In this paper, the initial displacement is solved by adding the excitation of the unbalanced force, and then the nonlinear contact stiffness and damping of the contact interface are obtained iteratively.Finally, the vibration characteristics of the positive pressure and the unbalanced force under the time-varying stiffness are discussed.

Selection of Computational Model
Contrate gear are classified according to their structural shapes, mainly including Hirth couplings and Curvic couplings, as shown in figure 1.The contrate gear between the rotor discs used in this paper are Hirth couplings.The contact surface of the contrate gear is flat and the tooth surface is triangular, as shown in figure 2.

Model and Constraint Conditions
The full structure of the rotor disc is adopted, and the circular symmetry is not used, but the full-circle model is adopted, which avoids the calculation result deviation caused by the setting of circular symmetric boundary conditions.The axial length is 286 mm, the inner radius is 240 mm, and the outer radius of the non-mortised side is 307.67 mm, and its structure is shown in figure 2. The rotor 3D model was established by using the numerical simulation finite element software ANSYS to simplify and consider the actual working situation.The upper surface of the tenon and groove end was set as a fixed surface, and the other surface was free.The equivalent stiffness of spring element and friction damping were added to the contact tooth surface to simulate the friction contact, and the shaft and blade were not considered in modeling.The three-dimensional model of the contrate gear disc is shown in figure 3. Model elastic modulus E=210 Gpa, density is 7830 kg/m 3 , summary points 166859, total elements 92791.

Global-local Sliding Model
For the friction interface between the contact points of the contrate gear, the micro-sliding friction model as shown in figure 4 was applied.This model is derived from the improvement of the Iwan model by Menq et al.Csaba [7] made a partial simplification on this basis, and the simplified model can still fully display the important characteristics of the micro-sliding friction interface.The friction interface is simulated by pressing the elastic rectangular plate on a semi-infinite rigid surface [8].The length of the rectangular plate is l, the uniform load q is applied to the upper surface of the rectangular plate, u is the displacement distance of the right end of the plate, and the right side F is the external excitation applied.When the elastic rectangular plate is in the local sliding stage, the external excitation is equal to the frictional damping force [9].
The internal friction factor μ is constant.Assuming that the friction action on the contact surface conforms to Coulomb's law of friction, the right end first begins to slide when the right end external excitation F appears.The sliding area gradually increases with the increase of F, until the sliding area extends to the whole, the whole sliding occurs, and the friction force is expressed as f(x)=μq(x).Its sliding area distribution is shown in the figure 4:  When F gradually increases from 0, the right end of the microelement begins to slide, a sliding region appears, and the sliding region gradually expands from right to left.At this time, the magnitude of friction force is equal to the magnitude of external excitation.When F continues to increase, ∆= , = , the sliding region expands to the left, the system reaches critical slide, and then begins to slide as a whole, at which time the sliding friction force is  =  = .
The functional relationship between force and displacement at the contrate gear of the rectangular plate can be expressed as equation (1) and equation ( 2), separately:

Vibration Analysis of Contrate Gear Disc
The vibration of gear disc with contrate gear teeth face is nonlinear vibration of frictional contact, showing nonlinear characteristics [10].The differential equation describing the structural dynamics can be expressed as equation (3): ̈() +  0 ()̇() +  0 () + ((), ̇()) = () When the speed is constant and the exciting force is harmonic, it can be expressed as equation (4): If the nonlinear system is weak and excited near the resonance point, the dynamic differential equation can be linearized and an approximate solution can be obtained by steady-state nonlinear calculation.Otherwise, the exact solution must be calculated by transient nonlinearity.

Analysis of Unbalanced Vibration Response under Time-varying Stiffness
In this section, the nonlinear process of equivalent stiffness and equivalent damping varying with the response of the rotor contrate gear plate is considered.In the actual process, the contact stiffness will change with the displacement and time of the contact point.When there is vibration, the contact surface will have relative displacement, and the contact stiffness will change with the displacement, resulting in the time variation of the contact stiffness.
The calculated model assumes no contact at the beginning.The contact point pair of each contact surface is selected in the apdl command flow to set the friction contact, and each point is changed from the initial state to the state where new equivalent stiffness and equivalent damping are added.The contact model is shown in figure 6 below: The multi-scale nonlinear frictional contact was calculated by apdl and matlab, and solved by ANSYS nonlinear solver.The equivalent stiffness and equivalent damping were calculated by finite element method using the self-programmed program jointly applied by ansys and matlab.The unbalance is applied on the wheel to solve the relationship between the contact points of the tooth surface with the amplitude of the unbalance force.Then, according to the results of vibration response calculation, the iterative cycle flow is shown as follows: First, the undamped response amplitude is extracted, and then the equivalent stiffness and equivalent damping of the disk are calculated, and then the current equivalent stiffness and amplitude under damping are calculated.The iterative cycle process is shown in figure 7: each step first calculates the stiffness value under the response, and updates the equivalent stiffness and equivalent damping.This section considers the effect of rotor mass unbalance on the response characteristics of the system.The response curve of the wheel in axial x and circumferential y.Different mass blocks are added at the same diameter position to simulate the unbalance of rotor mass.First, the loading position of disc mass unbalance is at the radius of 200mm.Load the unbalance mass of 0.1kg, 0.2kg and 0.5kg, that is, take the unbalance force of 20kg• mm, 40kg• mm and 100kg• mm as the external excitation conditions for analysis, and observe the influence of mass change on its amplitude under the same unbalance radius.
The figure 8 shows the amplitude-frequency characteristic curve of the system when different unbalanced forces act simultaneously.It can be seen that when the position of the unbalanced mass is the same, the resonance peak value increases with the increase of the mass.The peak value is basically maintained near the natural frequency.The radial amplitude response is greater than the axial amplitude response.When the size of the unbalance mass is unchanged, the influence of the radius of the unbalance force on the amplitude is considered, and 0.5kg of the unbalance mass is respectively loaded at 50mm, 100mm and 200mm, that is, the unbalance force of 25kg• mm, 50kg• mm and 100kg• mm are respectively used as external excitation conditions for analysis, and the same mass is observed.The influence of unbalance force variation on its amplitude at different radius.The results are shown in figure 9: It can be seen that the resonance peak value of the system increases with the increase of the unbalance radius, and the effect of the unbalance force is consistent with the effect of different unbalance masses on the amplitude peak value under the same unbalance radius.
Under the action of unbalance force, each direction of the wheel has different vibration response to the unbalance force.In the axial direction, when the frequency reaches the first natural frequency of 2200Hz, the unbalance response curves have a higher peak value.This is mainly due to the meshing action of the gear makes the meshing frequency of the gear close to the natural frequency of the system.With the increase of frequency, a secondary peak appears near the second frequency.In the radial direction, there is only one peak near the second natural frequency.
It can be seen from the above analysis that the response caused by the unbalance mass of multi-contact interface wheel with contrate gear teeth is mainly radial response, and the peak value of resonance response increases with the increase of unbalance force.

Conclusion
In this paper, a 3D finite element method is proposed to analyze the frictional contact of gas turbine contrate gear-face gears.The contact points between the multi-scale frictional contact interfaces are simulated by a micro-sliding model, and the calculation model is obtained.Taking a certain gas turbine wheel as the research object, the influence of different pre-unbalance forces on the vibration frequency of the wheel is studied.The main conclusions are as follows: (1) The unbalanced mass has an important impact on the contact stiffness and vibration frequency of the wheel, and the increase of the unbalanced mass will lead to the increase of the resonance peak under the premise of meeting the safety.
(2) For a wheel with transverse teeth with time-varying stiffness characteristics, the radial response is greater than the axial response, and the radial has only one resonance peak and the axial has two peaks.
In this paper, the contrate gear disk structure of a certain type of gas turbine is simplified, the finite element model is established, and the vibration response analysis method of the system with friction damping is developed.The vibration characteristics of the system under the action of unbalanced forces at different positions are analyzed, and the influence of the unbalanced forces on the vibration amplitude of the contrate gear-face teeth is studied.The theoretical basis is provided for the vibration damping effect of the contrate gear disc in the future.

Figure 2 .
Figure 2. Structure model and schematic diagram of contrate gear disc.

Figure 4 .
Figure 4. Schematic diagram of local sliding.Any element in the sliding area is shown by the force in the figure5:

Figure 6 .
Figure 6.Diagram of equivalent stiffness and equivalent damping of tooth surface.

Figure 7 .
Figure 7. Ansys and Matlab combined application flow chart.

Figure 8 .
Figure 8.(a) Response curve in the x direction (b) Response curve in the y direction.

Figure 9 .
Figure 9. (a) response curve in the x direction (b) response curve in the y direction.