Method for calculating influence of graphite face seal on rotor system dynamics

Abstract

The application discloses a kind of calculation methods for the influence of graphite end face seal on rotor system dynamics characteristics, as follows: the transverse excitation load model of graphite end face seal under the state of rotor whirl is established;The dynamic equation of the offset disc rotor-graphite end face seal system is established on the basis of the transverse excitation load model of graphite end face seal;Based on the established dynamic equation of the offset disc rotor-graphite end face seal system, the vibration response characteristics of the rotor are calculated, and the evolution law of the rotor dynamics behavior under the action of graphite end face seal is analyzed;And the influence mechanism of graphite end face seal on the vibration response of rotor system is further obtained by calculating the work of graphite end face seal friction. The application can obtain the influence law of graphite end face seal on the dynamics behavior and stability of rotor, and further obtain the mechanism of the influence of seal friction on the vibration response of rotor by analyzing the work of friction, which provides a theoretical basis for inhibiting friction excitation and stable operation of rotor.

Description

Technical Field

[0001] This invention belongs to the field of rotor dynamics technology, specifically relating to a method for calculating the influence of graphite end-face sealing on the dynamic characteristics of a rotor system. Background Technology

[0002] As a high-speed rotating power machine, aero engines have numerous rotating-stationary contact surfaces in their structure. To improve their performance and thermal efficiency, the engine's pressure ratio and temperature are constantly increasing, placing ever higher demands on overall engine efficiency. The gaps between these contact surfaces not only cause performance losses but also worsen operating conditions and the environment, directly affecting the lifespan and reliability of components, and even causing damage. Therefore, highly efficient sealing devices are needed to provide dynamic-static sealing for the mechanical interfaces.

[0003] Graphite seals are one of the most widely used sealing structures in aero-engines. Depending on their location, they can be divided into seals between the rotor and stator structures and seals between the two rotor shafts. Graphite end-face seals are used for rotor-stator sealing and are a type of contact seal. A graphite end-face seal structure consists of a stationary graphite component and a sealing ring that contacts it. The sealing ring is mounted on the shaft and rotates with the rotor shaft. The graphite ring is press-fitted onto a mounting base and fixed to the stator component. A wave spring between the mounting base and the stator component presses the graphite ring and the sealing ring together, providing a seal. However, when the rotor experiences lateral vibration and oscillation, relative movement occurs between the sealing ring and the graphite ring. The normal pressure between them becomes non-uniformly distributed. This non-uniform spring pressure generates a bending moment excitation on the sealing ring and a lateral excitation force on it. The excitation load exhibits time-varying nonlinear characteristics with the rotor's whirling motion, which will have a complex impact on the dynamic characteristics of the rotor system.

[0004] It is evident that the frictional load from the graphite end-face seal has a certain impact on the rotor's dynamic characteristics during operation. However, the relevant dynamic mechanisms have not yet been revealed, and a dynamic model for this rotor-seal system is lacking. Therefore, establishing a reasonable rotor-seal model and studying the influence of graphite end-face seal frictional excitation on rotor vibration from the perspective of single-disc rotor nonlinear motion is a valuable supplement to the cutting-edge theories of rotor dynamics and nonlinear dynamics. In-depth research into the mechanism of graphite end-face seal frictional excitation on rotor instability and analysis of rotor dynamic characteristics under graphite end-face seal frictional excitation can provide a theoretical basis for suppressing frictional vibration and ensuring smooth rotor operation, possessing significant practical scientific and engineering application value for the safety of aero-engines. Summary of the Invention

[0005] To address the shortcomings of the prior art, the present invention aims to provide a method for calculating the impact of graphite end-face seals on the dynamic characteristics of a rotor system, thereby solving the problem that the impact of contact graphite seals on rotors is difficult to quantify in the prior art.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] The present invention provides a method for calculating the impact of graphite end-face sealing on the dynamic characteristics of a rotor system, comprising the following steps:

[0008] 1) Establish a transverse excitation load model for graphite end face sealing under rotor whirl conditions: Based on the change in axial displacement Δz between the stationary and rotating rings caused by the vibration of the rotating ring at a certain micro-element, the frictional force df at the contact surface at the circumferential position of this micro-element can be obtained. s , to reduce frictional force df s By decomposing the element, we obtain the frictional force and bending moment excitation in the x and y directions. Integrating along the circumferential direction, we obtain the resultant force F of the stationary ring on the moving ring. sx F sy Resultant torque M sx M sy ;

[0009] 2) Based on the lateral excitation load model of the graphite end-face seal, establish the dynamic equation of the offset disk rotor-graphite end-face seal system; derive the resultant force F provided by the graphite end-face seal using the Lagrange energy method. sx F sy Substituting the external excitation force into the equation, and then substituting the kinetic energy E provided by the rotor translation and oscillation, we obtain the dynamic equation of the offset disk rotor-graphite end face sealing system.

[0010] 3) Based on the established dynamic equations of the offset disk rotor-graphite end-face seal system, the vibration response characteristics of the rotor are calculated, and the evolution law of the rotor dynamic behavior under the action of the graphite end-face seal is analyzed; and the influence mechanism of the graphite end-face seal on the vibration response of the rotor system is further obtained by calculating the work done by the friction force of the graphite end-face seal.

[0011] Furthermore, the modeling process in step 1) specifically includes:

[0012] 11) When the rotor is not vibrating, the clamping force between the moving ring and the stationary ring is uniformly distributed and subjected to an initial clamping force N0, with the initial clamping displacement represented by Z0. When the rotor moves at an angular velocity ω, whirling causes the clamping force between the stationary ring and the moving ring to exhibit a non-uniform distribution. The infinitesimal element d at any circumferential position α... α The axial displacement Δz is expressed as:

[0013] Δz=θ x (r ssinα-Y)-θ y (r s cosα-X);

[0014] Where X represents the translational degree of freedom along the x-direction, Y represents the translational degree of freedom along the y-direction, and θ x Let θ be the degree of freedom of the oscillation about the x-axis. y Let r be the degree of freedom of the oscillation motion about the y-axis. s Let k be the average radius of the stationary ring. s This refers to the axial stiffness of the stationary ring.

[0015] Then the normal force dN(α) of the infinitesimal element at the circumferential position α is:

[0016]

[0017] 12) When the rotor whirls, there is relative motion between the moving ring and the stationary ring. Based on Coulomb's law of friction, the frictional force df at the contact surface is obtained. s for:

[0018] df s =μdN(α)t(α)

[0019] Where μ is the friction coefficient between the moving ring and the stationary ring, and t(α) is the unit direction vector, whose direction is opposite to the direction of the relative velocity between the moving ring and the stationary ring; for the infinitesimal element at the circumferential position α, the relative velocity v between the moving ring and the stationary ring is:

[0020]

[0021] Where i and j are the unit vectors in the x and y directions, respectively, and t(α) is represented as:

[0022]

[0023] Decompose the frictional force into its components, and obtain the frictional force df of the infinitesimal element in the x-direction. sx Frictional force df in the y direction sy They are respectively:

[0024]

[0025]

[0026] In addition to friction, the non-uniform normal force also generates a bending moment excitation on the rotor, and the bending moment dM generated by the infinitesimal element at the circumferential position α about the ox axis. sx Bending moment dM about the oy axis sy for:

[0027] dM sx =-(r ssinα-y)dN(α)

[0028] dM sy =(r s cosα-x)dN(α);

[0029] Integrating the friction at the infinitesimal element along the circumferential direction, we obtain the resultant force and resultant torque of the stationary ring on the moving ring:

[0030]

[0031] Among them, F sx F is the frictional force exerted by the graphite end face seal on the moving ring in the x-direction. sy M represents the frictional force exerted by the graphite end face seal on the moving ring in the y-direction. sx M is the bending moment of the graphite end face seal relative to the moving ring along the ox axis. sy The bending moment of the graphite end face seal relative to the moving ring in the oy axis direction.

[0032] Furthermore, step 2) specifically includes:

[0033] The dynamic equations of the bias disk rotor under unbalanced conditions and graphite end-face sealing friction are derived based on the Lagrange energy method. The generalized coordinates of the rotor are q=[xy θ x θ y ] T The disk has a mass of m and a polar moment of inertia of J. p The rotational inertia of the diameter is J d The mass offset is e; the rotor kinetic energy E is the translational and oscillating kinetic energy of the disk, as shown in the following formula:

[0034]

[0035] The generalized force acting on the rotor system is shown in the following equation:

[0036]

[0037] Where, k ij Let i be the equivalent stiffness of the rotor shaft at the disk position, i = 4, j = 4, k 11 Let k be the force along the x-direction required to achieve a unit displacement of the disk center in the x-direction. 22 Let k be the force along the y-direction required to achieve a unit displacement of the disk center in the y-direction. 33 k is the torque about the ox axis that needs to be applied when the disk rotates a unit angle about the ox axis. 44 k is the torque about the oy axis that needs to be applied when the disk rotates a unit angle about the oy axis. 14 k is the force along the x-direction that needs to be applied when the disk rotates a unit angle about the oy axis. 23k is the force along the y-direction that needs to be applied when the disk rotates a unit angle about the ox axis. 32 k is the torque about the ox axis that needs to be applied when the center of the disk undergoes a unit displacement in the y direction. 41 c is the torque about the oy axis required to produce a unit displacement in the x-direction at the center of the disk, where the unit displacement or unit rotation is conditional upon the displacement or rotation in other directions being zero; 11 and c 22 F represents the damping coefficients of the rotor along the x and y directions, respectively. ux F uy These are the components of the rotor unbalance force in the x and y directions, respectively.

[0038] According to the Lagrange equation:

[0039]

[0040] The dynamic equations of the biased disk rotor-graphite end-face sealing system are as follows:

[0041]

[0042] It can be represented in matrix form as follows:

[0043]

[0044] in,

[0045]

[0046] Furthermore, step 3) specifically includes:

[0047] The dynamic equations of the offset disk rotor-graphite end-face sealing system were solved using the Newmark numerical integration method. In the calculation, the sealing effect was initially ignored, and the vibration response of the rotor without lateral load excitation was calculated as a reference for the vibration response of the offset disk rotor-graphite end-face sealing system. The time-domain responses of the two calculation cases—ignoring the sealing effect and considering the sealing effect—were extracted and expressed using bifurcation diagrams and three-dimensional waterfall plots. The vibration response amplitude and frequency of the rotor system were analyzed, and the influence of the graphite end-face sealing on the vibration response of the rotor system was obtained through comparison.

[0048] Calculate the frictional work done by the seal; for the excitation load F(t) acting on the vibration system, the work done on the system in one rotational speed cycle is as follows:

[0049]

[0050] Where q(t) and These represent the displacement and velocity of the vibrating system, respectively.

[0051] Calculate the work done by the frictional force of the graphite end-face seal and the work done by the system damping force; if the frictional force or damping force is in the same direction as the rotor precession, it inputs vibrational energy into the rotor system; otherwise, it dissipates system energy; the relationship between the frictional force and the damping force determines the increase and decrease of the system vibrational energy / vibration amplitude; based on the work done, provide a mechanistic explanation for the influence of the graphite end-face seal on the rotor system response.

[0052] The beneficial effects of this invention are:

[0053] This invention quantitatively represents the frictional excitation generated by the graphite end-face seal by establishing a lateral excitation load model for the graphite end-face seal under rotor vortex state; it also establishes the rotor dynamics equation for the offset disk rotor-graphite end-face seal interaction, supplementing the dynamics equation for this type of offset disk rotor-seal system; based on this, it calculates and analyzes the rotor's vibration response, deriving the influence law of the graphite end-face seal on the rotor's dynamic behavior and stability; and further derives the mechanism of the seal friction's influence on the rotor's vibration response through frictional work analysis; thus providing a theoretical basis for suppressing frictional excitation and ensuring the smooth operation of the rotor. Attached Figure Description

[0054] Figure 1 This is a schematic diagram illustrating the principle of the method of the present invention.

[0055] Figure 2a A schematic diagram of a transverse excitation load model for graphite end face sealing.

[0056] Figure 2b For infinitesimal element d α Schematic diagram of non-uniform normal pressure.

[0057] Figure 3 A schematic diagram is attached to establish the mechanical equations for the offset disk rotor-graphite end face sealing. Detailed Implementation

[0058] To facilitate understanding by those skilled in the art, the present invention will be further described below with reference to embodiments and accompanying drawings. The content mentioned in the embodiments is not intended to limit the present invention.

[0059] Reference Figure 1 As shown, the present invention provides a method for calculating the influence of graphite end-face sealing on the dynamic characteristics of a rotor system, comprising the following steps:

[0060] 1) Establish a transverse excitation load model for graphite end face sealing under rotor whirl conditions: Based on the change in axial displacement Δz between the stationary and rotating rings caused by the vibration of the rotating ring at a certain micro-element, the frictional force df at the contact surface at the circumferential position of this micro-element can be obtained. s, to reduce frictional force df s By decomposing the element, we obtain the frictional force and bending moment excitation in the x and y directions. Integrating along the circumferential direction, we obtain the resultant force F of the stationary ring on the moving ring. sx F sy Resultant torque M sx M sy ;

[0061] The modeling process in step 1) specifically includes:

[0062] 11) When the rotor is not vibrating, the clamping force between the moving ring and the stationary ring is uniformly distributed and subjected to an initial clamping force N0, with the initial clamping displacement represented by Z0. When the rotor moves at an angular velocity ω, whirling causes the clamping force between the stationary ring and the moving ring to exhibit a non-uniform distribution. The infinitesimal element d at any circumferential position α... α The axial displacement Δz is expressed as:

[0063] Δz=θ x (r s sinα-Y)-θ y (r s cosα-X);

[0064] Where X represents the translational degree of freedom along the x-direction, Y represents the translational degree of freedom along the y-direction, and θ x Let θ be the degree of freedom of the oscillation about the x-axis. y Let r be the degree of freedom of the oscillation motion about the y-axis. s Let k be the average radius of the stationary ring. s This refers to the axial stiffness of the stationary ring.

[0065] Then the normal force dN(α) of the infinitesimal element at the circumferential position α is:

[0066]

[0067] 12) When the rotor whirls, there is relative motion between the moving ring and the stationary ring. Based on Coulomb's law of friction, the frictional force df at the contact surface is obtained. s for:

[0068] df s =μdN(α)t(α)

[0069] Where μ is the friction coefficient between the moving ring and the stationary ring, and t(α) is the unit direction vector, whose direction is opposite to the direction of the relative velocity between the moving ring and the stationary ring; for the infinitesimal element at the circumferential position α, the relative velocity v between the moving ring and the stationary ring is:

[0070]

[0071] Where i and j are the unit vectors in the x and y directions, respectively, and t(α) is represented as:

[0072]

[0073] Decompose the frictional force into its components, and obtain the frictional force df of the infinitesimal element in the x-direction. sx Frictional force df in the y direction sy They are respectively:

[0074]

[0075]

[0076] In addition to friction, the non-uniform normal force also generates a bending moment excitation on the rotor, and the bending moment dM generated by the infinitesimal element at the circumferential position α about the ox axis. sx Bending moment dM about the oy axis sy for:

[0077] dM sx =-(r s sinα-y)dN(α)

[0078] dM sy =(r s cosα-x)dN(α);

[0079] Integrating the friction at the infinitesimal element along the circumferential direction, we obtain the resultant force and resultant torque of the stationary ring on the moving ring:

[0080]

[0081] Among them, F sx F is the frictional force exerted by the graphite end face seal on the moving ring in the x-direction. sy M represents the frictional force exerted by the graphite end face seal on the moving ring in the y-direction. sx M is the bending moment of the graphite end face seal relative to the moving ring along the ox axis. sy The bending moment of the graphite end face seal relative to the moving ring in the oy axis direction.

[0082] 2) Based on the lateral excitation load model of the graphite end-face seal, establish the dynamic equation of the offset disk rotor-graphite end-face seal system; derive the resultant force F provided by the graphite end-face seal using the Lagrange energy method. sx F sy Substituting the external excitation force into the equation, and then substituting the kinetic energy E provided by the rotor translation and oscillation, we obtain the dynamic equation of the offset disk rotor-graphite end face sealing system.

[0083] The dynamic equations of the bias disk rotor under unbalanced conditions and graphite end-face sealing friction are derived based on the Lagrange energy method. The generalized coordinates of the rotor are q=[xy θx θ y ] T The disk has a mass of m and a polar moment of inertia of J. p The rotational inertia of the diameter is J d The mass offset is e; the rotor kinetic energy E is the translational and oscillating kinetic energy of the disk, as shown in the following formula:

[0084]

[0085] The generalized force acting on the rotor system is shown in the following equation:

[0086]

[0087] Where, k ij Let i be the equivalent stiffness of the rotor shaft at the disk position, i = 4, j = 4, k 11 Let k be the force along the x-direction required to achieve a unit displacement of the disk center in the x-direction. 22 Let k be the force along the y-direction required to achieve a unit displacement of the disk center in the y-direction. 33 k is the torque about the ox axis that needs to be applied when the disk rotates a unit angle about the ox axis. 44 k is the torque about the oy axis that needs to be applied when the disk rotates a unit angle about the oy axis. 14 k is the force along the x-direction that needs to be applied when the disk rotates a unit angle about the oy axis. 23 k is the force along the y-direction that needs to be applied when the disk rotates a unit angle about the ox axis. 32 k is the torque about the ox axis that needs to be applied when the center of the disk undergoes a unit displacement in the y direction. 41 c is the torque about the oy axis required to produce a unit displacement in the x-direction at the center of the disk, where the unit displacement or unit rotation is conditional upon the displacement or rotation in other directions being zero; 11 and c 22 F represents the damping coefficients of the rotor along the x and y directions, respectively. ux F uy These are the components of the rotor unbalance force in the x and y directions, respectively.

[0088] According to the Lagrange equation:

[0089]

[0090] The dynamic equations of the biased disk rotor-graphite end-face sealing system are as follows:

[0091]

[0092] It can be represented in matrix form as follows:

[0093]

[0094] in,

[0095]

[0096] 3) Based on the established dynamic equations of the offset disk rotor-graphite end-face seal system, the vibration response characteristics of the rotor are calculated, and the evolution law of the rotor dynamic behavior under the action of the graphite end-face seal is analyzed; further, the influence mechanism of the graphite end-face seal on the vibration response of the rotor system is obtained by calculating the work done by the frictional force of the graphite end-face seal; specifically including:

[0097] The dynamic equations of the offset disk rotor-graphite end-face sealing system were solved using the Newmark numerical integration method. In the calculation, the sealing effect was initially ignored, and the vibration response of the rotor without lateral load excitation was calculated as a reference for the vibration response of the offset disk rotor-graphite end-face sealing system. The time-domain responses of the two calculation cases—ignoring the sealing effect and considering the sealing effect—were extracted and expressed using bifurcation diagrams and three-dimensional waterfall plots. The vibration response amplitude and frequency of the rotor system were analyzed, and the influence of the graphite end-face sealing on the vibration response of the rotor system was obtained through comparison.

[0098] Calculate the frictional work done by the seal; for the excitation load F(t) acting on the vibration system, the work done on the system in one rotational speed cycle is as follows:

[0099]

[0100] Where q(t) and These represent the displacement and velocity of the vibrating system, respectively.

[0101] Calculate the work done by the frictional force of the graphite end-face seal and the work done by the system damping force; if the frictional force or damping force is in the same direction as the rotor precession, it inputs vibrational energy into the rotor system; otherwise, it dissipates system energy; the relationship between the frictional force and the damping force determines the increase and decrease of the system vibrational energy / vibration amplitude; based on the work done, provide a mechanistic explanation for the influence of the graphite end-face seal on the rotor system response.

[0102] Reference Figure 2a As shown, the graphite end-face sealing structure used in this invention consists of a simple non-rotating component and a rotating component. The non-rotating component is simplified to an axial spring-rigid stationary ring unit, and the rotating component is simplified to a rigid rotating ring, which is an integral structure with the rotor. The axial spring presses the stationary ring and the rotating ring together to achieve a sealing effect.

[0103] Reference Figure 2bAs shown, when the rotor whirls, it compresses the spring, causing relative movement between the moving and stationary rings, resulting in a positive pressure between them as shown in the diagram. Figure 2b The non-uniform distribution characteristics shown lead to bending moment excitation and lateral excitation force on the sealing moving ring.

[0104] Reference Figure 3 As shown, the dynamic equation for the offset disk-graphite end-face sealing established in this invention is based on the structure shown in the figure; in the figure, the offset disk rotor consists of a massless shaft segment and an offset disk, the mass of which is m and the polar moment of inertia is J. p The rotational inertia of the diameter is J d The offset distance is e, and the offset disk and the graphite end-face sealing structure are integrated into one structure; the mass of the shaft section is negligible and torsion is not considered, it provides lateral bending stiffness for the offset disk rotor, and the lateral stiffness of the offset disk rotor can be obtained from beam theory. The right side of the disk is a simplified graphite end-face sealing non-rotating structure, and the axial stiffness of the stationary ring is k. s The coefficient of friction between the stationary ring and the wheel is μ, the preload of the stationary ring is N0, and the contact radius of the stationary ring is r. s The mass of the stationary ring is negligible, meaning its vibration is not considered. During operation, the rotor rotates at speed ω, and there is relative motion between the disk and the stationary ring. Due to the pre-clamping force between the stationary ring and the disk, the stationary ring will exert friction on the disk surface, affecting the lateral vibration of the rotor.

[0105] This invention has many specific applications. The above description is only a preferred embodiment of this invention. It should be noted that for those skilled in the art, several improvements can be made without departing from the principle of this invention, and these improvements should also be considered within the scope of protection of this invention.

Claims

1. A method for calculating the influence of graphite end-face sealing on the dynamic characteristics of a rotor system, characterized in that, The steps are as follows: 1) Establish a transverse excitation load model for graphite end face sealing under rotor whirl conditions: Based on the change in axial displacement Δz between the stationary and rotating rings caused by the vibration of the rotating ring at a certain micro-element, the frictional force df at the contact surface at the circumferential position of this micro-element can be obtained. s , to reduce frictional force df s By decomposing the element, we obtain the frictional force and bending moment excitation in the x and y directions. Integrating along the circumferential direction, we obtain the resultant force F of the stationary ring on the moving ring. sx F sy Resultant torque M sx M sy ; 2) Based on the lateral excitation load model of the graphite end-face seal, establish the dynamic equation of the offset disk rotor-graphite end-face seal system; derive the resultant force F provided by the graphite end-face seal using the Lagrange energy method. sx F sy Substituting the external excitation force into the equation, and then substituting the kinetic energy E provided by the rotor translation and oscillation, we obtain the dynamic equation of the offset disk rotor-graphite end face sealing system. 3) Based on the established dynamic equations of the offset disk rotor-graphite end-face seal system, the vibration response characteristics of the rotor are calculated, and the evolution law of the rotor dynamic behavior under the action of the graphite end-face seal is analyzed; and the influence mechanism of the graphite end-face seal on the vibration response of the rotor system is further obtained by calculating the work done by the friction force of the graphite end-face seal.

2. The method for calculating the influence of graphite end-face sealing on the dynamic characteristics of a rotor system according to claim 1, characterized in that, The modeling process in step 1) specifically includes: 11) When the rotor is not vibrating, the clamping force between the moving ring and the stationary ring is uniformly distributed and subjected to an initial clamping force N0, with the initial clamping displacement represented by Z0. When the rotor moves at an angular velocity ω, whirling causes the clamping force between the stationary ring and the moving ring to exhibit a non-uniform distribution. The infinitesimal element d at any circumferential position α... α The axial displacement Δz is expressed as: Δz=θ x (r s sinα-Y)-θ y (r s cosα-X); Where X represents the translational degree of freedom along the x-direction, Y represents the translational degree of freedom along the y-direction, and θ x Let θ be the degree of freedom of the oscillation about the x-axis. y Let r be the degree of freedom of the oscillation motion about the y-axis. s Let k be the average radius of the stationary ring. s This refers to the axial stiffness of the stationary ring. Then the normal force dN(α) of the infinitesimal element at the circumferential position α is: 12) When the rotor whirls, there is relative motion between the moving ring and the stationary ring. Based on Coulomb's law of friction, the frictional force df at the contact surface is obtained. s for: df s =μdN(α)t(α) Where μ is the friction coefficient between the moving ring and the stationary ring, and t(α) is the unit direction vector, whose direction is opposite to the direction of the relative velocity between the moving ring and the stationary ring; for the infinitesimal element at the circumferential position α, the relative velocity v between the moving ring and the stationary ring is: Where i and j are the unit vectors in the x and y directions, respectively, and t(α) is represented as: Decompose the frictional force into its components, and obtain the frictional force df of the infinitesimal element in the x-direction. sx Frictional force df in the y direction sy They are respectively: In addition to friction, the non-uniform normal force also generates a bending moment excitation on the rotor, and the bending moment dM generated by the infinitesimal element at the circumferential position α about the ox axis. sx Bending moment dM about the oy axis sy for: dM sx =-(r s sinα-y)dN(α) dM sy =(r s cosα-x)dN(α); Integrating the friction at the infinitesimal element along the circumferential direction, we obtain the resultant force and resultant torque of the stationary ring on the moving ring: Among them, F sx F is the frictional force exerted by the graphite end face seal on the moving ring in the x-direction. sy M represents the frictional force exerted by the graphite end face seal on the moving ring in the y-direction. sx M is the bending moment of the graphite end face seal relative to the moving ring along the ox axis. sy The bending moment of the graphite end face seal relative to the moving ring in the oy axis direction.

3. The method for calculating the influence of graphite end-face sealing on the dynamic characteristics of a rotor system according to claim 2, characterized in that, Step 2) specifically includes: The dynamic equations of the bias disk rotor under unbalanced conditions and graphite end-face sealing friction are derived based on the Lagrange energy method. The generalized coordinates of the rotor are q=[xy θ x θ y ] T The disk has a mass of m and a polar moment of inertia of J. p The rotational inertia of the diameter is J d The mass offset is e; the rotor kinetic energy E is the translational and oscillating kinetic energy of the disk, as shown in the following formula: The generalized force acting on the rotor system is shown in the following equation: Where, k ij Let i be the equivalent stiffness of the rotor shaft at the disk position, i = 4, j = 4, k 11 Let k be the force along the x-direction required to achieve a unit displacement of the disk center in the x-direction. 22 Let k be the force along the y-direction required to achieve a unit displacement of the disk center in the y-direction. 33 k is the torque about the ox axis that needs to be applied when the disk rotates a unit angle about the ox axis. 44 k is the torque about the oy axis that needs to be applied when the disk rotates a unit angle about the oy axis. 14 k is the force along the x-direction that needs to be applied when the disk rotates a unit angle about the oy axis. 23 k is the force along the y-direction that needs to be applied when the disk rotates a unit angle about the ox axis. 32 k is the torque about the ox axis that needs to be applied when the center of the disk undergoes a unit displacement in the y direction. 41 c is the torque about the oy axis required to produce a unit displacement in the x-direction at the center of the disk, where the unit displacement or unit rotation is conditional upon the displacement or rotation in other directions being zero; 11 and c 22 F represents the damping coefficients of the rotor along the x and y directions, respectively. ux F uy These are the components of the rotor unbalance force in the x and y directions, respectively. According to the Lagrange equation: The dynamic equations of the biased disk rotor-graphite end-face sealing system are as follows: It can be represented in matrix form as follows: in, 4. The method for calculating the influence of graphite end-face sealing on the dynamic characteristics of a rotor system according to claim 1, characterized in that, Step 3) specifically includes: The dynamic equations of the offset disk rotor-graphite end-face sealing system were solved using the Newmark numerical integration method. In the calculation, the sealing effect was initially ignored, and the vibration response of the rotor without lateral load excitation was calculated as a reference for the vibration response of the offset disk rotor-graphite end-face sealing system. The time-domain responses of the two calculation cases—ignoring the sealing effect and considering the sealing effect—were extracted and expressed using bifurcation diagrams and three-dimensional waterfall plots. The vibration response amplitude and frequency of the rotor system were analyzed, and the influence of the graphite end-face sealing on the vibration response of the rotor system was obtained through comparison. Calculate the frictional work done by the seal; for the excitation load F(t) acting on the vibration system, the work done on the system in one rotational speed cycle is as follows: Where q(t) and These represent the displacement and velocity of the vibrating system, respectively. Calculate the work done by the frictional force of the graphite end-face seal and the work done by the system damping force; if the frictional force or damping force is in the same direction as the rotor precession, it inputs vibrational energy into the rotor system; otherwise, it dissipates system energy; the work done by the frictional force and damping force determines the increase and decrease of the system vibrational energy / amplitude; based on the work done, provide a mechanistic explanation for the influence of the graphite end-face seal on the rotor system response.

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