Flywheel damping ring design method based on high-precision nonlinear friction dynamical model

Abstract

The invention relates to a flywheel damping ring design method based on a high-precision nonlinear friction dynamical model in the technical field of vibration reduction, which is characterized by comprising the following steps of: 1) obtaining a mode of a flywheel linear system; 2) selecting the material and size of the damping ring according to the size of the flywheel body; 3) determining the material and the size of the friction plate; 4) determining the material and the size of the rubber ring; 5) obtaining the mode of the flywheel-damping ring linear system; 6) establishing a high-dimensional nonlinear system dynamics model of the flywheel-damping ring containing multiple friction interfaces under basic random excitation; 7) predicting forced response of the flywheel under basic random excitation to obtain optimal positive pressure and friction coefficient; and 8) determining the height of the bolt sleeve. Through iterative analysis, the material and size of the damping ring, the material and size of the friction plate, the material and size of the rubber ring and the height of the bolt sleeve can be determined, and the optimal positive pressure and the optimal friction coefficient of the interface can be guaranteed. According to the design method, the calculation time is short, high-precision response can be obtained, and the design method has the advantage on rapid iterative design of the flywheel damping ring.

Description

Technical Field

[0001] The invention relates to the technical field of vibration reduction, and in particular to a flywheel damping ring design method based on a high-precision nonlinear friction dynamics model. Background Art

[0002] The flywheel is an important component of the satellite attitude and orbit control system and is the key to achieving precise satellite attitude control. Its working principle is that the control circuit controls the motor to rotate according to external requirements, driving the flywheel to run forward / reverse, accelerate / decelerate or stably, and generates the required angular momentum and reaction torque through the rotation or direction change of the flywheel to exchange angular momentum with the spacecraft, thereby achieving precise control of the spacecraft's attitude.

[0003] The working life cycle of a flywheel goes through two stages: rocket launch and on-orbit operation. During on-orbit operation, the flywheel is in working condition and rotates at high speed, and it itself becomes a vibration source. The vibration generated is mainly caused by micro-vibration caused by the imbalance of wheel mass and irregular bearings. During the launch phase, the flywheel does not work or rotate, and the vibration of the flywheel is mainly caused by the excitation of the rocket. The mechanical environment of the launch phase mainly includes steady-state acceleration, low-frequency vibration, broadband vibration (including random vibration and acoustic load) and impact.

[0004] The flywheel structure is usually an integral structure with very small structural damping. When excited, the response peak at its natural frequency is particularly prominent. As flywheels develop towards high torque and high speed, attention must be paid to the vibration amplification caused by the modal resonance of the flywheel structure. Because the vibration amplification of the flywheel is too large, a strong resonance is generated under the basic (rocket) excitation. According to the resonance intensity estimation of the flywheel, the force transmitted to the bearing assembly when the wheel body vibrates will exceed the static load capacity of the bearing assembly, aggravating the stress of the core components such as the bearing assembly, seriously affecting the function and reliability of the flywheel product. At the same time, there is almost no energy dissipation source inside the flywheel, and the ability of the bearing assembly to withstand these vibrations mainly depends on the external damping input. Therefore, it is urgent to develop new methods and new devices to suppress the vibration of the flywheel. For the flywheel structure, the vibration amplification suppression device is required to have a small additional mass and a small impact on the dynamic balance. The suppression method needs to cover a certain width of the excitation frequency range and have a suppression effect on the excitation amplitude within a certain range.

[0005] In the aerospace field, dry friction damping vibration reduction technology has been applied to flywheels to solve the vibration amplification problem of flywheels in the rocket launch stage. Typical momentum wheel dry friction damping components are as follows: Figure 1 The function of the dry friction damping component is to increase the damping of the flywheel structure by adding dry friction damping, consume the resonance energy in time, and protect the wheel structure and bearing components from damage when the flywheel is subjected to random vibration in the rocket launch section. Figure 1The dry friction damping assembly shown uses a damping ring to press the friction plate (made of epoxy phenolic glass cloth) onto the spoke of the flywheel to form a friction contact interface. By adjusting the bolt preload (P), the positive pressure of the friction interface can be adjusted to obtain an ideal friction damping vibration reduction effect. The damping ring is made of aluminum (LY12), which has good thermal conductivity and large heat capacity. The vibration reduction principle of the damping ring is: when the flywheel vibrates, the friction plate and the damping ring (the friction plate is bonded to the spoke, not to the damping ring) or the friction plate and the spoke (the friction plate is bonded to the damping ring, not to the spoke) move relative to each other on the contact interface, generating dry friction damping.

[0006] The outstanding advantages of dry friction damping components are: strong damping is achieved with small movements, energy is quickly released in the form of heat, and the heat does not change the friction characteristics of the friction pair material. In addition, the structural form and arrangement of the damping ring have advantages in reducing the weight of the flywheel, reducing the difficulty of dynamic balancing, and maximizing the moment of inertia. At the same time, the vibration amplification of the flywheel is suppressed from the aspect of energy consumption of external dry friction damping, and dry friction damping is provided through the relative movement of multiple contact friction interfaces to consume vibration energy in a timely manner. The damping and vibration reduction effect is insensitive to temperature changes and frequency changes, and is suitable for situations where both the excitation frequency and excitation amplitude change.

[0007] Although damping rings are used in some flywheels at home and abroad in slightly different forms (friction plates bonded to spokes, or bonded to damping rings), the design method for launch section damping rings is still imperfect. At present, the design of flywheel damping rings is based on experience and trial and error, or based on finite element nonlinear model analysis, which is time-consuming and labor-intensive, and has no advantages for rapid iterative design. Summary of the invention

[0008] In view of the defects in the prior art, the purpose of the present invention is to provide a dry friction damping vibration reduction mechanism based on a flywheel damping ring, and propose a flywheel damping ring design method based on a high-precision nonlinear friction dynamics model.

[0009] According to a flywheel damping ring design method based on a high-precision nonlinear friction dynamics model provided by the present invention, the specific steps are as follows:

[0010] S1. According to the elastic support structure characteristics of the flywheel, each component of the flywheel is modeled from bottom to top to obtain the mode of the flywheel linear system;

[0011] S2. Select the material and size of the damping ring according to the size of the flywheel body;

[0012] S3. Determine the material and size of the friction plate;

[0013] S4. Determine the material and size of the rubber ring;

[0014] S5. Obtain the modes of the flywheel-damper ring linear system;

[0015] S6. Establish a high-dimensional nonlinear system dynamics model of a flywheel-damping ring with multiple friction interfaces under basic random excitation;

[0016] S7. Predict the forced response of the flywheel under basic random excitation and determine the normal pressure and friction coefficient;

[0017] S8. Determine the height of the bolt sleeve.

[0018] Preferably, the step S1 further comprises the following steps:

[0019] S11. Modeling the base and shell:

[0020] The base and shell are modeled with solid elements, and the characteristic node on the base is coupled to a massless node and named as the base coupling node;

[0021] S12. Modeling the bearing assembly;

[0022] The bearing assembly is equivalent to a node with mass and named as the bearing mass node. The mass of this point is the real physical mass of the bearing assembly, and three linear spring connections are established between the bearing mass point and the base coupling node.

[0023] S13. Flywheel body modeling:

[0024] The flywheel body is modeled using three-dimensional solid elements, and the nodes of the inner side of the hub and the outer ring of the bearing assembly are coupled to a massless node, which is named the flywheel body coupling node.

[0025] S14. Support stiffness modeling:

[0026] Three linear spring connections are established between the bearing mass point and the flywheel body coupling node;

[0027] Modal analysis is performed on the above finite element model to obtain the modes of the flywheel linear system.

[0028] Preferably, the step S2 is specifically designed according to the actual inertia wheel body structure, the upper and lower damping rings are bonded to the friction plate and the friction plate is pressed against the spoke, a rubber ring is used to achieve hard elastic positioning with the wheel rim in the radial direction, and a friction plate groove is used to achieve rigid positioning with the spoke in the circumferential direction, so as to realize the constraint on the movement except along the length direction of the spoke.

[0029] Preferably, step S3 specifically determines the material and size of the friction plate. The friction plate is made of at least one material selected from silicone rubber, fluororubber, polytetrafluoroethylene, and epoxy resin. The thickness parameters of different friction plates are selected to meet the size of the friction plate to ensure a perfect fit with the damping ring.

[0030] Preferably, step S4 specifically determines the material and size of the rubber ring, the rubber ring is made of at least one material selected from silicone rubber, fluororubber, polytetrafluoroethylene, and epoxy resin, and the outer diameter parameters of different rubber rings are selected to meet the size of the rubber ring to ensure contact with the inner side of the wheel rim.

[0031] Preferably, the step S5 is to establish a finite element model of the flywheel-damping ring under basic excitation using finite element software according to the selected material and size of the damping ring, the material and size of the friction plate, and the material and size of the rubber ring, specifically:

[0032] The large mass point simulates the mass of the vibration table and serves as the point of action of the excitation force. The basic excitation force F is applied to the large mass point. ext (t) = Ma(t)g, where acceleration a(t) (in g, g = 9.80665) is used as the basic input acceleration data.

[0033] The vibration equation of the high-dimensional nonlinear system of flywheel-damping ring is:

[0034]

[0035] Where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the system respectively; x(t), are the displacement vector, velocity vector and acceleration vector of all nodes respectively; F nl (z(t),t) is the friction force vector at all contact nodes on the friction interface; F ext (t) is the excitation force vector acting on the large mass point;

[0036] Modal analysis is also carried out to obtain the modes of the flywheel-damper ring linear system.

[0037] Preferably, step S6 specifically includes establishing a high-dimensional nonlinear system dynamics model of a flywheel-damping ring containing multiple friction interfaces under basic random excitation, reducing the degrees of freedom of the high-dimensional nonlinear system using the modes of the linear system, and using Duhamel integration to solve the vibration equations of the system after the degree of freedom reduction.

[0038] Preferably, step S7 is specifically to predict the forced response of the flywheel under basic random excitation, obtain the fine local dynamic behavior of the friction interface, compare and calculate the acceleration response time history and acceleration response power spectrum density of the flywheel with / without the damping ring under different normal pressures and friction coefficients, and obtain the optimal normal pressure and friction coefficient according to the control effect of the damping ring on the acceleration response of the wheel rim under different normal pressures and friction coefficients.

[0039] Preferably: the step S8 specifically comprises clarifying the elastic deformation of the damping ring under different positive pressures based on the one-to-one correspondence between the bolt preload and the elastic deformation of the damping ring, and according to the elastic deformation of the damping ring = the height of the spoke + the thickness of the two friction plates - the height of the bolt sleeve. When the spoke height and the thickness of the two friction plates are determined, the height of the bolt sleeve can be designed to limit the specific elastic deformation of the damping ring, that is, to obtain the positive pressure required by the design;

[0040] The specific method to determine the height of the bolt sleeve is: establish a finite element model of the damping ring-bolt-sleeve, perform static analysis, calculate the sum of the elastic deformations of the two damping rings under the optimal positive pressure obtained by theoretical analysis, and then calculate the height of the bolt sleeve: the height of the bolt sleeve = the spoke height + the thickness of the two friction plates - the elastic deformation of the two damping rings.

[0041] Compared with the prior art, the technical solution of the present invention is as follows: based on the one-to-one correspondence between the bolt preload and the elastic deformation of the damping ring, the elastic deformation of the damping ring under different positive pressures is clarified. When the spoke height and the thickness of the two friction plates are determined, the height of the bolt sleeve can be designed to limit the damping ring to produce a specific elastic deformation, that is, to obtain the positive pressure required by the design. Through the above iterative analysis, the material and size of the damping ring, the material and size of the friction plate, the material and size of the rubber ring, and the height of the bolt sleeve can be determined, which can ensure the optimal positive pressure and a better friction coefficient of the interface. The method has a short calculation time and can obtain a high-precision response, which has advantages for the rapid iterative design of the flywheel damping ring. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Other features, objects and advantages of the present invention will become more apparent from the detailed description of non-limiting embodiments made with reference to the following drawings:

[0043] Figure 1 It is a schematic diagram of the flywheel damping ring design method of the present invention.

[0044] Figure 2 An embodiment of the present invention includes: a flywheel object and a three-dimensional model thereof;

[0045] (a) Actual photo of the flywheel; (b) 3D model of the flywheel; (c) Cross-sectional view of the 3D model of the flywheel

[0046] Figure 3 An implementation scheme of the present invention: a schematic diagram of the dynamic modeling of a flywheel and its finite element model

[0047] (a) Schematic diagram of flywheel dynamic modeling; (b) 3D solid finite element model of flywheel.

[0048] Figure 4 It is an embodiment of the present invention: the structure of the damping ring and the installation form on the flywheel;

[0049] (a) Structure of the damping ring; (b) Installation form of the damping ring; (c) Cross-sectional view of the three-dimensional model of the flywheel with the damping ring installed

[0050] Figure 5 An embodiment of the present invention: Finite element model of flywheel-damping ring under base excitation

[0051] Figure 6 An embodiment of the present invention: Acceleration response of the rim and bearing and acceleration response PSD without damping ring

[0052] (a) PSD of the acceleration response of the wheel rim; (b) PSD of the acceleration response of the bearing

[0053] Figure 7 An embodiment of the present invention: Acceleration response of the rim and bearing with a damping ring and acceleration response PSD

[0054] (a) PSD of the acceleration response of the wheel rim; (b) PSD of the acceleration response of the bearing

[0055] Figure 8 An embodiment of the present invention: the acceleration response of the rim and the bearing without / with a damping ring and the acceleration response PSD

[0056] (a) Acceleration response of wheel rim; (b) Acceleration response of bearing; (c) PSD of acceleration response of wheel rim; (d) PSD of acceleration response of bearing

[0057] Fig. 9 An embodiment of the present invention: Acceleration response of the wheel rim under vertical sine sweep excitation without / with damping ring

[0058] (a) Rim acceleration response (vertical) without damping ring; (b) Rim acceleration response (vertical) with damping ring

[0059] Fig.10 An embodiment of the present invention: Transverse sinusoidal sweep excitation signal and acceleration response of the wheel rim without / with damping ring

[0060] (a) Rim acceleration response (lateral) without damping ring; (b) Rim acceleration response (lateral) with damping ring DETAILED DESCRIPTION

[0061] The present invention is described in detail below in conjunction with specific embodiments. The following embodiments will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those of ordinary skill in the art, several changes and improvements can also be made without departing from the concept of the present invention. These all belong to the protection scope of the present invention.

[0062] See also Figure 1 , and Figure 2 Taking the flywheel shown in the figure as an example, the design method proposed by the present invention is briefly described as follows;

[0063] 1) Obtain the modes of the flywheel linear system;

[0064] Figure 2 In the figure, according to the elastic support relationship from bottom to top, the components of the flywheel are: mounting base, motor stator assembly, connecting shaft, disc spring, bearing assembly, wheel body (including the motor rotor assembly connected thereto) and housing. That is, the main and most important structural feature of the flywheel is that the disc spring and the bearing assembly elastically support the wheel body on the base in a series, with a clear elastic support structural feature of base → disc spring → bearing assembly → wheel body from bottom to top.

[0065] The schematic diagram of the dynamic modeling of the flywheel is as follows: Figure 3 As shown in (a). The “housing” marked in the figure includes the base and the housing; the “bearing stiffness” is the series stiffness of the disc spring and the bearing assembly; the “bearing mass point” represents the bearing assembly, and the mass of the mass point is the actual physical mass of the bearing assembly; the “support stiffness” is the parallel stiffness of the five spokes (which can be regarded as five cantilever beams that elastically support the mass of the wheel rim); the “wheel body” includes the flywheel body and the motor rotor assembly connected to it; the “transmission force” is the force transmitted to the bearing assembly when the wheel body vibrates. This study aims to reduce the transmission force by reducing the vibration of the wheel body. According to Figure 3 (a) shows the schematic diagram of the flywheel dynamic modeling. A coupled linear three-dimensional solid finite element model of the flywheel body solid structure-bearing assembly equivalent parameters-shell solid structure-foundation is established, as shown in Figure 3 (b) As shown. According to the material properties and parameters of the wheel body and the housing, the corresponding parameters are assigned to the support stiffness, the real mass of the bearing assembly and its equivalent stiffness. The finite element model of the flywheel is a dynamic model consisting of the three-dimensional solid finite element model of the flywheel body and the housing in the free state, the real mass and equivalent stiffness of the bearing assembly, and the spoke support stiffness in the installed state. The elasticity of the housing is considered during modeling, and the bearing stiffness is assumed to be linear.

[0066] According to the elastic support structure characteristics of the flywheel, following the bottom-up modeling sequence, the modeling method and modeling steps of the finite element model of the flywheel are as follows:

[0067] (A) Modeling of base and shell

[0068] The base and shell are not simplified or equivalent, and three-dimensional solid elements are used for modeling. All nodes supporting the disc spring on the base are coupled to a massless node, which is named the base coupling node.

[0069] Boundary conditions: When performing modal analysis, the six degrees of freedom of all nodes on the base mounting surface are fixed.

[0070] (B) Bearing assembly modeling

[0071] Consider the mass of the bearing assembly and the vertical and lateral stiffness of the elastically supported wheel. The bearing assembly is equivalent to a node with mass, and the node is named the bearing mass point. The mass of the mass point is the real physical mass of the bearing assembly. Three linear spring connections are established between the bearing mass point and the base coupling node to provide stiffness in three directions, including vertical (axial, Z direction) and lateral (radial, X, Y direction).

[0072] (C) Flywheel body modeling

[0073] The flywheel body is not simplified or treated equivalently, and is also modeled using three-dimensional solid elements. All nodes in contact between the inner side of the hub and the outer ring of the bearing assembly are coupled to a massless node, which is named the flywheel body coupling node.

[0074] (D) Support stiffness modeling

[0075] Considering that the spokes provide both vertical and lateral stiffness to support the rim, three linear spring connections are also established between the bearing mass point and the flywheel body coupling node to provide stiffness in three directions, including vertical and lateral directions.

[0076] Perform modal analysis on the established finite element model to obtain the modes of the flywheel linear system;

[0077] 2) Select the material and size of the damping ring according to the size of the flywheel body;

[0078] The structure of the damping ring is as follows Figure 4 As shown in (a), it mainly consists of two damping rings, ten friction plates, five bolt sleeves and ten rubber rings for installation and positioning. The damping ring is installed on the flywheel as shown in Figure 4 As shown in (b)(c).

[0079] The structural dimension parameters of the damping ring are (taking this flywheel as an example): the outer diameter of the damping ring r1=308mm, the inner diameter r2=260mm; the thickness of the friction plate t=1mm, the length of the rectangular friction interface in the groove l=24mm, the width w=16mm; the height of the bolt sleeve h=20mm, the outer diameter d1=8mm, the inner diameter d2=4mm.

[0080] The structural principle of the damping ring is as follows: the friction plate is clamped on the spoke through its own groove, and the two damping rings are connected by bolts to press the friction plate on the spoke, forming multiple contact friction interfaces between the friction plate and the spoke. Among them, the function of the bolt is to press the damping ring and provide positive pressure on the friction interface; the elastic deformation of the damping ring = the height of the spoke + the thickness of the two friction plates - the height of the bolt sleeve. When the height of the spoke and the thickness of the two friction plates are determined, the elastic deformation of the damping ring can be changed by adjusting the height of the bolt sleeve. Ten installation positioning rubber rings play a role in limiting the deformation of the damping ring.

[0081] The working principle of the damping ring is: when the flywheel is excited by the foundation, it is excited to resonate. Since the damping ring is restricted by ten installation and positioning rubber rings, the vibration displacement of the friction plate and the spoke is inconsistent, and the two move relative to each other on the contact interface, generating friction damping. The friction damping consumes the vibration energy of the flywheel in time, reduces the load (transmission force) transmitted to the bearing assembly when the flywheel vibrates, and plays a role in protecting the bearing assembly. That is, the damping ring's suppression mechanism is to consume energy through dry friction damping at the contact interface.

[0082] The design principle of the damping ring is: at the contact interface between the friction plate and the spoke, the friction force mainly depends on the friction coefficient of the friction plate and the positive pressure of the friction interface. Usually the friction coefficient is determined (depending on the material and surface roughness of the friction plate), and the ideal damping and vibration reduction effect can be obtained by simply adjusting the bolt preload to change the positive pressure of the friction interface. The method of adjusting the bolt preload is to change the height of the bolt sleeve to achieve the purpose of limiting the elastic deformation of the damping ring, thereby obtaining the positive pressure required by the design.

[0083] Select different damping rings such as outer diameter, inner diameter, thickness and other structural dimensions and inclined angles. The design is required to be based on the actual inertia wheel structure. The upper and lower damping rings are bonded to the friction plate and the friction plate is pressed against the spoke. The rubber ring is used to achieve hard elastic positioning with the wheel rim in the radial direction, and the friction plate groove is used to achieve rigid positioning with the spoke in the circumferential direction, so as to realize the constraint of movement except along the length direction of the spoke.

[0084] 3) Determine the material and size of the friction plate;

[0085] The friction plate is made of various materials such as silicone rubber, fluororubber, polytetrafluoroethylene, epoxy resin, etc., and the thickness parameters of different friction plates are selected. The requirement is that the size of the friction plate ensures a perfect fit with the damping ring;

[0086] 4) Determine the material and size of the rubber ring;

[0087] The rubber ring is made of various materials such as silicone rubber, fluororubber, polytetrafluoroethylene, epoxy resin, etc., and the outer diameter parameters of different rubber rings are selected. The requirement is that the size of the rubber ring ensures contact with the inner side of the wheel rim;

[0088] 5) Obtain the modes of the flywheel-damping ring linear system;

[0089] like Figure 5 As shown in the figure, the nodes on the bottom surface of the flywheel in the three-dimensional solid finite element model of the flywheel-damping ring are coupled to the node with a mass of M = 10 4 kg node (i.e., large mass point), we established Figure 5 The linear system finite element model of the flywheel-damper ring under base excitation is shown. A large mass point is used to simulate the mass of the vibration table and act as the point of action of the excitation force. The base excitation force F is applied to the large mass point. ext (t) = Ma (t) g. Here, acceleration a (t) (unit: g, g = 9.80665) is the basic input acceleration data.

[0090] Figure 5 The vibration equation of the high-dimensional nonlinear system of the flywheel-damper ring shown is:

[0091]

[0092] Where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the system respectively; x(t), are the displacement vector, velocity vector and acceleration vector of all nodes respectively; F nl (z(t),t) is the friction force vector at all contact nodes on the friction interface; F ext (t) is the excitation force vector acting on the large mass point.

[0093] Modal analysis is carried out on the established model to obtain the modes of the flywheel-damper ring linear system.

[0094] 6) Establish a high-dimensional nonlinear system dynamics model of the flywheel-damping ring with multiple friction interfaces under basic random excitation;

[0095] The elastic Coulomb friction force on each contact node is the relative displacement z of the contact node along the tangent direction of the friction interface. t (t) = x t (t)-y t(t) and the speed of the friction contact point A function where x t (t), y t (t) are the displacement vectors of the contact nodes on the spoke and friction plate along the tangential direction of the friction interface. Since the damping ring is constrained by ten positioning rubber rings, the tangential displacement of the contact node on the friction plate is 0, that is, y t (t) = 0. Therefore, the relative displacement of the flywheel and the damping ring along the friction interface is equal to the tangential vibration displacement of the contact node on the spoke, that is, z t (t) = x t (t). At each contact node, the friction force along the tangent direction of the friction interface is:

[0096]

[0097] In the formula, k t is the tangential stiffness; is the tangential displacement of the i-th contact node on the spoke; and are the tangential displacement and tangential velocity of the i-th friction contact point respectively; N i is the normal pressure at the i-th contact node.

[0098] Equation (1) is divided into linear degrees of freedom and nonlinear degrees of freedom (degrees of freedom of contact nodes on the friction interface). According to the positions of contact nodes and non-contact nodes on the finite element model, the vibration equation (4-1) of the flywheel-damping ring high-dimensional nonlinear system is rearranged according to the degrees of freedom, that is, written as the nonlinear degrees of freedom composed of contact nodes (with superscript " n ”) and the linear degrees of freedom composed of nodes outside the friction interface (denoted by superscript “ l ") indicates the form of:

[0099]

[0100] Equation (3) describes a high-dimensional nonlinear system, the dimension of which is equal to Figure 2 The number of nodes in the finite element model of the flywheel-damping ring is shown as × the number of degrees of freedom. Since the number of nodes in the finite element model of the flywheel-damping ring is large, the calculation amount of formula (3) is large.

[0101] In order to reduce the dimension of the equation group (3), the method of degree of freedom reduction is used to reduce the dimension, that is, the modal vibration matrix is ​​used to transform the coordinates of the coupled vibration equation group (3) to reduce the amount of calculation. When reducing the degree of freedom, all nonlinear degrees of freedom (degrees of freedom of all contact nodes of the friction interface) and the linear degrees of freedom of interest (degrees of freedom of the rim response node, the degree of freedom of the bearing response node, and the degree of freedom of the excitation force action node, i.e., the large mass point) are retained. The mass normalized vibration matrix obtained by calculation is used to transform the coordinates of equation (3):

[0102]

[0103] Where Φ is the mass normalized vibration matrix; q(t) is the modal coordinate vector; is the vibration mode matrix of the contact nodes on the spoke along the tangent direction of the friction interface.

[0104] Φ=[Φ f Φ M Φ resp Φ b ] T (5)

[0105] Φ f =[Φ f,1 ·Φ f,i ·Φ f,5 ] T (6)

[0106] In the formula, Φ f,i is the vibration mode matrix of all contact nodes on the i-th spoke; Φ M is the vibration mode matrix of the node (i.e., the large mass point) where the excitation force acts, Φ resp and Φ b They are the vibration mode matrices of the rim response node and the bearing response node respectively.

[0107] The vibration equation in the modal domain is obtained as follows:

[0108]

[0109] Where m is the modal order, that is, the first m-order continuous modes of the flywheel-damping ring (including the first three rigid body modes) are selected, and the sum of the modal masses of the first m-order modes is 99% of the total mass of the flywheel-damping ring; ω f,k , f,k are the circular frequency and damping ratio of the kth mode of the flywheel-damper ring respectively.

[0110] So far, a high-dimensional nonlinear system dynamic model of the flywheel-damper ring containing multiple friction interfaces under basic random excitation has been established, and the solution is now carried out.

[0111] 7) Predict the forced response of the flywheel under basic random excitation to obtain the optimal normal pressure and friction coefficient;

[0112] Rewrite equation (7) as:

[0113]

[0114] Where, {q(t)} m×1 is the modal coordinate vector; {f ext,i (t)} 1×t =Mg{a i (t)} 1×t is the excitation force vector time signal; {a i (t)} 1×t is the acceleration time signal (in g) of the measured vibration table, and a total of t acceleration values ​​are collected; M = 10 4 kg is the mass of the large mass point; g = 9.80665 is a constant; is the vertical vibration mode vector of the large mass point; is the matrix composed of the tangential friction forces on all contact nodes; is the vibration mode matrix of all contact nodes along the tangent direction of the friction interface.

[0115] By calculating the vibration equation group (8), the modal coordinate vector of the flywheel-damping ring system in the modal domain can be obtained. The displacement and velocity of each single-degree-of-freedom vibration system in equation (8) are calculated using Duhamel integration:

[0116]

[0117]

[0118]

[0119] In the formula, x0 and are the initial displacement and initial velocity respectively; in

[0120] The displacement x(t) and velocity of each single-degree-of-freedom vibration system are calculated by equations (8) and (9): Then, x(t) and Substituting into equation (8), calculate the acceleration of each single-degree-of-freedom vibration system

[0121] The friction force is closely related to the stick-slip behavior of the friction interface. From equation (2), we can see that the stick-slip behavior of each contact node on the friction interface is related to the tangential displacement of the contact node. Displacement of the friction contact point and speed Critical friction μ f N i and viscosity The transition of the friction interface from the viscous state to the slip state must satisfy The transition from slip state to sticky state must satisfy and In formula (14), the friction matrix of all contact nodes is It is calculated based on the tangential displacement of the contact node. The friction calculation process of the i-th contact node is as follows:

[0122] (A) At the start time t0 = 0, the displacement, velocity and acceleration of the flywheel are all 0, the friction force is 0, and the flywheel is subjected to the excitation force f ext (t0). The excitation force f ext (t0) is converted into the excitation force in the modal domain, and the modal coordinates q(t0) at time t0 and the displacement of the contact node in the physical space are calculated using Duhamel integration By judging Is it true? Obtain the sticking / slipping state of each contact node and calculate the friction force f at time t0 nl,i (t0).

[0123] (B) At time t1, the flywheel is simultaneously subjected to friction f nl,i (t0) and the excitation force f ext (t1) action. The friction force f in the time domain nl,i (t0) and the excitation force f ext (t1) is converted into friction and excitation force in the modal domain, and the modal coordinates q(t1) at time t1 and the displacement of the contact node in the physical space are calculated using Duhamel integration. By judging Is it true? Obtain the sticking / slipping state of each contact node and calculate the friction force f at time t1 nl,i (t1).

[0124] (C) And so on until t n At the same time, the flywheel is subjected to friction f nl,i (t n-1 ) and the incentive f ext (t n ) effect. The friction force f in the time domain nl,i (t n-1 ) and the incentive f ext (t n ) is converted into the excitation force in the modal domain and t is calculated using Duhamel integration. n The modal coordinates q(t n ) and the displacement of the contact node in physical space By judging Is it true? Obtain the sticking / slipping state of each contact node and calculate t n Friction force at time f nl,i (t n ).

[0125] Through the above steps, the forced response of the flywheel under basic random excitation can be obtained for prediction, the fine local dynamic behavior of the friction interface can be obtained, and the acceleration response time history and acceleration response power spectral density of the flywheel with / without damping ring under different normal pressures and friction coefficients can be compared and calculated. According to the control effect of the damping ring on the acceleration response of the wheel rim under different normal pressures and friction coefficients, the optimal normal pressure and friction coefficient can be obtained.

[0126] 8) Determine the height of the bolt sleeve.

[0127] Based on the one-to-one correspondence between the bolt preload and the elastic deformation of the damping ring, the elastic deformation of the damping ring under different positive pressures is clarified. According to the elastic deformation of the damping ring = the height of the spoke + the thickness of the two friction plates - the height of the bolt sleeve, when the spoke height and the thickness of the two friction plates are determined, the height of the bolt sleeve can be designed to limit the damping ring to produce a specific elastic deformation, that is, to obtain the positive pressure required by the design; the method for determining the height of the bolt sleeve is: establishing a finite element model of the damping ring-bolt-sleeve, performing static analysis, calculating the sum of the elastic deformations of the two damping rings under the optimal positive pressure obtained by theoretical analysis, and then calculating the height of the bolt sleeve: the height of the bolt sleeve = the height of the spoke + the thickness of the two friction plates - the elastic deformation of the two damping rings.

[0128] against Figure 2 The optimal bolt preload is 140 N for the flywheel shown in the figure. A high-dimensional nonlinear system dynamics model of a flywheel-damping ring with multiple friction interfaces under basic random excitation is established, and the calculation results show that the maximum preload (F nl (z(t), t) = 0) The acceleration response time history and acceleration response PSD of the wheel rim and bearing are compared with the test results, such as Figure 6 The acceleration response time history and acceleration response PSD of the wheel rim and bearing with the damping ring were calculated and compared with the test results, as shown in Figure 2. Figure 7 shown.

[0129] Depend on Figure 6 and Figure 7 It can be seen that the calculated acceleration response time history and acceleration response PSD of the rim and bearing with / without the damping ring are basically consistent with the test results, indicating that the established high-dimensional nonlinear dynamic model can accurately characterize the dynamic characteristics of the actual flywheel-damping ring system.

[0130] Compare the calculated acceleration time history and acceleration response PSD of the wheel rim and bearing without / with damping ring, as shown in Figure 2. Figure 8 As shown. Figure 8 (a)(b) It can be seen that the acceleration response time history of the flywheel and bearing with the damping ring is reduced compared with the calculation results without the damping ring. Figure 8 (c) (d) It can be seen that when there is no damping ring, the acceleration response PSD of the wheel rim at a frequency of 207.3 Hz is 1.3530 g. 2 / Hz; when there is a damping ring, the acceleration response PSD of the wheel rim at a frequency of 200.7Hz is 0.8296g 2 / Hz. It can be seen that after the damping ring is introduced, the PSD peak value of the acceleration response of the rim decreases by 4.25dB. Figure 8 (e) (f) It can be seen that when there is no damping ring, the acceleration response PSD of the bearing at a frequency of 207.3 Hz is 0.2960 g. 2 / Hz; when there is a damping ring, the acceleration response PSD of the bearing at a frequency of 200.7Hz is 0.1746g 2 / Hz. It can be seen that after the damping ring is introduced, the PSD peak value of the bearing acceleration response decreases by 4.58dB.

[0131] Design-based approach for Figure 2 The damping ring of the flywheel shown was designed and manufactured, and tested at the Testing Center of Shanghai Institute of Aerospace Control Technology. The flywheel with or without damping ring was subjected to vibration test under 0.5g sine sweep excitation on a vibration table, and the acceleration response of the wheel rim was measured with an acceleration sensor. Fig. 9 and Fig.10 The vertical and lateral acceleration responses of the wheel rim obtained from tests with and without a damping ring are given respectively.

[0132] Depend on Fig. 9 It can be seen that under the vertical sine sweep excitation of 0.5g, the acceleration response curve of the wheel rim has prominent peaks at 213.7Hz and 207.9Hz. From the modal analysis and test results of the flywheel, it can be seen that the emergence of the prominent peaks is due to the resonance of the umbrella mode under vertical excitation, resulting in vibration amplification. Fig. 9 (a) It can be seen that when there is no damping ring, the acceleration of the wheel rim at a frequency of 213.7 Hz is 38.00 g. Fig. 9 (b) It can be seen that when the damping ring is installed, the acceleration of the wheel rim at a frequency of 207.9 Hz is 16.71 g. The test results show that after the damping ring is installed, the peak value of the acceleration response of the wheel rim under the vertical sine sweep excitation of 0.5 g decreases by 7.14 dB.

[0133] Depend on Fig.10It can be seen that under the 0.5g lateral sinusoidal sweep excitation, the acceleration response curve of the wheel rim has prominent peaks at 266.2Hz and 254.0Hz. From the modal analysis and test results of the flywheel, it can be seen that the appearance of the prominent peaks is due to the resonance of the translational mode under lateral excitation, resulting in vibration amplification. Fig.10 (a) It can be seen that when there is no damping ring, the acceleration of the wheel rim at a frequency of 266.2 Hz is 14.007 g. Fig.10 (b) It can be seen that when the damping ring is installed, the acceleration of the wheel rim at a frequency of 254.0 Hz is 6.889 g. The test results show that after the damping ring is installed, the peak value of the acceleration response of the wheel rim decreases by 6.16 dB under the 0.5 g lateral sine sweep excitation.

[0134] The above embodiments show that the flywheel damping ring design method based on the high-precision nonlinear friction dynamics model of the present invention can quickly design a damping ring with high-efficiency vibration suppression performance.

[0135] The preferred embodiments of the present invention disclosed above are only used to help illustrate the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to the specific implementation methods described. Obviously, many modifications and changes can be made according to the content of this specification. This specification selects and specifically describes these embodiments in order to better explain the principles and practical applications of the present invention, so that those skilled in the art can understand and use the present invention well. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A flywheel damping ring design method based on a high-precision nonlinear friction dynamics model, characterized in that: The specific steps are as follows: S1. According to the elastic support structure characteristics of the flywheel, each component of the flywheel is modeled from bottom to top to obtain the mode of the flywheel linear system; S2. Select the material and size of the damping ring according to the size of the flywheel body; S3. Determine the material and size of the friction plate; S4. Determine the material and size of the rubber ring; S5. Obtain the modes of the flywheel-damper ring linear system; S6. Establish a high-dimensional nonlinear system dynamics model of a flywheel-damping ring with multiple friction interfaces under basic random excitation; S7. Predict the forced response of the flywheel under basic random excitation and determine the normal pressure and friction coefficient; S8. Determine the height of the bolt sleeve.

2. A flywheel damping ring design method based on a high-precision nonlinear friction dynamics model according to claim 1, characterized in that: The step S1 further comprises the following steps: S11. Modeling the base and shell: The base and shell are modeled with solid elements, and the characteristic node on the base is coupled to a massless node and named as the base coupling node; S12. Modeling the bearing assembly; The bearing assembly is equivalent to a node with mass and named as the bearing mass node. The mass of this point is the real physical mass of the bearing assembly, and three linear spring connections are established between the bearing mass point and the base coupling node. S13. Flywheel body modeling: The flywheel body is modeled using three-dimensional solid elements, and the nodes of the inner side of the hub and the outer ring of the bearing assembly are coupled to a massless node, which is named the flywheel body coupling node. S14. Support stiffness modeling: Three linear spring connections are established between the bearing mass point and the flywheel body coupling node; Modal analysis is performed on the above finite element model to obtain the modes of the flywheel linear system.

3. A flywheel damping ring design method based on a high-precision nonlinear friction dynamics model according to claim 1, characterized in that: The step S2 is specifically designed according to the actual inertia wheel body structure, the upper and lower damping rings are bonded to the friction plate and the friction plate is pressed against the spoke, a hard elastic positioning is achieved with the wheel rim in the radial direction through a rubber ring, and a rigid positioning is achieved with the spoke in the circumferential direction through a friction plate groove, so as to realize the constraint on the movement except along the length direction of the spoke.

4. The flywheel damping ring design method based on a high-precision nonlinear friction dynamics model according to claim 1 is characterized in that: The step S3 specifically determines the material and size of the friction plate. The friction plate is made of at least one material selected from silicone rubber, fluororubber, polytetrafluoroethylene, and epoxy resin. The thickness parameters of different friction plates are selected to meet the size of the friction plate to ensure a perfect fit with the damping ring.

5. The flywheel damping ring design method based on a high-precision nonlinear friction dynamics model according to claim 1 is characterized in that: The step S4 specifically determines the material and size of the rubber ring. The rubber ring is made of at least one material selected from silicone rubber, fluororubber, polytetrafluoroethylene, and epoxy resin. The outer diameter parameters of different rubber rings are selected to meet the size of the rubber ring to ensure contact with the inner side of the wheel rim.

6. A flywheel damping ring design method based on a high-precision nonlinear friction dynamics model according to claim 1, characterized in that: The step S5 is specifically to establish a finite element model of the flywheel-damping ring under basic excitation using finite element software according to the selected material and size of the damping ring, the material and size of the friction plate, and the material and size of the rubber ring, specifically: The large mass point simulates the mass of the vibration table and serves as the point of action of the excitation force. The basic excitation force F is applied to the large mass point. ext (t) = Ma(t)g, where acceleration a(t) (in g, g = 9.80665) is used as the basic input acceleration data. The vibration equation of the high-dimensional nonlinear system of flywheel-damping ring is: Where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the system respectively; x(t), are the displacement vector, velocity vector and acceleration vector of all nodes respectively; F nl (z(t),t) is the friction force vector at all contact nodes on the friction interface; F ext (t) is the excitation force vector acting on the large mass point; Modal analysis is also carried out to obtain the modes of the flywheel-damper ring linear system.

7. A flywheel damping ring design method based on a high-precision nonlinear friction dynamics model according to claim 1, characterized in that: The step S6 specifically includes establishing a high-dimensional nonlinear system dynamics model of a flywheel-damping ring containing multiple friction interfaces under basic random excitation, reducing the degrees of freedom of the high-dimensional nonlinear system using the modes of the linear system, and solving the vibration equations of the system after the degree of freedom reduction using Duhamel integration.

8. The flywheel damping ring design method based on a high-precision nonlinear friction dynamics model according to claim 1, characterized in that: The step S7 is specifically to predict the forced response of the flywheel under basic random excitation, obtain the fine local dynamic behavior of the friction interface, compare and calculate the acceleration response time history and acceleration response power spectrum density of the flywheel with / without the damping ring under different normal pressures and friction coefficients, and obtain the optimal normal pressure and friction coefficient according to the control effect of the damping ring on the acceleration response of the wheel rim under different normal pressures and friction coefficients.

9. A flywheel damping ring design method based on a high-precision nonlinear friction dynamics model according to claim 1, characterized in that: The step S8 specifically includes clarifying the elastic deformation of the damping ring under different positive pressures based on the one-to-one correspondence between the bolt preload and the elastic deformation of the damping ring. According to the elastic deformation of the damping ring = the height of the spoke + the thickness of the two friction plates - the height of the bolt sleeve, when the spoke height and the thickness of the two friction plates are determined, the height of the bolt sleeve can be designed to limit the elastic deformation of the damping ring to a specific value, that is, to obtain the positive pressure required by the design. The specific method to determine the height of the bolt sleeve is: establish a finite element model of the damping ring-bolt-sleeve, perform static analysis, calculate the sum of the elastic deformations of the two damping rings under the optimal positive pressure obtained by theoretical analysis, and then calculate the height of the bolt sleeve: the height of the bolt sleeve = the spoke height + the thickness of the two friction plates - the elastic deformation of the two damping rings.

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