Method for calculating axial displacement of thrust part of diesel engine

Abstract

The invention relates to a diesel engine thrust part axial displacement calculation method which comprises the steps that a right side shaft section of a thrust part is simplified into a multi-section Timoshenko beam connecting structure, and a shaft system dynamic model is established according to the multi-section Timoshenko beam connecting structure; establishing a transfer matrix of the shafting dynamic model by using a transfer matrix method; constructing a frequency equation according to the transfer matrix, and solving the frequency equation by using a Newton iteration method to determine a vibration mode function of the shafting dynamic model; determining kinetic energy and potential energy of the shafting dynamical model according to the vibration mode function, and substituting the kinetic energy and the potential energy into a second type of Lagrange equation to obtain an oscillatory differential equation set; and solving the oscillatory differential equation set by using a Newmark-beta method to obtain the axial displacement of the right shaft section of the thrust part. The axial displacement of the right shaft section of the thrust piece is deduced while the calculation precision is guaranteed, so that the influence of the axial displacement of the thrust piece on a shaft system is analyzed from the perspective of vibration.

Description

Technical Field

[0001] The present disclosure relates to the technical field of diesel engine shaft system vibration, and in particular to a method for calculating the axial displacement of a thrust member of a diesel engine. Background Art

[0002] The thrust piece of a diesel engine is a crucial positioning component that limits axial movement and primarily bears axial forces. During operation, the drive shaft is often subject to the axial force exerted by the clutch on the flywheel, causing it to move axially. This axial movement can sometimes cause the thrust piece to collide and wear against the engine body, leading to abnormal engine operation and, in severe cases, complete engine inoperability.

[0003] Existing research has focused on the harmful effects of thrust-side wear on diesel engine operation and proposed corresponding solutions, but the impact of thrust members on the shafting system from a vibration perspective has yet to be analyzed. If the axial displacement of a diesel engine's drive shafting exceeds the axial clearance on the thrust face under certain operating conditions, the thrust member will collide with the engine body, causing wear. Therefore, calculating the axial displacement of the thrust face is crucial for preventing vibration damage to diesel engines.

[0004] Therefore, it is necessary to provide a new technical solution to improve one or more problems existing in the above solutions.

[0005] It should be noted that the information disclosed in the above background technology section is only used to enhance the understanding of the background of the present disclosure, and therefore may include information that does not constitute prior art known to ordinary technicians in the field. Summary of the Invention

[0006] The purpose of the embodiments of the present disclosure is to provide a method for calculating the axial displacement of a diesel engine thrust member, which can derive the axial displacement of the right shaft section of the thrust member, so as to analyze the influence of the thrust member on the shaft system from the perspective of vibration.

[0007] According to a first aspect of an embodiment of the present disclosure, a method for calculating the axial displacement of a thrust member of a diesel engine is provided, comprising:

[0008] The right shaft section of the thrust member is simplified into a multi-segment Timoshenko beam connection structure, and a shaft system dynamics model is established based on the multi-segment Timoshenko beam connection structure;

[0009] Establishing a transfer matrix of the shafting dynamics model using a transfer matrix method;

[0010] Constructing a frequency equation according to the transfer matrix, and solving the frequency equation using a Newton iteration method to determine a vibration mode function of the shafting dynamics model;

[0011] determining the kinetic energy and potential energy of the shaft system dynamics model according to the mode shape function, and substituting the kinetic energy and potential energy into the second kind Lagrangian equation to obtain a set of vibration differential equations;

[0012] The vibration differential equations are solved using the Newmark-β method to obtain the axial displacement of the right shaft section of the thrust member.

[0013] In an exemplary embodiment of the present disclosure, solving the vibration differential equations using the Newmark-β method to obtain the axial displacement of the right shaft segment of the thrust member includes:

[0014] Solving the vibration differential equations using the Newmark-β method to obtain the lateral displacement of the right shaft segment of the thrust member;

[0015] Based on the coupling stiffness theory, the change in spring potential energy caused by the lateral displacement is determined, and the change in spring potential energy is substituted into the second-kind Lagrangian equation. The Newmark-β method is used to solve the vibration differential equation group to obtain the axial force and the axial displacement of the right shaft segment of the thrust member.

[0016] In an exemplary embodiment of the present disclosure, the right shaft section of the thrust member includes a first transmission shaft, a flywheel, an elastic coupling, and a second transmission shaft arranged in sequence along the axial direction;

[0017] The method of simplifying the right shaft section of the thrust member into a multi-section Timoshenko beam connection structure includes:

[0018] The first transmission shaft is simplified into a first section beam unit, the flywheel and the highly elastic left end of the elastic coupling are simplified into a second section beam unit, the highly elastic right end of the elastic coupling is simplified into a third section beam unit, and the second transmission shaft is simplified into a fourth section beam unit; the multi-section Timoshenko beam connection structure includes the first section beam unit, the second section beam unit, the third section beam unit, and the fourth section beam unit.

[0019] In an exemplary embodiment of the present disclosure, the step of establishing a shaft system dynamics model according to the multi-segment Timoshenko beam connection structure includes:

[0020] Setting constraint conditions for the multi-segment Timoshenko beam connection structure;

[0021] Among them, the constraints include the elastic constraint condition of the left end of the first beam section unit, the rigid connection constraint condition between the first beam section unit and the second beam section unit, the elastic connection constraint condition between the second beam section unit and the third beam section unit, the rigid connection constraint condition between the third beam section unit and the fourth beam section unit, and the elastic constraint condition of the right end of the fourth beam section unit.

[0022] In an exemplary embodiment of the present disclosure, the use of a transfer matrix method to establish a transfer matrix of the shaft system dynamics model includes:

[0023] The lateral transfer relationship and axial transfer relationship of each beam unit of the multi-segment Timoshenko beam connection structure are expressed by using the transfer matrix method;

[0024] determining a spring boundary relationship of the multi-segment Timoshenko beam connection structure;

[0025] According to the lateral transfer relationship and the axial transfer relationship of each beam unit segment, combined with the constraint conditions and the spring boundary relationship of the multi-segment Timoshenko beam connection structure, the transfer matrix of the shaft system dynamics model is determined.

[0026] In an exemplary embodiment of the present disclosure, the lateral transfer relationship of each segment beam unit is expressed as:

[0027]

[0028] Among them, T xy represents the lateral transfer matrix, y l , y0 represents the lateral displacement, θ l , θ0 represents the rotation angle, M zl 、M z0 represents the bending moment, Q yl , Q y0 represents the lateral force, 0 and l represent the coordinate position of each beam element in the length direction;

[0029] The axial transfer relationship of each beam element is expressed as:

[0030]

[0031] Among them, T x represents the axial transfer matrix; U(0) and F(0) represent the axial displacement and axial force for the left end point of the straight beam when x=0; U(l) and F(l) represent the axial displacement and axial force for the right end point of the straight beam when x=l.

[0032] In an exemplary embodiment of the present disclosure, the transfer matrix of the shaft system dynamics model includes a lateral transfer matrix and an axial transfer matrix; wherein the lateral transfer matrix and the axial transfer matrix are expressed as:

[0033]

[0034] Among them, S y represents the lateral transfer matrix, S x represents the axial transfer matrix, K Y represents the lateral transfer matrix between springs, K X represents the axial transfer matrix between springs, T 4Y represents the bending moment of the fourth beam element in lateral vibration, T 3Y represents the bending moment of the third beam element in lateral vibration, T 2Y represents the bending moment of the second beam element in lateral vibration, T 1Y represents the bending moment of the first beam element in lateral vibration.

[0035] In an exemplary embodiment of the present disclosure, the kinetic energy and potential energy of the shaft system dynamics model include the kinetic energy and potential energy of lateral vibration and the kinetic energy and potential energy of axial vibration; wherein the kinetic energy and potential energy of the lateral vibration are expressed as:

[0036]

[0037] Among them, T h Represents the kinetic energy of transverse vibration, V h represents the potential energy of lateral vibration, l represents the length of the beam element, E represents the elastic modulus, A represents the cross-sectional area of ​​the beam element, ρ represents the density, represents the square of the transverse component of the mode function, express The second derivative of represents the square of the generalized coordinates of the i-th mode, represents the square of the generalized velocity of the i-th mode, h represents the sign of the lateral vibration, i represents the mode number, and t represents the time;

[0038] The kinetic energy and potential energy of the axial vibration are expressed as:

[0039]

[0040] Among them, T x represents the kinetic energy of axial vibration, x represents the sign of axial vibration, V x represents the potential energy of axial vibration, represents the square of the axial mode function, express The first derivative of k zhrepresents the coupling stiffness, and Δx represents the relative displacement between the two ends of the elastic coupling.

[0041] In an exemplary embodiment of the present disclosure, the vibration differential equations are expressed as:

[0042]

[0043] Where [M] represents the generalized mass matrix and [K] represents the generalized stiffness matrix.

[0044] In an exemplary embodiment of the present disclosure, the generalized mass matrix and the generalized stiffness matrix of the vibration differential equation system are established according to the mode shape function; wherein the generalized mass matrix and the generalized stiffness matrix are expressed as:

[0045]

[0046] Among them, M h The generalized mass matrix representing the lateral vibration, K h The generalized stiffness matrix representing the lateral vibration, M x The generalized mass matrix representing axial vibration, K x represents the generalized stiffness matrix of axial vibration, and x1 and x2 represent the coordinates of the two ends of the highly elastic coupling.

[0047] The technical solutions provided by the present disclosure may have the following beneficial effects:

[0048] In an embodiment of the present disclosure, the right side shaft section of the thrust member is simplified into a multi-segment Timoshenko beam connection structure, and a shaft system dynamics model is established based on the simplified multi-segment Timoshenko beam connection structure. Based on the constructed shaft system dynamics model, the transfer matrix of the shaft system dynamics model is established using the transfer matrix method, and the vibration mode function of the shaft system dynamics model is further determined by using the transfer matrix method. The kinetic energy and potential energy of the shaft system dynamics model are determined according to the vibration mode function, and the kinetic energy and potential energy are substituted into the second-kind Lagrangian equation to obtain a group of vibration differential equations, and then the Newmark-β method is used to solve the group of vibration differential equations to obtain the axial displacement of the right side shaft section of the thrust member. This embodiment is based on the continuum transfer matrix theory and the calculation method of the second-kind Lagrangian equation. While ensuring the calculation accuracy, the axial displacement of the right side shaft section of the thrust member is derived, so as to analyze the influence of the axial displacement of the thrust member on the shaft system from the perspective of vibration.

[0049] It is to be understood that the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the disclosure. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] The accompanying drawings are incorporated into and constitute a part of the specification, illustrate embodiments consistent with the present disclosure, and together with the specification, serve to explain the principles of the present disclosure. Obviously, the drawings described below are only some embodiments of the present disclosure, and those skilled in the art can derive other drawings based on these drawings without inventive effort.

[0051] Figure 1 A flowchart showing the steps of a method for calculating the axial displacement of a thrust member of a diesel engine in an exemplary embodiment of the present disclosure is provided;

[0052] Figure 2 A schematic diagram showing a right shaft section of a thrust member in an exemplary embodiment of the present disclosure simplified as a multi-segment Timoshenko beam connection structure;

[0053] Figure 3 A schematic diagram showing a shaft system dynamics model in an exemplary embodiment of the present disclosure;

[0054] Figure 4 A schematic diagram showing the transverse force and deformation of a Timoshenko beam unit in an exemplary embodiment of the present disclosure is shown;

[0055] Figure 5 A schematic diagram showing the axial force and deformation of a Timoshenko beam unit in an exemplary embodiment of the present disclosure is shown;

[0056] Figure 6 A schematic diagram showing an equivalent boundary of an axial spring in an exemplary embodiment of the present disclosure is shown;

[0057] Figure 7 A schematic diagram of axial spring coupling in an exemplary embodiment of the present disclosure is shown;

[0058] Figure 8 A schematic diagram showing changes in lateral displacement calculated in an exemplary embodiment of the present disclosure is shown;

[0059] Figure 9 A schematic diagram showing changes in axial force calculated in an exemplary embodiment of the present disclosure is shown;

[0060] Figure 10 A schematic diagram showing changes in axial displacement calculated in an exemplary embodiment of the present disclosure is shown. DETAILED DESCRIPTION

[0061] Example embodiments will now be described more fully with reference to the accompanying drawings. However, example embodiments can be embodied in many forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete and will fully convey the concepts of the example embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0062] In addition, the accompanying drawings are merely schematic illustrations of the present disclosure and are not necessarily drawn to scale. Identical reference numerals in the figures denote identical or similar parts, and thus repetitive descriptions thereof will be omitted. Some of the block diagrams shown in the accompanying drawings are functional entities that do not necessarily correspond to physically or logically separate entities. These functional entities may be implemented in software, in one or more hardware modules or integrated circuits, or in different networks and / or processor devices and / or microcontroller devices.

[0063] This example embodiment first provides a method for calculating the axial displacement of a thrust member of a diesel engine, referring to Figure 1 As shown in , the method may include the following steps: Step S101: simplifying the right shaft section of the thrust member into a multi-segment Timoshenko beam connection structure, and establishing a shaft system dynamics model based on the multi-segment Timoshenko beam connection structure; Step S102: establishing a transfer matrix of the shaft system dynamics model using a transfer matrix method; Step S103: constructing a frequency equation based on the transfer matrix, and solving the frequency equation using the Newton iteration method to determine the vibration mode function of the shaft system dynamics model; Step S104: determining the kinetic energy and potential energy of the shaft system dynamics model based on the vibration mode function, and substituting the kinetic energy and potential energy into the second-kind Lagrangian equation to obtain a group of vibration differential equations; Step S105: solving the group of vibration differential equations using the Newmark-β method to obtain the axial displacement of the right shaft section of the thrust member.

[0064] In an embodiment of the present disclosure, the right side shaft section of the thrust member is simplified into a multi-segment Timoshenko beam connection structure, and a shaft system dynamics model is established based on the simplified multi-segment Timoshenko beam connection structure. Based on the constructed shaft system dynamics model, the transfer matrix of the shaft system dynamics model is established using the transfer matrix method, and the vibration mode function of the shaft system dynamics model is further determined by using the transfer matrix method. The kinetic energy and potential energy of the shaft system dynamics model are determined according to the vibration mode function, and the kinetic energy and potential energy are substituted into the second-kind Lagrangian equation to obtain a group of vibration differential equations, and then the Newmark-β method is used to solve the group of vibration differential equations to obtain the axial displacement of the right side shaft section of the thrust member. This embodiment is based on the continuum transfer matrix theory and the calculation method of the second-kind Lagrangian equation. While ensuring the calculation accuracy, the axial displacement of the right side shaft section of the thrust member is derived, so as to analyze the influence of the axial displacement of the thrust member on the shaft system from the perspective of vibration.

[0065] Below, each step of the above method in this exemplary embodiment will be described in more detail.

[0066] In one embodiment, reference Figure 2 As shown in , in step S101, the thrust member is a thrust plate, and the right shaft section of the thrust member includes a first transmission shaft, a flywheel, an elastic coupling, and a second transmission shaft arranged in sequence along the axial direction;

[0067] The method of simplifying the right shaft section of the thrust member into a multi-section Timoshenko beam connection structure includes:

[0068] The first transmission shaft is simplified into a first section beam unit, the flywheel and the highly elastic left end of the elastic coupling are simplified into a second section beam unit, the highly elastic right end of the elastic coupling is simplified into a third section beam unit, and the second transmission shaft is simplified into a fourth section beam unit; the multi-section Timoshenko beam connection structure includes the first section beam unit, the second section beam unit, the third section beam unit, and the fourth section beam unit.

[0069] By simplifying the right shaft section of the thrust member into a multi-segment Timoshenko beam connection structure, the lateral and axial transfer relationships of each beam unit of the multi-segment Timoshenko beam connection structure can be used to determine the transfer matrix of the entire system of the shaft system dynamics model.

[0070] It should be noted that the axial direction in this embodiment is Figure 3 The x direction in the horizontal direction is Figure 3 The y direction and z direction in .

[0071] In one embodiment, reference Figure 3As shown in , in step S101, the step of establishing a shaft system dynamics model according to the multi-segment Timoshenko beam connection structure includes:

[0072] Setting constraint conditions for the multi-segment Timoshenko beam connection structure;

[0073] Among them, the constraints include the elastic constraint condition of the left end of the first beam section unit, the rigid connection constraint condition between the first beam section unit and the second beam section unit, the elastic connection constraint condition between the second beam section unit and the third beam section unit, the rigid connection constraint condition between the third beam section unit and the fourth beam section unit, and the elastic constraint condition of the right end of the fourth beam section unit.

[0074] The constraints set for the first, second, third, and fourth beam units of the multi-segment Timoshenko beam connection structure enable the shaft system dynamics model constructed in step S101 to fit the actual vibration conditions of the right shaft segment of the thrust member, thereby facilitating the accurate calculation of the axial displacement of the right shaft segment of the thrust member.

[0075] What needs to be explained is that Figure 3 The intermediate stepped shaft shown in Figure 2 The flywheel and elastic coupling between the first and second transmission shafts are shown in detail with their geometry and dimensions.

[0076] Example, reference Figure 4 As shown in , a single Timoshenko beam transverse force and deformation diagram is shown, with the x direction being the axial direction and the y and z directions being the transverse directions. The Timoshenko beam is a beam that takes into account the shear effect on bending. For the multi-segment Timoshenko beam connection structure in step S101, the elastic modulus is set to E, the shear modulus is set to G, the length of the beam is set to L, the cross-sectional area is set to A, the moment of inertia is set to I, and the polar moment of inertia is set to I. p The displacement, rotation, bending moment and force of the beam are represented by Y, θ, M and Q respectively.

[0077] It should be noted that, based on the basic knowledge of vibration dynamics, the lateral vibration differential equation and axial vibration differential equation of a single Timoshenko beam are used to obtain the vibration mode function of a section of the shaft, and the lateral transfer relationship and axial transfer relationship of each beam unit are respectively expressed by the transfer matrix method.

[0078] In one embodiment, in step S102, establishing the transfer matrix of the shafting dynamics model using the transfer matrix method includes:

[0079] Step S1021: using a transfer matrix method to express the lateral transfer relationship and the axial transfer relationship of each beam unit of the multi-segment Timoshenko beam connection structure;

[0080] Step S1022: determining the spring boundary relationship of the multi-segment Timoshenko beam connection structure;

[0081] Step S1023: Determine the transfer matrix of the shafting dynamics model according to the lateral transfer relationship and the axial transfer relationship of each beam unit segment, combined with the constraint conditions and spring boundary relationship of the multi-segment Timoshenko beam connection structure.

[0082] Each beam element segment of the multi-segment Timoshenko beam connection structure in step S1021 corresponds to the first beam element segment, the second beam element segment, the third beam element segment, and the fourth beam element segment. That is, in step S1021, the transfer matrix method is used to represent the lateral and axial transfer relationships of the first beam element segment, the lateral and axial transfer relationships of the second beam element segment, the lateral and axial transfer relationships of the third beam element segment, and the lateral and axial transfer relationships of the fourth beam element segment.

[0083] Specifically, according to Figure 4 As shown in the transverse force and deformation diagram of the Timoshenko beam, for the transverse vibration of the shaft segment, it can be seen from vibration dynamics that the transfer relationship between the boundary conditions at the left and right ends of the beam element can be expressed in matrix form, that is, the transverse transfer relationship of the beam elements of each segment in step S1021 is expressed as:

[0084]

[0085] Among them, T xy represents the lateral transfer matrix, y l , y0 represents the lateral displacement, θ l , θ0 represents the rotation angle, M zl 、M z0 represents the bending moment, Q yl , Q y0 represents the lateral force, 0 and l represent the coordinate position of each beam element in the length direction;

[0086] Among them, the transverse transfer matrix T of each beam element is xy for:

[0087]

[0088] Among them, σ, τ, c0, c1, c2, c3, c4, c5, represent intermediate variables, E represents elastic modulus, I z represents the moment of inertia;

[0089] Among them, the intermediate variables c0, c1, c2, c3, c4, and c5 are:

[0090]

[0091] in, μ represents the linear density of the beam, and ω represents the free vibration circular frequency of the beam.

[0092] It should be explained that since the axial sections of the beam elements are all circular cross-section structures and the transverse section inertia moments are equal, the transverse transfer matrix of the beam element is applicable to both the in-plane transverse direction and the out-of-plane transverse direction. Therefore, the in-plane transverse transfer matrix of the beam element is equal to the out-of-plane transverse transfer matrix, that is, T xy =T xz , where T xy represents the in-plane transverse transfer matrix of the beam element, T xz It should be noted that the axial direction in this embodiment is Figure 3 In the x direction, Figure 3 The xy plane is considered to be in-plane and the xz plane is considered to be out-of-plane.

[0093] Specifically, according to Figure 5 As shown in the axial force and deformation diagram of the Timoshenko beam, for the axial vibration of the shaft segment, it can be known from vibration dynamics that the axial transmission relationship of the beam elements of each segment in step S1021 can be expressed in matrix form as follows:

[0094]

[0095] Among them, T x represents the axial transfer matrix,

[0096] U(0) and F(0) represent the axial displacement and axial force at the left end point of the straight beam when x = 0;

[0097] U(l) and F(l) represent the axial displacement and axial force at the right end point of the straight beam when x=l;

[0098] Among them, the axial transfer matrix T of each beam element is x for:

[0099]

[0100] Where l is the length of the beam element, E is the elastic modulus, ρ is the density, and A is the cross-sectional area of ​​the beam element.

[0101] Furthermore, based on the constraints set for the multi-segment Timoshenko beam connection structure in step S101, in the above step S1022, it is necessary to determine corresponding spring boundary relationships for the left end of the first beam unit, between the second beam unit and the third beam unit, and the right end of the fourth beam unit.

[0102] For example, according to Figure 6 The axial spring boundary equivalent diagram shown in FIG is a schematic diagram of a beam element. For the beam element, the corresponding spring boundary relationship at different ends in the longitudinal direction can be treated as the following equations (11) and (12):

[0103]

[0104] Where ky(0,t) represents the spring boundary relationship at the left end of the beam element, y(0,t) represents the lateral vibration displacement at the axial position x = 0 as a function of time t, and I(x) represents the moment of inertia of the cross section of the shaft at the axial position x about the neutral axis.

[0105]

[0106] Where ky(l,t) represents the spring boundary relationship at the right end of the beam element, y(l,t) represents the lateral vibration displacement at the axial position x = l as a function of time t, and I(x) represents the moment of inertia of the cross section of the shaft at the axial position x about the neutral axis.

[0107] It should also be noted that the reference Figure 2 As shown in , the structure studied in this application is a beam extended in one direction, and there is a force coupling mechanism between the axial direction and the in-plane lateral direction, and between the axial direction and the out-of-plane lateral direction. Therefore, in the subsequent analysis and solution of its dynamic response, it is necessary to solve them separately, that is, in step S105, the lateral motion in the in-plane lateral direction and the out-of-plane lateral direction should be solved first, and then the axial motion should be solved. Here, the lateral vibration state vector φ=[u θM z Q y ] T The lateral vibration state vector describes the key parameters of the lateral vibration state of the shaft system, which is directly used in constructing the transfer matrix of the shaft system dynamic model and subsequently establishing the vibration differential equation group.

[0108] Furthermore, based on the constraints set for the first beam unit, the second beam unit, the third beam unit, and the fourth beam unit of the multi-segment Timoshenko beam connection structure in the above step S101, in step S1023, after determining the lateral transfer relationship and the axial transfer relationship of each beam unit and the spring boundary relationship of the multi-segment Timoshenko beam connection structure, the transfer matrix of the shafting dynamics model is further determined in combination with the constraints in step S101.

[0109] For example, based on the above analysis and determination of the vibration transmission of each beam unit in the transverse direction and the axial direction, when determining the transfer matrix of the shaft system dynamics model, the present application respectively determines the transverse transfer matrix and the axial transfer matrix of the shaft system dynamics model, that is, the transfer matrix of the shaft system dynamics model includes the transverse transfer matrix and the axial transfer matrix; wherein, the transverse transfer matrix and the axial transfer matrix are expressed as:

[0110]

[0111] Among them, S y represents the lateral transfer matrix, S x represents the axial transfer matrix, K Y represents the lateral transfer matrix between springs, K X represents the axial transfer matrix between springs, T 4Y represents the bending moment of the fourth beam element in the lateral (y-direction) vibration, T 3Y represents the bending moment of the third beam element in the lateral (y-direction) vibration, T 2Y represents the bending moment of the second beam element in the lateral (y-direction) vibration, T 1Y Represents the bending moment of the first beam segment in the lateral (y-direction) vibration.

[0112] Among them, the lateral transfer matrix K between springs is Y Expressed as:

[0113]

[0114] Among them, K y represents the stiffness coefficient in the lateral direction (y direction);

[0115] Among them, the axial transfer matrix K between the springs X Expressed as:

[0116]

[0117] Among them, K x Indicates the stiffness coefficient in the axial direction (x direction).

[0118] It should be noted that in step S103, a frequency equation is constructed based on the lateral transfer matrix and axial transfer matrix obtained above, and the frequency equation is solved using the Newton iteration method to obtain the natural frequency, and the vibration mode function of the shaft system dynamic model is further determined based on the natural frequency.

[0119] It should also be noted that the above-mentioned lateral transfer matrix and axial transfer matrix constitute the overall transfer matrix of the system. By making the determinant of the overall transfer matrix equal to zero, the frequency equation is obtained. The purpose of constructing the above-mentioned frequency equation is to solve the natural frequency and mode function of the system by analyzing the vibration characteristics of the shaft system dynamics model. This process requires combining the lateral transfer matrix and the longitudinal transfer matrix, while considering the boundary conditions and connection conditions of the system. Among them, the boundary conditions: for both ends of the system, elastic constraints are usually adopted according to the actual constraints; connection conditions: between each segment of the beam unit, rigid connection or elastic connection conditions need to be considered.

[0120] The Newton iteration method (chord intercept method) is further used to solve the frequency equation, gradually approaching the roots of the frequency equation through iteration. The iteration method can be understood as first selecting an initial frequency value, calculating the frequency value and its derivative of the frequency equation based on the initial frequency value, and then continuously updating the frequency value until the convergence condition is met, that is, the frequency value of the frequency equation is small enough or the change in frequency is small enough, and the root of the frequency equation is obtained. The root of the frequency equation can be regarded as the natural frequency of the system. Finally, the natural frequency is substituted into the mode function equation, and the expression of the mode function is obtained by solving the linear equation system (mode function equation). That is, the mode function equation is the linear equation system used to solve the mode function. Its specific form needs to be determined in combination with the system's dynamic model and boundary conditions.

[0121] In vibration analysis, the mode function equation is usually related to the stiffness matrix and mass matrix of the system, and its general form can be expressed as: ([K]-ω 2 [M])φ=0, where [K] is the stiffness matrix, [M] is the mass matrix, ω is the natural frequency, and φ is the mode function. Solving this system of linear equations yields the specific expression for the mode function.

[0122] Considering the coupling mechanism between the axial direction and the in-plane lateral direction, and between the axial direction and the out-of-plane lateral direction, the coupling stiffness between the axial direction and the lateral direction is calculated before step S104. Figure 7 As shown in , the angle θ is a small quantity that satisfies sinθ=θ=tanθ. The angle θ on the left and the angle θ on the right satisfy the following equations (15) and (16), respectively:

[0123]

[0124] Where Δy represents the lateral deformation, Δu represents the axial deformation, and l represents the length of the beam element.

[0125] If we express the axial force in N, we get:

[0126]

[0127] Where E represents the elastic modulus and A represents the cross-sectional area of ​​the beam element;

[0128] Combining the above formulas (15), (16) and (17), the coupling stiffness can be obtained:

[0129]

[0130] Among them, k zh represents the coupling stiffness.

[0131] In one embodiment, in step S104, the kinetic energy and potential energy of the shafting dynamics model are determined according to the mode shape function, wherein the kinetic energy and potential energy of the shafting dynamics model include the kinetic energy and potential energy of the lateral vibration and the kinetic energy and potential energy of the axial vibration; wherein the kinetic energy and potential energy of the lateral vibration are expressed as:

[0132]

[0133] Among them, T h Represents the kinetic energy of transverse vibration, V h represents the potential energy of lateral vibration, l represents the length of the beam element, E represents the elastic modulus, A represents the cross-sectional area of ​​the beam element, ρ represents the density, represents the square of the transverse component of the mode function, express The second derivative of represents the square of the generalized coordinates of the i-th mode, represents the square of the generalized velocity of the i-th mode, h represents the sign of lateral vibration, x represents axial vibration, i represents the mode number, and t represents the time;

[0134] The kinetic energy and potential energy of the axial vibration are:

[0135]

[0136] Among them, T x represents the kinetic energy of axial vibration, x represents the sign of axial vibration, V x represents the potential energy of axial vibration, represents the square of the axial mode function, express The first derivative of k zhrepresents the coupling stiffness, and Δx represents the relative displacement between the two ends of the elastic coupling.

[0137] It should be noted that the subscript h represents lateral vibration, and the subscript x represents axial vibration. Δx represents the relative displacement between the two ends of the elastic coupling. In step S104, the kinetic energy and potential energy of the lateral vibration and the kinetic energy and potential energy of the axial vibration determined above are substituted into the second-kind Lagrangian equation to obtain a set of vibration differential equations. The specific expression of the second-kind Lagrangian equation is:

[0138]

[0139] Where L represents the Lagrangian function, L = TV, T represents kinetic energy, V represents potential energy, q j Represents generalized coordinates, j represents the subscript, j=1,2...,s,Q j Represents the generalized force acting on the entire system of the shaft system dynamics model.

[0140] It's important to explain that the Lagrangian equations of the second kind are a common approach to solving engineering dynamics problems. They are derived by expressing the universal equations of dynamics in generalized coordinates. The Lagrangian equations provide a set of mutually independent differential equations with the same number of defined generalized coordinates. For solving complex dynamics problems, the Lagrangian equations are often more convenient, effective, and analytically sound.

[0141] In step S104, the kinetic energy and potential energy of the lateral vibration, and the kinetic energy and potential energy of the axial vibration determined above are substituted into the second kind Lagrangian equations, and the obtained vibration differential equations are expressed as follows:

[0142]

[0143] Where [M] represents the generalized mass matrix, [K] represents the generalized stiffness matrix, F represents the exciting force, and x0 represents the reference position.

[0144] It should be noted that the generalized mass matrix [M] and generalized stiffness matrix [K] of the above-mentioned vibration differential equations are established based on the mode function, which are specifically expressed as follows:

[0145]

[0146] Among them, M h The generalized mass matrix representing the lateral vibration, K h The generalized stiffness matrix representing the lateral vibration, M x The generalized mass matrix representing axial vibration, K x represents the generalized stiffness matrix of axial vibration, and x1 and x2 represent the coordinates of the two ends of the highly elastic coupling.

[0147] It should be noted that the kinetic energy and potential energy of the system are composed of the kinetic energy and potential energy of the lateral vibration and the kinetic energy and potential energy of the axial vibration. Through the modal coordinate transformation of the vibration shape function, the kinetic energy and potential energy of the system are converted from physical coordinates to modal coordinates, thereby establishing the generalized mass matrix and generalized stiffness matrix. The establishment of these matrices not only simplifies the form of the vibration differential equations but also makes the solution process more efficient and stable.

[0148] Among them, the generalized mass matrix is ​​related to the kinetic energy of the system. Using the vibration mode function, the kinetic energy expression of the system can be converted from physical coordinates to modal coordinates. By integrating the kinetic energy at the axial position x and combining the generalized coordinates q of the modal coordinates i (t), and obtain the generalized mass matrix of the system.

[0149] Among them, the generalized stiffness matrix is ​​related to the potential energy of the system. Similar to the generalized mass matrix, the potential energy expression is converted from physical coordinates to modal coordinates by integrating the potential energy at the axial position x and combining the generalized coordinates q of the modal coordinates i (t), and obtain the generalized stiffness matrix of the system.

[0150] In one embodiment, in step S105, solving the vibration differential equations using the Newmark-β method to obtain the axial displacement of the right shaft segment of the thrust member includes:

[0151] Step S1051: using the Newmark-β method to solve the vibration differential equations to obtain the lateral displacement of the right shaft segment of the thrust member;

[0152] Step S1052: Determine the change in spring potential energy caused by the lateral displacement based on the coupling stiffness theory, substitute the change in spring potential energy into the second-kind Lagrangian equation, and use the Newmark-β method to solve the vibration differential equations to obtain the axial force and axial displacement of the right shaft segment of the thrust member.

[0153] In the above step S1051, when using the Newmark-β method to solve the vibration differential equations to calculate the lateral displacement of the right shaft section of the thrust member, the influence of the eccentric mass and the fluctuating speed on the shaft system will be converted into the influence of the exciting force. The centrifugal force generated by the eccentric mass during the rotation process is modeled as the exciting force, thereby affecting the vibration characteristics of the shaft system. The speed fluctuation will cause the angular velocity of the rotating shaft system to change, thereby changing the magnitude of the centrifugal force. This change is modeled as the exciting force input into the system.

[0154] In the above step S1052, when the Newmark-β method is used to solve the vibration differential equations to calculate the axial displacement and axial force of the right shaft segment of the thrust member, the change in spring potential energy is substituted into the second kind Lagrangian equation as excitation.

[0155] It should be explained that the Newmark-β method can make the calculation of time domain response easier. Taking a single degree of freedom linear elastic system as an example, the conclusions drawn are also applicable to multi-degree of freedom systems and nonlinear systems. Taking a single degree of freedom as an example, the equation for single degree of freedom vibration is:

[0156]

[0157] Among them, m represents the mass of the system, c represents the damping of the system, k represents the stiffness characteristics of the system; ü represents the acceleration response of the system, represents the velocity response of the system, u represents the displacement response of the system, and p represents the external load. The initial condition can be expressed as

[0158] Discretize the time domain [0, T] into t0, t1, ..., t n ,remember:

[0159] Δt i =t i+1 -t i (twenty one)

[0160] u i =u(t i ), p i =p(t i ) i=0,1,…,n (22)

[0161] Where t represents the time;

[0162] Under the premise of knowing all the reactions at previous moments, in order to obtain the dynamic reaction at time i+1, the Newmark-β method assumes that:

[0163]

[0164] Among them, γ and β represent parameters that control the accuracy and stability of numerical analysis. Generally, γ = 1 / 2. When β = 1 / 4, it is the average constant acceleration mode, and when β = 1 / 6, it is the linear acceleration mode. This is the most commonly used form of the Newmark-β method.

[0165] It's also worth explaining that the mechanism of the Newmark-β method has been well-studied. Its basic idea is to assume that the acceleration response within a step Δt follows a certain distribution, and then obtain the velocity and displacement responses at the end of that step through integration. For ease of explanation, a local coordinate system T is established within the step Δt, with the origin at the beginning of the step Δt. Clearly, the acceleration distribution within the step Δt can be described by a function related to τ, namely:

[0166] ü(τ)=ü i +η(τ)·Δü i (25)

[0167] Where η(τ) = 1 / 2, and the integrated velocity and displacement are:

[0168]

[0169] Where τ = Δt, and Δü i =ü i+1 -ü i .

[0170] The Newmark-β method assumes the following relationship for load distribution:

[0171]

[0172] Among them, m represents the mass of the system, c represents the damping of the system, k represents the stiffness characteristics of the system; ü represents the acceleration response of the system, represents the velocity response of the system, and u represents the displacement response of the system.

[0173] Considering the average constant acceleration mode, the load distribution mode within the step distance △t can be obtained, that is:

[0174] p(τ)=A0+A1τ+A2τ 2 (29)

[0175]

[0176] Among them, k represents the stiffness characteristics of the system; ü represents the acceleration response of the system, represents the velocity response of the system, and u represents the displacement response of the system.

[0177] Here, the relationship is used Obviously, in the load expression within the step △t, the acceleration response increment Δü within the step △t is i is unknown. In order to find △t, rewrite the equation into the form of an incremental equation, that is:

[0178]

[0179] Among them, △p i Indicates the external load increment, Δü i represents the acceleration response increment of the system, Represents the rate response increment of the system, Δu i Represents the displacement response increment of the system.

[0180] To accommodate the incremental form, the following formula is used:

[0181]

[0182] Among them, γ and β represent parameters that control the accuracy and stability of numerical analysis.

[0183] After sorting, we get the following formula:

[0184]

[0185]

[0186] in, represents the corrected equivalent mass coefficient, represents the corrected force increment.

[0187] The acceleration response increment can be obtained

[0188] Following the same steps, the Newmark-β method can be used to obtain the load distribution pattern within the step △t:

[0189] p(τ)=B0+B1τ+B2τ 2 +B3τ 3 (38)

[0190]

[0191] Among them, B0 represents the initial load, B1 represents the load increment related to velocity and acceleration, B2 represents the quadratic term coefficient related to acceleration, and B3 represents the cubic term coefficient related to acceleration increment; k represents the stiffness characteristics of the system, and ü represents the acceleration response of the system. represents the velocity response of the system, and u represents the displacement response of the system;

[0192] Taking γ = 1 / 2 and β = 1 / 6, we can obtain the mode of the segmented load corresponding to the Newmark-β method. The generalized coordinate vector q obtained by solving can be used to calculate the displacement using the modal superposition method. If the calculation coordinate is x1, then the corresponding displacement result is:

[0193] y=φ T (x1)q (40)

[0194] Where y represents displacement, q represents the generalized coordinate vector, φ T (x1) represents the transposed vector of the mode function at position x1.

[0195] It should be noted that the above formula (40) can be used to calculate the lateral displacement, wherein when formula (40) is used to calculate the lateral displacement, y represents the lateral displacement. This formula calculates the lateral displacement response at a specific position by the modal superposition method. It should be explained that the calculation of axial displacement and axial force is relatively complicated, and the coupling effect of axial vibration and lateral vibration needs to be considered; in step S1052, the change in spring potential energy caused by lateral displacement is determined based on the coupling stiffness theory and substituted into the second-order Lagrangian equation; the Newmark-β method is used again to solve the vibration differential equation group to obtain the axial force and axial displacement of the shaft system dynamics model.

[0196] It should also be noted that the axial clearance oil film of the thrust shaft is treated as a multi-gradient intermittent spring and solved as a specific external exciting force, which is specifically reflected in formulas (14) and (37).

[0197] The parameters of the shaft system dynamics model are set as: me = 2.96 kgm, K x止推 =1.05×10 8 N / m, K y止推 =2.0×10 8 N / m, K z止推 =2.5×10 8 N / m, K y联轴器 =0.65×10 7 N / m, K z联轴器 =0.65×10 7 N / m, K x中止 =1.01×10 7 N / m, K y中止 =2.0×10 8 N / m, K z中止 =2.5×10 8 N / m. The calculation method of the above embodiment can be used to solve the lateral displacement, axial displacement and axial force of the right shaft section of the thrust plate; wherein the lateral displacement is as follows: Figure 8 As shown in , the axial force and axial displacement are Figure 9 and Figure 10 As shown in .

[0198] Those skilled in the art will readily appreciate other embodiments of the present disclosure after considering the specification and practicing the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the present disclosure that follow the general principles of the present disclosure and include common knowledge or customary techniques in the art not disclosed herein. The description and examples are to be considered as exemplary only, with the true scope and spirit of the present disclosure being indicated by the appended claims.

Claims

1. A method for calculating the axial displacement of a thrust member of a diesel engine, characterized in that: include: The right shaft section of the thrust member is simplified into a multi-segment Timoshenko beam connection structure, and a shaft system dynamics model is established based on the multi-segment Timoshenko beam connection structure; Establishing a transfer matrix of the shafting dynamics model using a transfer matrix method; Constructing a frequency equation according to the transfer matrix, and solving the frequency equation using a Newton iteration method to determine a vibration mode function of the shafting dynamics model; determining the kinetic energy and potential energy of the shaft system dynamics model according to the mode shape function, and substituting the kinetic energy and potential energy into the second kind Lagrangian equation to obtain a set of vibration differential equations; The vibration differential equations are solved using the Newmark-β method to obtain the axial displacement of the right shaft section of the thrust member.

2. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 1, characterized in that: The method of using the Newmark-β method to solve the vibration differential equations to obtain the axial displacement of the right shaft section of the thrust member includes: Solving the vibration differential equations using the Newmark-β method to obtain the lateral displacement of the right shaft segment of the thrust member; Based on the coupling stiffness theory, the change in spring potential energy caused by the lateral displacement is determined, and the change in spring potential energy is substituted into the second-kind Lagrangian equation. The Newmark-β method is used to solve the vibration differential equation group to obtain the axial force and the axial displacement of the right shaft segment of the thrust member.

3. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 1, characterized in that: The right shaft section of the thrust member includes a first transmission shaft, a flywheel, an elastic coupling and a second transmission shaft arranged in sequence along the axial direction; The method of simplifying the right shaft section of the thrust member into a multi-section Timoshenko beam connection structure includes: The first transmission shaft is simplified into a first section beam unit, the flywheel and the highly elastic left end of the elastic coupling are simplified into a second section beam unit, the highly elastic right end of the elastic coupling is simplified into a third section beam unit, and the second transmission shaft is simplified into a fourth section beam unit; the multi-section Timoshenko beam connection structure includes the first section beam unit, the second section beam unit, the third section beam unit, and the fourth section beam unit.

4. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 3, characterized in that: The step of establishing a shaft system dynamics model according to the multi-segment Timoshenko beam connection structure includes: Setting constraint conditions for the multi-segment Timoshenko beam connection structure; Among them, the constraints include the elastic constraint condition of the left end of the first beam section unit, the rigid connection constraint condition between the first beam section unit and the second beam section unit, the elastic connection constraint condition between the second beam section unit and the third beam section unit, the rigid connection constraint condition between the third beam section unit and the fourth beam section unit, and the elastic constraint condition of the right end of the fourth beam section unit.

5. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 4, characterized in that: The method of establishing the transfer matrix of the shafting dynamics model by using the transfer matrix method includes: The lateral transfer relationship and axial transfer relationship of each beam unit of the multi-segment Timoshenko beam connection structure are expressed by using the transfer matrix method; determining a spring boundary relationship of the multi-segment Timoshenko beam connection structure; According to the lateral transfer relationship and the axial transfer relationship of each beam unit segment, combined with the constraint conditions and the spring boundary relationship of the multi-segment Timoshenko beam connection structure, the transfer matrix of the shaft system dynamics model is determined.

6. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 5, characterized in that: The lateral transfer relationship of each beam element is expressed as: Among them, T xy represents the lateral transfer matrix, y l , y0 represents the lateral displacement, θ l , θ0 represents the rotation angle, M zl 、M z0 represents the bending moment, Q yl , Q y0 represents the lateral force, 0 and l represent the coordinate position of each beam element in the length direction; The axial transfer relationship of each beam element is expressed as: Among them, T x represents the axial transfer matrix; U(0) and F(0) represent the axial displacement and axial force for the left end point of the straight beam when x=0; U(l) and F(l) represent the axial displacement and axial force for the right end point of the straight beam when x=l.

7. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 5, characterized in that: The transfer matrix of the shaft system dynamics model includes a lateral transfer matrix and an axial transfer matrix; wherein the lateral transfer matrix and the axial transfer matrix are expressed as: Among them, S y represents the lateral transfer matrix, S x represents the axial transfer matrix, K Y represents the lateral transfer matrix between springs, K X represents the axial transfer matrix between springs, T 4Y represents the bending moment of the fourth beam element in lateral vibration, T 3Y represents the bending moment of the third beam element in lateral vibration, T 2Y represents the bending moment of the second beam element in lateral vibration, T 1Y represents the bending moment of the first beam element in lateral vibration.

8. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 7, characterized in that: The kinetic energy and potential energy of the shaft system dynamics model include the kinetic energy and potential energy of the lateral vibration and the kinetic energy and potential energy of the axial vibration; wherein the kinetic energy and potential energy of the lateral vibration are expressed as: Among them, T h Represents the kinetic energy of transverse vibration, V h represents the potential energy of lateral vibration, l represents the length of the beam element, E represents the elastic modulus, A represents the cross-sectional area of ​​the beam element, ρ represents the density, represents the square of the transverse component of the mode function, express The second derivative of represents the square of the generalized coordinates of the i-th mode, represents the square of the generalized velocity of the i-th mode, h represents the sign of lateral vibration, x represents axial vibration, i represents the mode number, and t represents the time; The kinetic energy and potential energy of the axial vibration are expressed as: Among them, T x represents the kinetic energy of axial vibration, x represents the sign of axial vibration, V x represents the potential energy of axial vibration, represents the square of the axial mode function, express The first derivative of k zh represents the coupling stiffness, and Δx represents the relative displacement between the two ends of the elastic coupling.

9. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 8, characterized in that: The vibration differential equations are expressed as: Where [M] represents the generalized mass matrix and [K] represents the generalized stiffness matrix.

10. The method for calculating the axial displacement of a thrust member of a diesel engine according to claim 9, characterized in that: The generalized mass matrix and the generalized stiffness matrix of the vibration differential equation system are established according to the mode shape function; wherein the generalized mass matrix and the generalized stiffness matrix are expressed as: Among them, M h The generalized mass matrix representing the lateral vibration, K h The generalized stiffness matrix representing the lateral vibration, M x The generalized mass matrix representing axial vibration, K x represents the generalized stiffness matrix of axial vibration, and x1 and x2 represent the coordinates of the two ends of the highly elastic coupling.

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