Generalized multi-symplectic method for dynamics of bladed disk system with multi-surface friction behavior

Abstract

The application discloses a generalized multi-symplectic calculation method for a blade disc system dynamics containing multi-joint surface friction behavior, and is implemented according to the following steps: step 1, a substitution method is used to convert a vibration differential equation of a blade disc system containing a dry friction damping structure joint surface into a first-order format, and is written in a matrix form; step 2, a step length h and a total time t are set, and then a fourth-order implicit symplectic RK method is used to solve at a kth time; step 3, according to initial conditions of the vibration differential equation and the result obtained in step 2, a value of z at a corresponding time can be further recursively obtained, and then an analytical solution of the equation is obtained. The vibration response of the blade disc is calculated by using a symplectic RK algorithm with long-term tracking ability and high precision, and the algorithm has an unconditional stable property, and the calculation precision is significantly improved.

Description

Technical Field

[0001] This invention belongs to the field of high-precision calculation technology of linear structural dynamic equations of bladed disk systems, core components in aero-engines, which are at the forefront of intelligent manufacturing of high-end equipment. Specifically, it relates to a generalized multisymplectic calculation method for the dynamics of bladed disk systems containing friction behavior of multiple mating surfaces. Background Technology

[0002] The bladed disk system of high-end aero-engines needs to operate under high speed, high centrifugal force, and blade vibration. Meanwhile, blade manufacturing errors and wear during operation are often random, frequently leading to random high-cycle fatigue accidents in the bladed disk system. Furthermore, the contact surfaces of dry friction damping structures such as braces, shrouds, and blade crowns undergo slip-viscous micro-motions, resulting in complex nonlinear dry friction forces at the contact surfaces, making it more difficult to accurately predict the vibration response of detuned bladed disks. Therefore, researching algorithms with long-term tracking capabilities and higher accuracy to predict the vibration response of detuned bladed disks with dry friction damping structures is helpful in predicting the vibration hazards of detuned bladed disk systems, providing a theoretical basis for the research and analysis of high-cycle fatigue in bladed disk systems, and improving the ability to predict the vibration safety of bladed disks. Calculating the long-term dynamic response of detuned bladed disks with dry friction damping structures using traditional calculation algorithms cannot maintain the energy and momentum conservation of the system, resulting in a non-conservation of the system's total energy, meaning the physical characteristics of the high-end aero-engine bladed disk system cannot be maintained, thus the calculation process cannot accurately reflect the long-term evolution of the bladed disk system. Therefore, exploring algorithms that can accurately reflect the long-term dynamic behavior of bladed disk systems containing multi-interface friction is a prerequisite for bladed disk vibration control. Summary of the Invention

[0003] The purpose of this invention is to provide a generalized multisymplectic calculation method for the dynamics of bladed disk systems with multi-interface friction behavior. This method addresses the problem that traditional calculation algorithms cannot maintain the energy conservation of the bladed disk system when calculating the long-term dynamic response of detuned bladed disks with dry friction damping structures. This results in changes in the total energy of the bladed disk dynamic system, leading to the calculation process failing to accurately reflect the long-term evolutionary behavior of the bladed disk system.

[0004] To achieve the above objectives, the technical solution adopted in this invention is: a generalized multisymplectic calculation method for the dynamics of an impeller system containing multi-interface friction behavior, specifically implemented according to the following steps:

[0005] Step 1: Use the substitution method to transform the vibration differential equation of the dry friction damping structure combined with the bladed disk system into a first-order scheme and write it in matrix form.

[0006] Step 2: Set the step size h and the total time t, and then use the 4th-order implicit symplectic RK method to solve at the k-th time.

[0007] Step 3: Based on the initial conditions of the vibration differential equation and the results obtained in Step 2, the value of z at the corresponding time can be further recursively obtained, and then the numerical solution of the equation can be obtained.

[0008] As a preferred embodiment of the present invention, in step 1, the vibration differential equation of the bladed disk system containing the dry friction damping structure is:

[0009]

[0010] The initial conditions are:

[0011]

[0012] In the formula, M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the bladed disk system, respectively. F(t) and x(t) are the nonlinear terms introduced by the dry friction behavior of the multi-combined surface, and the column vectors of the airflow excitation force load and displacement, respectively. This represents the derivative of displacement with respect to time. This represents the second derivative of displacement with respect to time.

[0013] As a preferred technical solution of the present invention, in step 1, the matrix form is specifically as follows:

[0014] Let q = x, Equation (1) can be written as:

[0015]

[0016] make Equation (3) can be transformed into:

[0017]

[0018] in:

[0019]

[0020] As a preferred technical solution of the present invention, in step 2, the solution obtained by using the 4th-order implicit generalized dossin method at the k-th time is as follows:

[0021]

[0022] In the formula:

[0023]

[0024] As a preferred technical solution of the present invention, in step 3, the analytical solution of the equation is specifically as follows:

[0025] z k+1=z k +h(b1f(t k +c1h,Z1)+b2f(t k +c2h,Z2)) (6).

[0026] The beneficial effects of the present invention are as follows: The present invention provides a generalized multisymplectic calculation method for the dynamics of a bladed disk system with multi-joint surface friction behavior. The principle is to calculate the vibration response of the bladed disk with multi-joint surface friction behavior by using a generalized multisymplectic algorithm with long-term tracking capability and high accuracy. The algorithm has the following properties: (1) unconditional stability, (2) significantly improved calculation accuracy, and (3) preservation of the dynamic physical characteristics of the bladed disk system. Thus, the goal of numerical discretization does not affect the basic characteristics. Detailed Implementation

[0027] The present invention will be further described in detail below with reference to specific embodiments. The specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0028] The present invention is further described below through embodiments, but is not limited to the following implementation examples.

[0029] Example 1

[0030] The present invention provides a generalized multisymplectic calculation method for the dynamics of an bladed disk system containing multi-interface friction behavior, which is implemented according to the following steps:

[0031] Step 1: Use the substitution method to transform the vibration differential equation of the dry friction damping structure combined with the bladed disk system into a first-order scheme and write it in matrix form.

[0032] Step 2: Set the step size h and the total time t, and then use the 4th-order implicit symplectic RK method to solve at the k-th time.

[0033] Step 3: Based on the initial conditions of the vibration differential equation and the results obtained in Step 2, the value of z at the corresponding time can be further recursively obtained, and thus the analytical solution of the equation can be obtained.

[0034] This invention presents a generalized multisymplectic algorithm for calculating the microequations of bladed disk vibration involving frictional behavior at the mating surfaces. This algorithm boasts long-term tracking capability, strong adaptability, and high accuracy. The slip-viscous micro-motions at multiple mating surfaces of detuned bladed disks result in complex nonlinear dry frictional forces, hindering long-term stable and accurate prediction of the vibration response. This invention combines the highly adaptable and low-complexity RK algorithm with the symplectic algorithm, which possesses long-term tracking capability and stability, to develop a stable and highly accurate symplectic RK algorithm for calculating the microequations of bladed disk vibration with complex frictional behavior. The RK scheme of the nonlinear vibration differential equations of the bladed disk is transformed into a symplectic scheme, combining the advantages of both algorithms. This algorithm exhibits unconditional stability, significantly improving computational accuracy. This invention provides a computational foundation for safety analysis during the design phase of bladed disk systems in high-end equipment structures.

[0035] Example 2

[0036] Unlike Example 1, this example specifies step 1 of the generalized symplectic RK calculation method for bladed disk systems with multi-interface friction behavior according to the present invention:

[0037] The vibration differential equation of the bladed disk system with dry friction damping structure is:

[0038]

[0039] The initial conditions are:

[0040]

[0041] In the formula, M, C, and K are the mass matrix, damping matrix, and stiffness matrix, respectively. F(t) and x(t) are column vectors of the nonlinear term, the airflow excitation force load, and the displacement, respectively. This represents the derivative of displacement with respect to time. This represents the second derivative of displacement with respect to time;

[0042] In matrix form, it is specifically as follows:

[0043] Let q = x, Equation (1) can be written as:

[0044]

[0045] make Equation (3) can be transformed into:

[0046]

[0047] in:

[0048]

[0049] Example 3

[0050] Unlike Example 2, this example provides specific limitations on steps 2 and 3 of the generalized symplectic RK calculation method for bladed disk systems with multi-interface friction behavior according to the present invention:

[0051] The solution obtained using the 4th-order implicit symplectic RK method at time k is as follows:

[0052]

[0053] In the formula:

[0054]

[0055]

[0056] The analytical solution to the equation in step 3 is as follows:

[0057] z k+1 =z k +h(b1f(t k +c1h,Z1)+b2f(t k +c2h,Z2)).

Claims

1. A generalized multisymplectic calculation method for the dynamics of bladed disk systems containing multi-interface friction behavior, characterized in that, The specific steps are as follows: Step 1: Use the substitution method to transform the vibration differential equation of the dry friction damping structure combined with the bladed disk system into a first-order scheme and write it in matrix form. Step 2, set the step size h And the total time t, then on the t k The solution at each time step is obtained using the fourth-order implicit symplectic RK method. Step 3: Based on the initial conditions of the vibration differential equation and the results obtained in Step 2, the corresponding time can be further recursively derived. The value of is then used to obtain the analytical solution to the equation; In step 1, the vibration differential equation of the bladed disk system containing the dry friction damping structure is: (1) The initial conditions are: (2) In the formula These are the mass matrix, damping matrix, and stiffness matrix, respectively. , , These are column vectors representing the nonlinear term, the airflow excitation force load, and the displacement, respectively. This represents the derivative of displacement with respect to time. This represents the second derivative of the displacement with respect to time. In step 1, the matrix form is as follows: make , Equation (1) can be written as: （3） make Equation (3) can be simplified to: (4) in: ； In step 2, at the k The solution obtained using the 4th-order implicit symplectic RK method at each time step is as follows: (5) In the formula: ； In step 3, the analytical solution of the equation is specifically as follows: (6)。

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