Method for obtaining lateral vibration of fixed-length axial moving string system under complex boundary

Abstract

The application discloses a method for obtaining lateral vibration of a fixed-length axial moving string system under a complex boundary, which comprises the following steps: step one, a dynamic model of the fixed-length axial moving string system under mass-damping-spring boundary and Dirichlet boundary conditions is established according to dynamic characteristics and mechanical principles; step two, a calculation domain is converted into a characteristic line domain, and a wave reflection equation under the Dirichlet boundary condition and a wave reflection equation under the mass-damping-spring complex boundary condition are established at the end points of the characteristic line domain; step three, the Duhamel integral is used for solving, and an analytical solution of the wave reflection equation is obtained; step four, according to the linear assumption of the wave form in a small step, a semi-analytical solution of the wave reflection equation in the characteristic line domain is obtained by using a fast recursion method; and step five, according to a superposition method of left and right traveling waves, the calculation results of the characteristic line domain are superposed to obtain an original time-space domain, and the vibration results of the fixed-length axial moving string system are obtained.

Description

Technical Field

[0001] This invention belongs to the field of mechanical vibration measurement technology, specifically a method for obtaining lateral vibration of a fixed-length axially moving string system under complex boundaries. Background Technology

[0002] Axial movement systems are widely used in mechanical engineering systems, such as wire saws, high-speed elevators, mine hoists, drive belts, and textile machinery. In these applications, axial movement systems are typically modeled as chord models, assuming small deformations and neglecting bending stiffness. However, additional vibrations can have detrimental effects, potentially leading to safety hazards, especially under complex boundary conditions in specific engineering projects.

[0003] For classic axial movement systems, there are two main types of dynamic analysis methods: numerical methods and analytical methods. Numerical methods can be further divided into two types: finite difference methods and various weighted residual methods, as well as Galerkin methods or hypothetical modal methods. However, the stability conditions and ability to reveal vibration mechanisms of finite difference methods are limited. The accuracy of Galerkin methods or hypothetical modal methods largely depends on the selection and order of trial functions. When fewer trial functions are used, the entire computational domain will be disturbed, which does not conform to the actual wave characteristics. From the perspective of revealing vibration mechanisms, analytical solutions are a powerful tool. Current analytical solution techniques are generally limited to calculations within a single period. For example, Chinese Patent CN109614745A proposes a method for obtaining the lateral vibration of an axially moving rope device under mixed boundary conditions. This method proposes a method for obtaining the reflected wave response equations for each stage in one vibration period T of the axially moving rope, but this method cannot perform calculations for arbitrary long periods. Summary of the Invention

[0004] In view of this, the purpose of the present invention is to provide a method for obtaining the lateral vibration of a fixed-length axially moving string system under complex boundary conditions, which aims to obtain the lateral vibration of the axially moving string system under fixed length and complex boundary conditions.

[0005] To achieve the above objectives, the present invention provides the following technical solution:

[0006] A method for obtaining lateral vibration of a fixed-length axially moving chord system under complex boundary conditions includes the following steps:

[0007] Step 1: Based on the dynamic characteristics and mechanical principles, establish a dynamic model of the fixed-length axially moving string system under mass-damped-spring boundary and Dirichlet boundary conditions;

[0008] Of the two ends of the chord in the fixed-length axially movable chord system, the first end is the excitation input end, and the second end is the equivalent concentrated mass end. A fixed coordinate system OXY is established with the first end of the chord as the origin. The dynamic model of the fixed-length axially movable chord system under mass-damped-spring boundary and Dirichlet boundary conditions is as follows:

[0009]

[0010] Where: ρ, T, and V are the linear density, static tension, and constant velocity of the string, respectively; x represents the distance of a point on the string from the origin in the X direction; w(x, t) represents the lateral displacement of a point on the string at a distance x from the origin; l0 is the initial length of the string; l(t) is the real-time length of the string; and t represents time.

[0011] The boundary conditions are:

[0012]

[0013] Where, m e c e and k e These represent the equivalent lumped mass, equivalent damping, and equivalent stiffness, respectively; e(t) represents the displacement excitation input at the first end. This represents the first derivative of the lateral displacement with respect to time. w' represents the second derivative of the lateral displacement with respect to time; w' represents the first derivative of the lateral displacement with respect to space; w″ represents the second derivative of the lateral displacement with respect to space. This represents the second-order mixed derivative of the lateral displacement with respect to time and space.

[0014] Step 2: Transform the computational domain into a characteristic line domain, and establish the elastic wave reflection equation at the endpoints of the characteristic line domain. The elastic wave reflection equation includes the wave reflection equation under the Dirichlet boundary condition and the wave reflection equation under the complex boundary condition of mass-damped-spring.

[0015] Step 3: Solve using Duhamel integration to obtain the analytical solution of the wave reflection equation;

[0016] Step 4: Based on the assumption of waveform linearity within small step sizes, a rapid recursive method is used to obtain the semi-analytical solution of the wave reflection equation within the characteristic line domain;

[0017] Step 5: Based on the superposition method of the left and right traveling waves, the calculation results of the characteristic line domain are superimposed to obtain the original spatiotemporal domain, and the vibration results of the fixed-length axially moving string system are obtained.

[0018] Furthermore, in step two, the computational domain is transformed into a feature line domain, and the solution w of the dynamic model is expressed as the superposition of two traveling waves:

[0019] w=F(xV r t)+G(x+V l t)

[0020] V r =c+V

[0021] V l =cV

[0022] Where: V r and V l This represents the velocity of the second traveling wave moving towards the second end of the chord and the velocity of the first traveling wave moving towards the first end in a fixed coordinate system. F(xV) represents the wave speed of a string under static conditions. r t) is the velocity V r The second traveling wave; G(x+V) l t) is the velocity V l The first traveling wave;

[0023] Determine the initial lateral displacement and velocity conditions for string vibration:

[0024]

[0025] Where: the function φ(x) is the initial lateral displacement at different positions on the chord in the fixed coordinate system; the function ψ(x) is the initial velocity at different positions on the chord in the fixed coordinate system;

[0026] The expressions for the first and second traveling waves at the initial time are obtained as follows:

[0027]

[0028] Where C is the integration constant;

[0029] In the xt domain, the first and second transverse waves are used to represent the transverse displacement at any point in time within the chord:

[0030] w(x,t)=F(ξ)+G(η)

[0031]

[0032] Where: x represents the position of the point in the fixed coordinate system; t represents any time; ξ represents the second traveling wave parameter introduced, that is, the second coordinate in the feature line domain; η represents the first traveling wave parameter introduced, that is, the first coordinate in the feature line domain;

[0033] In the ξ-η domain, the traveling wave is represented, and the problem of solving the dynamic equations is transformed into solving the first and second traveling waves on the boundary, where:

[0034] The first traveling wave parameter ξ is:

[0035] ξ=-V r t

[0036] The second traveling wave parameter η is:

[0037] η = l0 + V l t.

[0038] Furthermore, in step two, the wave reflection equation under the Dirichlet boundary condition is:

[0039] F(-V r t)+G(V l t)=e(t)

[0040] Where: F(-V r t) represents the second traveling wave at the first endpoint; G(V) l t) represents the first traveling wave at the first endpoint;

[0041] The relationship between the first traveling wave and the second reflected traveling wave is as follows:

[0042]

[0043]

[0044] Where: F(ξ) represents the second traveling wave; F′(ξ) represents the first derivative of the second traveling wave with respect to the second traveling wave parameter; F″(ξ) represents the second derivative of the second traveling wave with respect to the second traveling wave parameter; Indicates the first traveling wave; Substituting the first derivative of the first traveling wave with respect to the first traveling wave parameters into... Substituting the second derivative of the first traveling wave with respect to its parameters into... Y(ξ) represents the displacement excitation of the first endpoint in the characteristic line domain; Y′(ξ) represents the first derivative of the displacement excitation of the first endpoint with respect to the second traveling wave parameter in the characteristic line domain; Y″(ξ) represents the second derivative of the displacement excitation of the first endpoint with respect to the second traveling wave parameter in the characteristic line domain.

[0045] Furthermore, in step two, the wave reflection equation under the complex boundary conditions of mass-damped-spring is:

[0046] AG″+BC′+CG=R

[0047]

[0048] Where A represents the coefficient of the second-order differential term of the reflection equation at the complex boundary; B represents the coefficient of the first-order differential term of the reflection equation at the complex boundary; C represents the coefficient of the zero-order differential term of the reflection equation at the complex boundary; R represents the non-homogeneous term of the reflection equation at the complex boundary; F represents the second traveling wave at the second endpoint boundary; F′ represents the first derivative of the second traveling wave at the second endpoint boundary with respect to the second traveling wave parameter; F″ represents the second derivative of the second traveling wave at the second endpoint boundary with respect to the second traveling wave parameter; G represents the first traveling wave at the second endpoint boundary; G′ represents the first derivative of the first traveling wave at the second endpoint boundary with respect to the first traveling wave parameter; G″ represents the second derivative of the first traveling wave at the second endpoint boundary with respect to the first traveling wave parameter.

[0049] The arguments of F, G, and R are:

[0050]

[0051] The relationship between the arguments of the first and second waves at the second boundary is as follows:

[0052] ξ=2l0-2Vt-η.

[0053] Furthermore, in step three, the relationship between the arguments of the first and second traveling waves at the second boundary is substituted into the wave reflection equation under the complex boundary conditions of mass-damped-spring, resulting in:

[0054]

[0055] in: The characteristic equation can be obtained by substituting the non-homogeneous terms of R(ξ) with ξ = 2l0 - 2Vt - η:

[0056] Aλ 2 +Bλ+C=0

[0057] The characteristic roots of the characteristic equation are obtained:

[0058]

[0059] Based on the four cases of characteristic roots, the complete solutions of the unit impulse response function and the expression are obtained respectively:

[0060] First scenario: A≠0; B 2 -4AC>0, the characteristic equation has two unequal real characteristic roots, thus we get:

[0061]

[0062]

[0063] Where C1 and C2 are both parameters;

[0064] Using unit pulse excitation With l0 = 0, we obtain the complete solution of the unit impulse response function and its expression:

[0065]

[0066]

[0067] Where: H(η) represents the unit impulse response function of the reflection equation; H′(η) represents the first derivative of the unit impulse response function of the reflection equation with respect to the first traveling wave parameter;

[0068] The second scenario: A≠0; B 2 -4AC=0, the characteristic equation has two equal real characteristic roots, thus:

[0069]

[0070]

[0071] Using unit pulse excitation With l0 = 0, we obtain the complete solution of the unit impulse response function and its expression:

[0072]

[0073]

[0074] The third scenario: A≠0; B 2 -4AC < 0, the characteristic equation has a pair of complex conjugate roots, that is:

[0075]

[0076] Where: abs() is the function for calculating the absolute value. It is the imaginary unit; α and β are the real and imaginary parts of the eigenvalues, respectively;

[0077] get:

[0078]

[0079]

[0080] Using unit pulse excitation With l0 = 0, we obtain the complete solution of the unit impulse response function and its expression:

[0081]

[0082]

[0083] Fourth case: A = 0; m e=0, the characteristic roots of the characteristic equation are:

[0084]

[0085] get:

[0086]

[0087] C1 = G(l0)

[0088] Using unit pulse excitation With l0 = 0, we obtain the complete solution of the unit impulse response function and its expression:

[0089]

[0090]

[0091] Based on the first to third cases, point η n Substituting them in, we can get G″(η) n Using Duhamel integration, we derive G′(η) in the fourth case. n ) and G″(η n ); where η n This represents any first traveling wave parameter.

[0092] Furthermore, in step four,

[0093] Assume terms related to waveform It is linear:

[0094]

[0095]

[0096] Where: Δη represents the step size of the discrete first traveling wave parameter; K(η) n-1 ) represents the slope of the external non-homogeneous term within a first traveling wave parameter step;

[0097] Duhamel integral expansion yields:

[0098]

[0099]

[0100]

[0101] Where H1(Δη) represents the integral of the unit impulse response function within a first traveling wave parameter step; H1′(Δη) represents the integral of the first derivative of the unit impulse response function with respect to the first traveling wave parameter within a discrete first traveling wave parameter step; H2(Δη) represents the integral of the unit impulse response function multiplied by the first traveling wave parameter within a first traveling wave parameter step; and H′2(Δη) represents the integral of the unit impulse response function multiplied by the first traveling wave parameter within a first traveling wave parameter step.

[0102] Furthermore, the vibration results are as follows:

[0103] w(x1,t1)=F(ξ)+G(η).

[0104] Where: x1 represents the position coordinates of any point; t1 represents any time.

[0105] The beneficial effects of this invention are as follows:

[0106] The present invention provides a method for obtaining lateral vibration of a fixed-length axially moving string system under complex boundary conditions. This method addresses the wave problem of a fixed-length axially moving string system under complex boundary conditions by employing the characteristic line method and Duhamel integral. First, a mathematical model of the system is established based on its dynamic characteristics and mechanical principles, and the computational domain is transformed into a characteristic line domain. Elastic wave reflection equations are established at the endpoints of the characteristic line domain, including wave reflection equations under Dirichlet boundary conditions and under complex mass-damped-spring boundary conditions. Second, Duhamel integrals are used to solve the wave reflection equations analytically. Based on the assumption of waveform linearity within small steps, a rapid recursive method is used to obtain a semi-analytical solution to the wave reflection equations within the characteristic line domain. Finally, the calculation results of the characteristic line domain are superimposed using the superposition method of the first and second traveling waves to obtain the vibration result of the axially moving string in the original spatiotemporal domain. Thus, the method for obtaining lateral vibration of a fixed-length axially moving string system under complex boundary conditions can acquire the lateral vibration of the axially moving string system under fixed length and complex boundary conditions. Attached Figure Description

[0107] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0108] Figure 1 This is a schematic diagram of a typical existing fixed-length axially movable chord system under complex boundary conditions.

[0109] Figure 2 This is a schematic diagram of the traveling wave superposition method in the xt domain;

[0110] Figure 3 This is a schematic diagram of the superposition method of traveling waves in the ξ-η domain. Detailed Implementation

[0111] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0112] like Figure 1 The diagram shows a typical fixed-length axially moving string system under complex boundary conditions. Specifically, in this embodiment, the left end of the string is the first end, and the right end is the second end. A fixed coordinate system OXY is established with the left end of the string as the origin; e(t) is the displacement excitation input of the left end of the string; l0 is the initial length of the string; x represents the position of the point in the fixed coordinate system at a distance of x axial length from the left fixed end of the string; w(x,t) represents the lateral displacement of the string at x in the fixed coordinate system; V is the constant moving speed of the string; m e c is the equivalent concentrated mass at the right end; e The equivalent damping at the right end; k e This represents the equivalent stiffness at the right end.

[0113] The following describes the specific implementation of the method for obtaining lateral vibration of a fixed-length axially moving string system under complex boundary conditions (hereinafter referred to as "the system") of the present invention.

[0114] The method for obtaining lateral vibration of a fixed-length axially moving chord system under complex boundaries in this embodiment includes the following steps.

[0115] Step 1: Describe the mechanical properties of the system in a fixed coordinate system. Based on the dynamic properties and mechanical principles of the system, obtain the dynamic model of the system under mass-damped-spring boundary and Dirichlet boundary conditions.

[0116]

[0117] Where ρ, T, and V are the linear density, static tension, and constant velocity of the string, respectively; x represents the position of a point in the fixed coordinate system at a distance x from the left fixed end of the string along the axial direction; w(x, t) represents the lateral displacement of the string at x in the fixed coordinate system; l0 is the initial length of the string; l(t) is the real-time length of the string; and t represents time.

[0118] Its boundary conditions are as follows:

[0119]

[0120] Where, m e ce and k e These are the equivalent lumped mass, equivalent damping, and equivalent stiffness, respectively; e(t) represents the displacement excitation input at the left end (first end); This represents the first derivative of the lateral displacement with respect to time. w' represents the second derivative of the lateral displacement with respect to time; w' represents the first derivative of the lateral displacement with respect to space; w″ represents the second derivative of the lateral displacement with respect to space. It represents the second-order mixed derivative of the lateral displacement with respect to time and space.

[0121] Step 2: Convert the computational domain into a characteristic line domain, and establish the elastic wave reflection equation at the endpoints of the characteristic line domain. The elastic wave reflection equation includes the wave reflection equation under the Dirichlet boundary condition and the wave reflection equation under the complex boundary condition of mass-damped-spring.

[0122] 21) Transform the computational domain into a characteristic line domain and determine the initial conditions of the chord:

[0123] The solution w of equation (1) can be expressed as the superposition of two traveling waves, as shown in equation (3):

[0124] w=F(xV r t)+G(x+V l t) (3)

[0125] In equation (3); V r =c + V and V l =cV represents the velocities of the rightward-moving and leftward-moving waves in a chord relative to a fixed coordinate system. It is the wave speed of a string under static conditions; F(xV) r t) is the velocity V r The right-traveling wave; G(x+V) l t) is the velocity V l The left-facing wave.

[0126] The initial transverse displacement and velocity conditions for the string vibration are determined as shown in equation (4):

[0127]

[0128] In equation (4): the function φ(x) is the initial lateral displacement at different positions on the chord in the fixed coordinate system; the function ψ(x) is the initial velocity at different positions on the chord in the fixed coordinate system.

[0129] Perform feature line transformation:

[0130] Substituting equation (3) into equation (4) and integrating, we obtain the expressions for the left and right traveling waves at the initial time, as shown in equation (5):

[0131]

[0132] In equation (5): C is the integration constant, which can be taken as 0.

[0133] according to Figure 2 In the xt domain, left and right transverse waves are used to represent the transverse displacement of any point at any time in the chord, as shown in equation (6):

[0134] w(x1, t1)=F(ξ)+G(η) (6)

[0135] In equation (6): x1 represents the position of the point in the fixed coordinate system; t1 represents any time; ξ represents the specific second traveling wave parameter; η represents the specific first traveling wave parameter; and the expressions for ξ and η are as shown in equation (7):

[0136]

[0137] Where: x represents the position of the point in the fixed coordinate system; t represents any time; ξ represents the second traveling wave parameter introduced, that is, the second coordinate in the characteristic line domain; η represents the first traveling wave parameter introduced, that is, the first coordinate in the characteristic line domain.

[0138] according to Figure 3 By introducing left-traveling wave parameters ξ and right-traveling wave parameters η, and representing the traveling wave in the ξ-v domain, the problem of solving the dynamic equations is transformed into solving the problem of left and right traveling waves on the boundary, where:

[0139] The left-traveling wave parameter ξ is given by equation (8):

[0140] ξ=-V r t (8)

[0141] The right-traveling wave parameter η is given by equation (9):

[0142] η = l0 + V l t (9)

[0143] 22) Determine the wave reflection equation under the Dirichlet boundary conditions:

[0144] Substituting equation (3) into the first term of equation (2), we derive the wave reflection equation under the Dirichlet boundary condition, as shown in equation (10):

[0145] F(-V r t)+G(V l t)=e(t) (10)

[0146] Where: F(-V r t) represents the second traveling wave at the first endpoint, G(V) l t) represents the first traveling wave at the first endpoint;

[0147] The relationship between the left traveling wave and the right reflected traveling wave is determined as shown in equation (11):

[0148]

[0149] In equation (11): F(ξ) represents the second traveling wave; F′(ξ) represents the first derivative of the second traveling wave with respect to the second traveling wave parameter; F″(ξ) represents the second derivative of the second traveling wave with respect to the second traveling wave parameter; Indicates the first traveling wave; Substituting the first derivative of the first traveling wave with respect to the first traveling wave parameters into... Substituting the second derivative of the first traveling wave with respect to its parameters into... Y(ξ) represents the displacement excitation of the first endpoint in the characteristic line domain; Y′(ξ) represents the first derivative of the displacement excitation of the first endpoint with respect to the second traveling wave parameter in the characteristic line domain; Y″(ξ) represents the second derivative of the displacement excitation of the first endpoint with respect to the second traveling wave parameter in the characteristic line domain. Y(ξ) is the displacement excitation of the left endpoint, and its expression is given by equation (12):

[0150]

[0151] 23) Determine the wave reflection equation under the complex boundary conditions of mass-damped-spring:

[0152] Substituting equation (3) into the second term of equation (2), we obtain the wave reflection equation under the complex boundary conditions of mass-damped-spring, as shown in equation (13):

[0153] AG″+BG′+CG=R (13)

[0154] The parameters in equation (13) are as shown in equation (14):

[0155]

[0156] Where A represents the coefficient of the second-order differential term of the reflection equation at the complex boundary; B represents the coefficient of the first-order differential term of the reflection equation at the complex boundary; C represents the coefficient of the zero-order differential term of the reflection equation at the complex boundary; R represents the non-homogeneous term of the reflection equation at the complex boundary; F represents the second traveling wave at the second endpoint boundary; F′ represents the first derivative of the second traveling wave at the second endpoint boundary with respect to the second traveling wave parameter; F″ represents the second derivative of the second traveling wave at the second endpoint boundary with respect to the second traveling wave parameter; G represents the first traveling wave at the second endpoint boundary; G′ represents the first derivative of the first traveling wave at the second endpoint boundary with respect to the first traveling wave parameter; G″ represents the second derivative of the first traveling wave at the second endpoint boundary with respect to the first traveling wave parameter.

[0157] The arguments of F, G, and R are as follows: as in equation (15):

[0158]

[0159] The relationship between the arguments of the left and right traveling waves on the right boundary is determined as shown in equation (16):

[0160] ξ=2l0-2Vt-η (16)

[0161] Step 3: Use Duhamel integration to solve for the analytical solution of the wave reflection equation.

[0162] Substituting equation (16) into equation (13) yields equation (17):

[0163]

[0164] in: The characteristic equation can be obtained by substituting the non-homogeneous terms of R(ξ) with ξ = 2l0 - 2Vt - η:

[0165] The characteristic equation of formula (18) is:

[0166] Aλ 2 +Bλ+C=0 (18)

[0167] The characteristic roots of equation (19) are:

[0168]

[0169] Based on the four cases of characteristic roots, the complete solutions of the unit impulse response function and the expression are obtained respectively:

[0170] First scenario: A≠0; B 2 -4AC>0, the characteristic equation has two unequal real characteristic roots, and the general solution of equation (17) is derived as shown in equation (20):

[0171]

[0172] In equation (20):

[0173]

[0174] Where C1 and C2 are both parameters;

[0175] Assuming the above system uses a unit impulse excitation, i.e. Given l0 = 0, the initial conditions for the response are derived as shown in equation (21):

[0176]

[0177] Substituting equation (21) into equation (20), we derive the unit impulse response function, as shown in equation (22):

[0178]

[0179] Where: H(η) represents the unit impulse response function of the reflection equation; H′(η) represents the first derivative of the unit impulse response function of the reflection equation with respect to the first traveling wave parameter;

[0180] The complete solution of equation (17) is obtained as shown in equation (23):

[0181]

[0182] The second scenario: A≠0; B 2 -4AC=0, the characteristic equation has two equal real characteristic roots, and the general solution of equation (17) is derived as shown in equation (24):

[0183]

[0184] In equation (24):

[0185]

[0186] Following the same steps as the first case, the complete solution of the unit impulse response function and equation (17) is derived, as shown in equations (25) and (26):

[0187]

[0188]

[0189] The third scenario: A≠0; B 2 -4AC < 0, the characteristic equation has a pair of complex conjugate roots, as shown in equation (27):

[0190]

[0191] Where: abs() is the function for finding the absolute value. It is the imaginary unit; α and β are the real and imaginary parts of the eigenvalues, respectively;

[0192] The general solution of equation (17) is derived as shown in equation (28):

[0193]

[0194] In equation (28):

[0195]

[0196] Where: α represents the real part of the characteristic root; β represents the imaginary part of the characteristic root;

[0197] Following the same steps as the first case, the complete solution of the unit impulse response function and equation (17) is derived, as shown in equations (29) and (30):

[0198]

[0199]

[0200] Fourth case: A = 0; m e =0, the characteristic roots of the characteristic equation are as shown in equation (31):

[0201]

[0202] The general solution of equation (17) is derived as shown in equation (32):

[0203]

[0204] In equation (32):

[0205] C1 = G(l0)

[0206] Following the same steps as the first case, the complete solution of the unit impulse response function and equation (17) is derived, as shown in equations (33) and (34):

[0207]

[0208]

[0209] Using a precise recursive algorithm, the complete analytical expression of the integral part of the Duhamel analysis is obtained, as shown in equation (35);

[0210]

[0211] In equation (35): and These represent the first parts of equations (24), (26), (30), and (34), respectively. Based on the first to third cases, point η... n Substituting into equation (17), we can derive G″(η) n Using Duhamel integration, we derive G′(η) in the fourth case. n ) and G″(η n ), where η n This represents any first traveling wave parameter.

[0212] Step 4: Based on the assumption of waveform linearity within a small step size, a fast recursive method is used to determine the semi-analytical solution of the wave reflection equation.

[0213] Assume that the term R(τ) related to the waveform is linear, as shown in equation (36);

[0214]

[0215] In equation (36):

[0216]

[0217] Where: Δη represents the step size of the discrete first traveling wave parameter; K(η) n-1 ) represents the slope of the external non-homogeneous term within a first traveling wave parameter step;

[0218] Duhamel integral expansion, as shown in equation (37):

[0219]

[0220] In equation (37):

[0221]

[0222]

[0223] Where H1(Δη) represents the integral of the unit impulse response function within a first traveling wave parameter step; H′1(Δη) represents the integral of the first derivative of the unit impulse response function with respect to the first traveling wave parameter within a discrete first traveling wave parameter step; H2(Δη) represents the integral of the unit impulse response function multiplied by the first traveling wave parameter within a first traveling wave parameter step; and H′2(Δη) represents the integral of the unit impulse response function multiplied by the first traveling wave parameter within a first traveling wave parameter step.

[0224] Step 5: Based on the superposition method of the left and right traveling waves, the calculation results of the characteristic line domain are superimposed to obtain the vibration result of the axially moving string in the original spatiotemporal domain, as shown in equation (39):

[0225] w(x1, t1)=F(ξ)+G(η) (39)

[0226] Where: x1 represents the position coordinates of any point under consideration; t1 represents any moment under consideration.

[0227] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A method for obtaining lateral vibration of a fixed-length axially moving string system under complex boundaries, characterized in that: Includes the following steps: Step 1: Based on the dynamic characteristics and mechanical principles, establish a dynamic model of the fixed-length axially moving string system under mass-damped-spring boundary and Dirichlet boundary conditions; Of the two ends of the chord in the fixed-length axially movable chord system, the first end is the excitation input end, and the second end is the equivalent concentrated mass end. A fixed coordinate system OXY is established with the first end of the chord as the origin. The dynamic model of the fixed-length axially movable chord system under mass-damped-spring boundary and Dirichlet boundary conditions is as follows: Where: ρ, T, and V are the linear density, static tension, and constant velocity of the string, respectively; x represents the distance of a point on the string from the origin in the X direction; w(x, t) represents the lateral displacement of a point on the string at a distance x from the origin; l0 is the initial length of the string; l(t) is the real-time length of the string; and t represents time. The boundary conditions are: Where, m e c e and k e These represent the equivalent lumped mass, equivalent damping, and equivalent stiffness, respectively; e(t) represents the displacement excitation input at the first end. This represents the first derivative of the lateral displacement with respect to time. w' represents the second derivative of the lateral displacement with respect to time; w' represents the first derivative of the lateral displacement with respect to space; w″ represents the second derivative of the lateral displacement with respect to space. This represents the second-order mixed derivative of the lateral displacement with respect to time and space. Step 2: Transform the computational domain into a characteristic line domain, and establish the elastic wave reflection equation at the endpoints of the characteristic line domain. The elastic wave reflection equation includes the wave reflection equation under the Dirichlet boundary condition and the wave reflection equation under the complex boundary condition of mass-damped-spring. Step 3: Solve using Duhamel integration to obtain the analytical solution of the wave reflection equation; Step 4: Based on the assumption of waveform linearity within small step sizes, a rapid recursive method is used to obtain the semi-analytical solution of the wave reflection equation within the characteristic line domain; Step 5: Based on the superposition method of the left and right traveling waves in the characteristic line domain, the calculation results of the characteristic line domain are superimposed to obtain the original spatiotemporal domain, and the vibration results of the fixed-length axially moving string system are obtained.

2. The method for obtaining lateral vibration of a fixed-length axially moving chord system under complex boundaries according to claim 1, characterized in that: In step two, the computational domain is transformed into a feature line domain, and the solution w of the dynamic model is expressed as the superposition of two traveling waves: w = F(x - V r t) + G(x + V l t) In r =c+V In l =cV Where: V r and V l This represents the velocity of the second traveling wave moving towards the second end of the chord and the velocity of the first traveling wave moving towards the first end in a fixed coordinate system. F(xV) represents the wave speed of a string under static conditions. r t) is the velocity V r The second traveling wave; G(x+V) l t) is the velocity V l The first traveling wave; Determine the initial lateral displacement and velocity conditions for string vibration: Where: the function φ(x) is the initial lateral displacement at different positions on the chord in the fixed coordinate system; the function ψ(x) is the initial velocity at different positions on the chord in the fixed coordinate system; The expressions for the first and second traveling waves at the initial time are obtained as follows: Where C is the integration constant; In the xt domain, the first and second transverse waves are used to represent the transverse displacement at any point in time within the chord: w(x,t)=F(ξ)+G(η) Where: x represents the position of the point in the fixed coordinate system; t represents any time; ξ represents the second traveling wave parameter; η represents the first traveling wave parameter. Representing traveling waves in the ξ-η domain, the problem of solving the dynamic equations is transformed into solving the first and second traveling waves on the boundary: The second traveling wave parameter ξ at the boundary of the first end is: ξ=-V r t The first traveling wave parameter η at the boundary of the second end is: η=l0+V l t。 3. The method for obtaining lateral vibration of a fixed-length axially moving chord system under complex boundaries according to claim 2, characterized in that: In step two, the wave reflection equation under the Dirichlet boundary condition is: F(-V r t)+G(V l t)=e(t) Where: F(-V r t) represents the second traveling wave at the first endpoint, G(V) l t) represents the first traveling wave at the first endpoint; The relationship between the first traveling wave and the second reflected traveling wave is as follows: Where: F(ξ) represents the second traveling wave; F′(ξ) represents the first derivative of the second traveling wave with respect to the second traveling wave parameter; F″(ξ) represents the second derivative of the second traveling wave with respect to the second traveling wave parameter; Indicates the first traveling wave; Substituting the first derivative of the first traveling wave with respect to the first traveling wave parameters into... Substituting the second derivative of the first traveling wave with respect to its parameters into... Y′(ξ) represents the displacement excitation of the first endpoint in the characteristic line domain; Y″(ξ) represents the first derivative of the displacement excitation of the first endpoint with respect to the second traveling wave parameter in the characteristic line domain; Y″(ξ) represents the second derivative of the displacement excitation of the first endpoint with respect to the second traveling wave parameter in the characteristic line domain.

4. The method for obtaining lateral vibration of a fixed-length axially moving chord system under complex boundaries according to claim 2, characterized in that: In step two, the wave reflection equation under the complex boundary conditions of mass-damped-spring is: AG″+BG′+CG=R Where A represents the coefficient of the second-order differential term of the reflection equation at the complex boundary; B represents the coefficient of the first-order differential term of the reflection equation at the complex boundary; C represents the coefficient of the zero-order differential term of the reflection equation at the complex boundary; R represents the non-homogeneous term of the reflection equation at the complex boundary; F represents the second traveling wave at the second endpoint boundary; F′ represents the first derivative of the second traveling wave at the second endpoint boundary with respect to the second traveling wave parameter; F″ represents the second derivative of the second traveling wave at the second endpoint boundary with respect to the second traveling wave parameter; G represents the first traveling wave at the second endpoint boundary; G′ represents the first derivative of the first traveling wave at the second endpoint boundary with respect to the first traveling wave parameter; G″ represents the second derivative of the first traveling wave at the second endpoint boundary with respect to the first traveling wave parameter. The arguments of F, G, and R are: The relationship between the arguments of the first and second waves at the second boundary is as follows: ξ=2l0-2Vt-η.

5. The method for obtaining lateral vibration of a fixed-length axially moving chord system under complex boundaries according to claim 4, characterized in that: In step three, the relationship between the arguments of the first and second traveling waves at the second boundary is substituted into the wave reflection equation under the complex boundary conditions of mass-damped-spring, resulting in: in: The characteristic equation can be obtained by substituting the non-homogeneous terms of R(ξ) with ξ = 2l0 - 2Vt - η: Al 2 +Bλ+C=0 The characteristic roots of the characteristic equation are obtained: Based on the four cases of characteristic roots, the complete solutions of the unit impulse response function and the expression are obtained respectively: First scenario: A≠0; B 2 -4AC>0, the characteristic equation has two unequal real characteristic roots, thus we get: Where C1 and C2 are both parameters; Using unit pulse excitation With l0 = 0, we obtain the complete solution of the unit impulse response function and its expression: Where: H(η) represents the unit impulse response function of the reflection equation; H′(η) represents the first derivative of the unit impulse response function of the reflection equation with respect to the first traveling wave parameter; The second scenario: A≠0; B 2 -4AC=0, the characteristic equation has two equal real characteristic roots, thus: Using unit pulse excitation With l0 = 0, we obtain the complete solution of the unit impulse response function and its expression: The third scenario: A≠0; B 2 -4AC<0, the characteristic equation has a pair of complex conjugate roots, i.e. Here, abs() is the function for calculating the absolute value. It is the imaginary unit; α and β are the real and imaginary parts of the eigenvalues, respectively; get: Using unit pulse excitation With l0 = 0, we obtain the complete solution of the unit impulse response function and its expression: Fourth case: A = 0; m e =0, the characteristic roots of the characteristic equation are: get: C1 = G(l0) Using unit pulse excitation With l0 = 0, we obtain the complete solution of the unit impulse response function and its expression: Based on the first to third cases, point η n Substituting them in, we can get G″(η) n Using Duhamel integration, we derive G′(η) in the fourth case. n ) and G″(η n ); where η n This represents any first traveling wave parameter.

6. The method for obtaining lateral vibration of a fixed-length axially moving chord system under complex boundaries according to claim 5, characterized in that: In step four, Assume terms related to waveform It is linear: Where: Δη represents the step size of the discrete first traveling wave parameter; K(η) n-1 ) represents the slope of the external non-homogeneous term within a first traveling wave parameter step; Duhamel integral expansion yields: Where H1(Δη) represents the integral of the unit impulse response function within a first traveling wave parameter step; H′1(Δη) represents the integral of the first derivative of the unit impulse response function with respect to the first traveling wave parameter within a discrete first traveling wave parameter step; H2(Δη) represents the integral of the unit impulse response function multiplied by the first traveling wave parameter within a first traveling wave parameter step; and H′2(Δη) represents the integral of the unit impulse response function multiplied by the first traveling wave parameter within a first traveling wave parameter step.

7. The method for obtaining lateral vibration of a fixed-length axially moving chord system under complex boundaries according to claim 6, characterized in that: The vibration results are as follows: w(x1, t1) = F(ξ) + G(η) Where: x1 represents the position coordinates of any point; t1 represents any time.

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