Method for obtaining transverse vibration and vibration suppression of belt model under complex boundary conditions

Abstract

The application discloses a method for obtaining transverse vibration and vibration suppression of a conveyor belt model under complex boundary conditions, which comprises the following steps: 1. obtaining a motion equation of the conveyor belt model according to the Hamilton principle, and expressing the solution as a form of left and right traveling wave superposition; 2. deducing initial left and right traveling waves according to initial conditions of the motion of the conveyor belt model; 3. solving each traveling wave expression in a reflection period according to a traveling wave reflection equation; 4. superimposing the left and right traveling waves on the boundary obtained by solving according to the traveling wave superposition method; and 5. obtaining a transverse vibration response. The application can solve the problem that it is difficult to solve the vibration of a conveyor belt device with complex boundaries by using the traveling wave method, the traveling wave reflection process is described by using a traveling wave reflection period diagram, and the method is simple and easy to understand, so that the problem that the vibration response is lost due to the increase of the moving speed when the vibration response is solved by using a numerical method can be solved, and the transverse vibration of the conveyor belt can be suppressed by boundary control.

Description

Technical Field

[0001] This invention belongs to the field of mechanical system dynamics modeling and vibration control, specifically relating to a method for obtaining and suppressing the lateral vibration of a conveyor belt model under complex boundary conditions. Background Technology

[0002] Belt drive systems, as an important means of transmitting power and motion in mechanical transmission, are characterized by their simple structure, free speed variation, lack of distance limitations, and ease of adjustment and replacement. Maintaining high precision, stability, reliability, and low noise under high-speed operation has become an important indicator for evaluating the transmission quality of belt drive systems. Furthermore, they are used in many real-world applications, such as power transmission belts, paper belts, aerial cableways, high-rise elevators, cables, and automotive drive belts.

[0003] Lateral vibration of conveyor belts is a challenging problem that has been studied for many years and continues to attract widespread attention. Traditional research techniques rely on partial differential equations of motion based on Hamilton's principle and finite element dynamics equations based on Lagrange's equations, using numerical calculations to obtain the lateral vibration response. However, these existing methods suffer from problems such as complex solution processes, low accuracy, and poor stability when solving lateral vibration problems of conveyor belts under complex boundary conditions.

[0004] In practical engineering, transmission belts are mostly of fixed length. When using traveling waves to solve the motion equations in a fixed-length axial motion model, under complex boundary conditions, the lateral vibration of the conveyor belt will continue to vibrate after one cycle. Furthermore, in a fixed-length conveyor belt, the reflection and superposition of traveling waves have multiple periods. The multiple reflections of traveling waves at complex boundaries make the solution process more complicated. Summary of the Invention

[0005] To overcome the shortcomings of the prior art, this invention provides a method for obtaining lateral vibration and vibration suppression of a conveyor belt model under complex boundary conditions, aiming to accurately obtain lateral vibration under (mass-spring-damping) fixed boundary conditions; and to extend its vibration response to any period by applying the traveling wave reflection superposition method; and to add a suppressing force to reduce the lateral motion of the reflected wave, thereby suppressing the overall lateral vibration of the conveyor belt.

[0006] The present invention adopts the following technical solution to solve the technical problem:

[0007] This invention discloses a method for obtaining lateral vibration and vibration suppression of a conveyor belt model under complex boundary conditions. The complex boundary conditions refer to a situation where, among the two boundary points of the conveyor belt model, the right boundary is a fixed boundary, and the other end has a complex boundary consisting of a mass-damping-spring boundary. The method comprises the following steps:

[0008] Step 1: Based on Hamilton's principle, obtain the motion equation of the conveyor belt model using equation (1):

[0009] u tt +2vu xt +(v 2 -c 2 )u xx =0 (1)

[0010] In equation (1), x represents the axial coordinate of the conveyor belt model, t represents time, and v represents the axial velocity of the conveyor belt model; u tt It is the second partial derivative of the lateral displacement function u of the conveyor belt model with respect to time t; u xx It is the second partial derivative of the lateral displacement function u of the conveyor belt model with respect to the axial coordinate x; u xt These are the first-order partial derivatives of the lateral displacement function u of the conveyor belt model with respect to the axial coordinate x and time t, respectively. ρ represents the traveling wave velocity; P represents the tension in the conveyor belt model; ρ is the linear density of the conveyor belt model.

[0011] Solving equation (1) yields the transverse vibration response u(x,t) in the form of superposition of left and right traveling waves, as shown in equation (2):

[0012] u(x,t)=F(xv r t)+G(x+v l t) (2)

[0013] In equation (2), v r v is the velocity of the right-traveling wave in the conveyor belt model relative to a fixed coordinate system. l Let F(xv) be the velocity of the left-hand traveling wave in the conveyor belt model relative to a fixed coordinate system. r t) represents a velocity of v r = c+v right-traveling wave; G(x+v) l t) represents a velocity of v l = cv's left-facing wave; wherein, the fixed coordinate system is a coordinate system established with the complex boundary as the origin, the axial movement direction of the transmission belt as the x-direction, and the lateral vibration direction as the u-direction;

[0014] Step 2: Determine the initial conditions and mass-damping-spring boundary conditions of the conveyor belt model:

[0015] Step 2.1: Obtain the initial motion conditions of the conveyor belt model using equation (3):

[0016]

[0017] In equation (3), the function φ(x) represents the initial lateral displacement at different positions on the conveyor belt model in the fixed coordinate system; the function ψ(x) represents the initial velocity at different positions on the conveyor belt model in the fixed coordinate system; l0 is the distance between the two boundaries; u t It is the first-order partial derivative of the lateral displacement function u of the conveyor belt model with respect to time t;

[0018] The complex boundary condition is a mass-damped-spring boundary at x = 0 and a fixed boundary at x = 10, and the complex boundary condition is obtained using equation (4):

[0019]

[0020] In equation (4), m is the left boundary mass of the conveyor belt model; k is the spring stiffness coefficient of the conveyor belt model at the left boundary; η is the damping coefficient of the conveyor belt model at the left boundary; f(t) is the damping force of the conveyor belt model at the left boundary; u x It is the first-order partial derivative of the lateral displacement function u of the conveyor belt model with respect to time x;

[0021] Step 3: Based on the motion law of the traveling wave in the conveyor belt model and the reflection law of the traveling wave at both ends of the conveyor belt model, obtain the reflection period diagram of the traveling wave, and then use equation (5) to determine the reflection period T:

[0022]

[0023] Step 4: Substitute equation (2) into equation (3) to obtain the initial left and right traveling waves as shown in equation (6):

[0024]

[0025] In equation (6), This represents the initial right-traveling wave expression relative to the axial coordinate x; The expression for the initial left-traveling wave relative to the axial coordinate x;

[0026] Step 5: Based on the traveling wave reflection pattern and continuity condition in the nth period of the traveling wave reflection period diagram, obtain the expressions for the left and right traveling waves in the nth period;

[0027] Step 5.1: At the mass-damping-spring boundary on the left side at x=0 within the nth period, use equation (7) to obtain the result from the first left traveling wave. The second right-traveling wave reflected from the left boundary

[0028]

[0029] In equation (7), β represents the first intermediate variable parameter, and r represents the second intermediate variable parameter, and r = -v r t; This represents the second right-traveling wave relation after two reflections in the nth period, following the transformation by the second intermediate variable r. express The first derivative; express The second derivative; This represents the first intermediate transformation relation, which is related to the first left-traveling wave. Related, and obtained from equation (8):

[0030]

[0031] In equation (8), This represents the first left-traveling wave that undergoes one reflection in the nth cycle. express The first derivative; express The second derivative;

[0032] Step 5.2: Solve equation (7) to obtain the second right-traveling wave relation as shown in equation (9).

[0033]

[0034] In equation (9), μ1 and μ2 are two characteristic solutions of equation (7); C1 and C2 are two constants. ξ represents the expression for the integrand; ξ represents the integration variable;

[0035] Step 5.3: Based on the continuity of reflection, the continuity condition is obtained using equation (10):

[0036]

[0037] In equation (10), This represents the first right-traveling wave after one reflection in the nth period. yes The first derivative;

[0038] Step 5.4: Substituting equation (9) into equation (10), we obtain two constants C1 and C2 using equation (11):

[0039]

[0040] Step 5.5: Let t b Indicates the second left-traveling wave The time it takes for the conveyor belt model to move from its right end point to its left end point, and t b=l0 / v l , in t b After a certain time, equation (12) is used to obtain the result from the second left-traveling wave. The third right-traveling wave reflected from the left boundary

[0041]

[0042] In equation (12), This represents the third right-traveling wave relation after transformation by the second intermediate variable r, which undergoes three reflections in the nth period. Indicated First derivative; express The second derivative; This represents the second intermediate transformation relation, which is related to the second left-traveling wave. Related, and:

[0043]

[0044] In equation (13), This represents the second left-traveling wave that undergoes two reflections in the nth cycle. express The first derivative, express The second derivative;

[0045] Step 5.6: Solve equation (12) to obtain the third right-traveling wave relation as shown in equation (14).

[0046]

[0047] In equation (14), Let C3 and C4 represent the integrand, ξ represent the integration variable, and C3 and C4 represent two constants.

[0048] Step 5.7: Based on the continuity of reflection, the continuity condition is obtained using equation (15):

[0049]

[0050] In equation (15), This represents the third right-traveling wave after three reflections in the nth period; express The first derivative;

[0051] Step 5.8: Substituting equation (14) into equation (15), we obtain two constants C3 and C4 using equation (16):

[0052]

[0053] Step 5.9: At the fixed boundary x = l0 on the right side within the nth period, use equation (17) to obtain the first right traveling wave F1 n The second left-traveling wave reflected from the right boundary

[0054]

[0055] In equation (17), s represents the third intermediate variable parameter, and s = l0 + v l t; This represents the expression for the second left-traveling wave that undergoes two reflections in the nth cycle after transformation by the intermediate variable parameter s.

[0056] Using equation (18), we obtain the second right-traveling wave. The third left-traveling wave reflected from the right boundary

[0057]

[0058] In equation (18), This represents the expression for the third left-facing wave that undergoes three reflections in the nth cycle after transformation by the intermediate variable parameter s.

[0059] The transverse vibration response of the nth cycle in the conveyor belt model is obtained based on the traveling wave reflection diagram and equations (2), (6), (9), (11), (14), (16), (17), and (18).

[0060] The method for obtaining lateral vibration and vibration suppression of a conveyor belt model under complex boundary conditions, as described in this invention, is also characterized by:

[0061] When the inhibitory force f(t) satisfies f(t) = mu tt (0,t)+ku(0,t)+ηu t (0,t)-η o u t At (0,t), the damping force f(t) counteracts the effects of the left-end boundary mass and spring stiffness. At this point, the left-end boundary damping η is adjusted to the optimal value η. o ,Right now This results in no transverse vibration of the reflected wave at the left boundary, and the axial rope-moving system after one traveling wave reflection cycle no longer has transverse vibration without excitation.

[0062] Compared with existing technologies, the beneficial effects of this invention are reflected in:

[0063] 1. This invention solves the problem of difficulty in solving the vibration of complex boundary conveyor belt equipment using the traveling wave method by superimposing the traveling wave reflection and using the traveling wave reflection period diagram. The traveling wave reflection process is explained by the traveling wave reflection period diagram, which is simple and easy to understand.

[0064] 2. Compared with the commonly used numerical methods, the method of this invention has the advantages of accuracy and good stability. This invention uses an analytical method to solve the problem, which obtains an accurate analytical expression. It solves the problem of instability caused by the increase of moving speed in the numerical method for solving the vibration response. It also has high computational efficiency and greatly shortens the design cycle of vibration reduction design for conveyor belt equipment.

[0065] 3. This invention provides a boundary control method to suppress the lateral vibration of the conveyor belt, which can reduce the noise during equipment operation, extend its service life, etc. Attached Figure Description

[0066] Figure 1 For the mass-spring-damped fixed boundary model;

[0067] Figure 2a The initial right-traveling wave reflection period diagram for the nth period;

[0068] Figure 2b This is the initial left-traveling wave reflection period diagram for the nth period. Detailed Implementation

[0069] In this embodiment, complex boundary conditions refer to a situation where, in the boundary between the two ends of the conveyor belt model, the right end boundary is a fixed boundary, while the other end's complex boundary is a mass-damped-spring boundary; for example... Figure 1 The conveyor belt model shown here does not consider the influence of conveyor belt bending moment. A method for obtaining the lateral vibration and vibration suppression of the conveyor belt model under complex boundary conditions is as follows: The motion equations of the conveyor belt model are obtained according to Hamilton's principle; the solution of the obtained motion equations is expressed as the superposition of left and right traveling waves; the initial conditions and complex boundary conditions of the conveyor belt model are determined; the traveling wave reflection period diagram is obtained according to the motion law of the traveling waves in the conveyor belt model and the superposition law of the traveling waves at both ends of the conveyor belt model; the initial left and right traveling waves are derived according to the initial motion conditions of the conveyor belt model; the expressions of each traveling wave within the reflection period are solved according to the traveling wave reflection equation; the left and right traveling waves on the boundary of the model are superimposed according to the traveling wave superposition method; the lateral vibration response of the model is obtained. Specifically, this method is carried out according to the following steps:

[0070] Step 1: Based on Hamilton's principle, obtain the motion equation of the conveyor belt model using equation (1):

[0071] u tt +2vu xt +(v 2 -c2 )u xx =0 (1)

[0072] In equation (1), x represents the axial coordinate of the conveyor belt model, t represents time, and v represents the axial velocity of the conveyor belt model; u tt It is the second partial derivative of the lateral displacement function u of the conveyor belt model with respect to time t; u xx It is the second partial derivative of the lateral displacement function u of the conveyor belt model with respect to the axial coordinate x; u xt These are the first-order partial derivatives of the lateral displacement function u of the conveyor belt model with respect to the axial coordinate x and time t, respectively. ρ represents the traveling wave velocity; P represents the tension in the conveyor belt model; ρ is the linear density of the conveyor belt model.

[0073] Solving equation (1) yields the transverse vibration response u(x,t) in the form of superposition of left and right traveling waves, as shown in equation (2):

[0074] u(x,t)=F(xv r t)+G(x+v l t) (2)

[0075] In equation (2), v r v is the velocity of the right-traveling wave in the conveyor belt model relative to a fixed coordinate system. l Let F(xv) be the velocity of the left-hand traveling wave in the conveyor belt model relative to a fixed coordinate system. r t) represents a velocity of v r = c+v right-traveling wave; G(x+v) l t) represents a velocity of v l = cv's left-traveling wave; fixed coordinate system such as Figure 1 As shown, a fixed coordinate system is established with the complex boundary as the origin, the axial movement direction of the conveyor belt as the x-direction, and the lateral vibration direction as the u-direction.

[0076] Step 2: Determine the initial conditions and mass-damping-spring boundary conditions for the conveyor belt model:

[0077] Step 2.1: Obtain the initial motion conditions of the conveyor belt model using equation (3):

[0078]

[0079] In equation (3), the function φ(x) represents the initial lateral displacement at different positions on the conveyor belt model in the fixed coordinate system; the function ψ(x) represents the initial velocity at different positions on the conveyor belt model in the fixed coordinate system; l0 is the distance between the two boundaries; u t It is the first-order partial derivative of the lateral displacement function u with respect to time t in the conveyor belt model.

[0080] The complex boundary conditions are a mass-damped-spring boundary at x = 0 and a fixed boundary at x = l0, and the complex boundary conditions are obtained using equation (4):

[0081]

[0082] In equation (4), m is the left boundary mass of the conveyor belt model; k is the spring stiffness coefficient of the conveyor belt model at the left boundary; η is the damping coefficient of the conveyor belt model at the left boundary; f(t) is the damping force of the conveyor belt model at the left boundary; u x It is the first-order partial derivative of the lateral displacement function u with respect to time x in the conveyor belt model.

[0083] Step 3: Based on the motion law of the traveling wave in the conveyor belt model and the reflection law of the traveling wave at both ends of the conveyor belt model, obtain the reflection period diagram of the traveling wave, such as... Figure 2a and Figure 2b As shown, where according to Figure 2a and Figure 2b The reflection patterns of the left and right traveling waves within the nth period can be obtained; for example... Figure 2a As shown, we can obtain:

[0084] The domain is {(x,t)|0≤x≤l0,0≤t≤x / v} r};

[0085] The domain is {(x,t)|0≤x≤l0,(l0-x) / v} l ≤t≤(l0-x) / v l +l0 / v r};

[0086] The domain is {(x,t)|0≤x≤l0,x / v} r +l0 / v l ≤t≤T};

[0087] Second left-traveling wave It is the first right-traveling wave Reflected from the right boundary; the third right-traveling wave It is the second left-traveling wave

[0088] It was reflected from the left boundary;

[0089] like Figure 2b As shown, we can obtain:

[0090] The domain is {(x,t)|0≤x≤l0,0≤t≤(l0-x) / v} l};

[0091] The domain is {(x,t)|0≤x≤l0,x / v} r ≤t≤x / v r +l0 / v l};

[0092] The domain is {(x,t)|0≤x≤l0,(l0-x) / v} l +l0 / v r ≤t≤T};

[0093] Second right-traveling wave It is the first left-traveling wave Reflected from the left boundary; the third left-traveling wave It is the second right-traveling wave

[0094] It is reflected from the left boundary; the reflection period T is determined using equation (5):

[0095]

[0096] Step 4: Substitute equation (2) into equation (3) to obtain the initial left and right traveling waves as shown in equation (6):

[0097]

[0098] In equation (6), This represents the initial traveling wave expression of the right-hand traveling wave relative to the axial coordinate x; The initial traveling wave expression representing the left traveling wave relative to the axial coordinate x;

[0099] Step 5: Based on the traveling wave reflection period diagram, substitute equation (2) into equation (4) for the complex boundary of the mass-damped-spring boundary to obtain:

[0100]

[0101]

[0102] In equation (7), β represents the first intermediate variable parameter, and r represents the second intermediate variable parameter, and r = -v r t; R(r) represents the intermediate relation used to simplify equation (7); its R(r) is related to the left-traveling wave G in detail as shown below:

[0103]

[0104] In equation (8), s represents the third intermediate variable parameter, and s = l0 + v l t;

[0105] Using equation (9), we obtain the characteristic equation corresponding to equation (7):

[0106]

[0107] Solving the characteristic equation (9) yields:

[0108]

[0109] In this embodiment, β≠1 is used for the solution explanation. Then, by using equation (10) to solve equation (7), the expression of the right-traveling wave is obtained:

[0110]

[0111] In equation (10), F(r) represents the expression of the right-traveling wave; C1 and C2 represent two constants, R(ξ) represents the integrand, and ξ represents the integration variable; its relational expression is consistent with the relational expression of R(r); a is a certain constant;

[0112] Based on the traveling wave reflection pattern in the nth period and the continuity condition, the expressions for the left and right traveling waves in the nth period are obtained.

[0113] Step 5.1: At the mass-damping-spring boundary on the left side at x=0 within the nth period, use equation (11) to obtain the result from the first left traveling wave. The second right-traveling wave reflected from the left boundary

[0114]

[0115] In equation (11), This represents the second right-hand traveling wave relation after transformation by the second intermediate variable r, which undergoes two reflections within the nth period. express The first derivative; express The second derivative; This represents the first intermediate transformation relation; it is related to the first left-traveling wave. Related, and obtained from equation (12):

[0116]

[0117] In equation (12), This indicates that the first left-traveling wave is reflected once in the nth cycle. express The first derivative; express The second derivative;

[0118] Step 5.2: Solve equation (11) to obtain the second right-traveling wave relation as shown in equation (9).

[0119]

[0120] In equation (13), C1 and C2 are two constants. Let ξ represent the expression of the integrand, which is the same as the first intermediate transformation relation, and let ξ represent the integration variable.

[0121] Step 5.3: Based on the continuity of reflection, the continuity condition is obtained using equation (14):

[0122]

[0123] In equation (10), This represents the first right-traveling wave after one reflection in the nth period. yes The first derivative;

[0124] Step 5.4: Substitute equation (13) into equation (14), and then use equation (15) to obtain C1 and C2:

[0125]

[0126] Step 5.5: Let t b Indicates the second left-traveling wave The time it takes for the conveyor belt to move from the right end to the left end, and t b =l0 / v l , in t b After a certain time, equation (16) is used to obtain the result from the second left-traveling wave. The third right-traveling wave reflected from the left boundary

[0127]

[0128] In equation (13), This represents the third right-traveling wave relation after three reflections in the nth period, following the transformation by the second intermediate variable r. Indicated First derivative; express The second derivative This represents the second intermediate transformation relation, which is related to the second left-traveling wave. Related, and obtained from equation (17):

[0129]

[0130] In equation (17), , This represents the second left-traveling wave that undergoes two reflections in the nth cycle. express The first derivative, express The second derivative;

[0131] Step 5.6: Solve equation (16) to obtain the third right-traveling wave relation as shown in equation (18).

[0132]

[0133] In equation (18), Let ξ represent the integrand, whose function expression is the same as the second intermediate transformation relation, ξ represent the integration variable, and C3 and C4 represent two constants.

[0134] Step 5.7: Based on the continuity of reflection, the continuity condition is obtained using equation (19):

[0135]

[0136] In equation (19), This represents the third right-traveling wave after three reflections in the nth period. express The first derivative;

[0137] Step 5.8: Substitute equation (18) into equation (19), and then use equation (20) to obtain C3 and C4;

[0138]

[0139] Step 5.9: At the fixed boundary x = l0 on the right side within the nth period, use equation (21) to obtain the result from the first right traveling wave. The second left-traveling wave reflected from the right boundary

[0140]

[0141] In equation (21), s represents the third intermediate variable parameter, and s = l0 + v l t; This represents the expression for the second left-traveling wave that undergoes two reflections in the nth cycle after transformation by the intermediate variable parameter s.

[0142] Using equation (22), we obtain the second right-traveling wave. The third left-traveling wave reflected from the right boundary

[0143]

[0144] In equation (22), This represents the expression for the third left-traveling wave after three reflections in the nth cycle, transformed by the intermediate variable parameter s.

[0145] The transverse vibration response of the nth cycle in the conveyor belt model is obtained based on the traveling wave reflection diagram and equations (2), (6), (13), (15), (18), (20), (21), and (22).

[0146] In this embodiment, when the inhibitory force f(t) satisfies f(t) = mu tt (0,t)+ku(0,t)+ηu t (0,t)-η o u t At (0,t), the damping force f(t) counteracts the effects of the left-end boundary mass and spring stiffness. At this point, the left-end boundary damping η is adjusted to the optimal value η. o ,Right now This results in no transverse vibration of the reflected wave at the left boundary, and the axial rope-moving system after one traveling wave reflection cycle no longer has transverse vibration without excitation.

Claims

1. A method for obtaining lateral vibration and vibration suppression of a conveyor belt model under complex boundary conditions, wherein the complex boundary conditions refer to a situation where, among the two boundary points of the conveyor belt model, the right boundary is a fixed boundary, and the other complex boundary is a mass-damped-spring boundary; characterized in that, The acquisition method includes the following steps: Step 1: Based on Hamilton's principle, obtain the motion equation of the conveyor belt model using equation (1): u tt +2vu xt +(v 2 -c 2 )u xx =0(1) In equation (1), x represents the axial coordinate of the conveyor belt model, t represents time, and v represents the axial velocity of the conveyor belt model; u tt It is the second partial derivative of the lateral displacement function u of the conveyor belt model with respect to time t; u xx It is the second partial derivative of the lateral displacement function u of the conveyor belt model with respect to the axial coordinate x; u xt These are the first-order partial derivatives of the lateral displacement function u of the conveyor belt model with respect to the axial coordinate x and time t, respectively. ρ represents the traveling wave velocity; P represents the tension in the conveyor belt model; ρ is the linear density of the conveyor belt model. Solving equation (1) yields the transverse vibration response u(x,t) in the form of superposition of left and right traveling waves, as shown in equation (2): u(x,t)=F(x-v r t)+G(x+v l t)(2) In equation (2), v r v is the velocity of the right-traveling wave in the conveyor belt model relative to a fixed coordinate system. l Let F(xv) be the velocity of the left-hand traveling wave in the conveyor belt model relative to a fixed coordinate system. r t) represents a velocity of v r = c+v right-traveling wave; G(x+v) l t) represents a velocity of v l = cv's left-traveling wave; wherein, the fixed coordinate system is a coordinate system established with the complex boundary as the origin, the axial movement direction of the conveyor belt as the x-direction, and the lateral vibration direction as the u-direction; Step 2: Determine the initial conditions and mass-damping-spring boundary conditions of the conveyor belt model: Step 2.1: Obtain the initial motion conditions of the conveyor belt model using equation (3): In equation (3), the function φ(x) represents the initial lateral displacement at different positions on the conveyor belt model in the fixed coordinate system; the function ψ(x) represents the initial velocity at different positions on the conveyor belt model in the fixed coordinate system; l0 is the distance between the two boundaries; u t It is the first-order partial derivative of the lateral displacement function u of the conveyor belt model with respect to time t; The complex boundary condition is a mass-damped-spring boundary at x = 0 and a fixed boundary at x = 10, and the complex boundary condition is obtained using equation (4): In equation (4), m is the left boundary mass of the conveyor belt model; k is the spring stiffness coefficient of the conveyor belt model at the left boundary; η is the damping coefficient of the conveyor belt model at the left boundary; f(t) is the damping force of the conveyor belt model at the left boundary; u x It is the first-order partial derivative of the lateral displacement function u of the conveyor belt model with respect to time x; Step 3: Based on the motion law of the traveling wave in the conveyor belt model and the reflection law of the traveling wave at both ends of the conveyor belt model, obtain the reflection period diagram of the traveling wave, and then use equation (5) to determine the reflection period T: Step 4: Substitute equation (2) into equation (3) to obtain the initial left and right traveling waves as shown in equation (6): In equation (6), F1 1 (x) represents the initial right-traveling wave expression relative to the axial coordinate x; The expression for the initial left-traveling wave relative to the axial coordinate x; Step 5: Based on the traveling wave reflection pattern and continuity condition in the nth period of the traveling wave reflection period diagram, obtain the expressions for the left and right traveling waves in the nth period; Step 5.1: At the mass-damping-spring boundary on the left side at x=0 within the nth period, use equation (7) to obtain the first left traveling wave G1 n The second right-traveling wave F2 reflected from the left boundary n : In equation (7), β represents the first intermediate variable parameter, and r represents the second intermediate variable parameter, and r = -v r t; This represents the second right-traveling wave relation after two reflections in the nth period, following the transformation by the second intermediate variable r. express The first derivative; express The second derivative; This represents the first intermediate transformation relation, which is related to the first left-traveling wave. Related, and obtained from equation (8): In equation (8), This represents the first left-traveling wave that undergoes one reflection in the nth cycle. express The first derivative; express The second derivative; Step 5.2: Solve equation (7) to obtain the second right-traveling wave relation as shown in equation (9). In equation (9), μ1 and μ2 are two characteristic solutions of equation (7); C1 and C2 are two constants. ξ represents the expression for the integrand; ξ represents the integration variable; Step 5.3: Based on the continuity of reflection, the continuity condition is obtained using equation (10): In equation (10), F1 n F1 represents the first right-traveling wave after one reflection in the nth period. n ′ is F1 n The first derivative; Step 5.4: Substituting equation (9) into equation (10), we obtain two constants C1 and C2 using equation (11): Step 5.5: Let t b Indicates the second left-traveling wave The time it takes for the conveyor belt model to move from its right end point to its left end point, and t b =l0v l , in t b After a certain time, equation (12) is used to obtain the result from the second left-traveling wave. The third right-traveling wave reflected from the left boundary In equation (12), This represents the third right-traveling wave relation after transformation by the second intermediate variable r, which undergoes three reflections in the nth period. Indicated First derivative; express The second derivative; This represents the second intermediate transformation relation, which is related to the second left-traveling wave. Related, and: In equation (13), This represents the second left-traveling wave that undergoes two reflections in the nth cycle. express The first derivative, express The second derivative; Step 5.6: Solve equation (12) to obtain the third right-traveling wave relation as shown in equation (14). In equation (14), Let C3 and C4 represent the integrand, ξ represent the integration variable, and C3 and C4 represent two constants. Step 5.7: Based on the continuity of reflection, the continuity condition is obtained using equation (15): In equation (15), F3 n This represents the third right-traveling wave after three reflections in the nth period; F3 n′ Indicates F3 n The first derivative; Step 5.8: Substituting equation (14) into equation (15), we obtain two constants C3 and C4 using equation (16): Step 5.9: At the fixed boundary x = l0 on the right side within the nth period, use equation (17) to obtain the first right traveling wave F1 n The second left-traveling wave reflected from the right boundary In equation (17), s represents the third intermediate variable parameter, and s = l0 + v l t; This represents the expression for the second left-traveling wave that undergoes two reflections in the nth cycle after transformation by the intermediate variable parameter s. Using equation (18), we obtain the second right-traveling wave F2 n The third left-traveling wave reflected from the right boundary In equation (18), This represents the expression for the third left-facing wave that undergoes three reflections in the nth cycle after transformation by the intermediate variable parameter s. The transverse vibration response of the nth cycle in the conveyor belt model is obtained based on the traveling wave reflection diagram and equations (2), (6), (9), (11), (14), (16), (17), and (18).

2. The method for obtaining lateral vibration and vibration suppression of a conveyor belt model under complex boundary conditions according to claim 1, characterized in that: When the inhibitory force f(t) satisfies f(t) = mu tt (0,t)+ku(0,t)+ηu t (0,t)-η o u t At (0,t), the damping force f(t) counteracts the effects of the left-end boundary mass and spring stiffness. At this point, the left-end boundary damping η is adjusted to the optimal value η. o ,Right now This results in no transverse vibration of the reflected wave at the left boundary, and the axial rope-moving system after one traveling wave reflection cycle no longer has transverse vibration without excitation.

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