Solution method for axial moving rope lateral vibration controlled by bilateral magneto-rheological dampers

Abstract

The application provides a kind of axial moving rope lateral vibration solving method of double-sided magneto-rheological damping control, at least comprising the following steps: installing magneto-rheological damper at the both ends of axial moving rope model. According to Hamilton principle, the motion equation of axial moving rope model and the equation set of damping boundary constraint are obtained. The solution of the motion equation of axial moving rope model is expressed as the form of left traveling wave, right traveling wave and load wave superimposed on each other. According to the form of load, the expression of load wave is solved. According to the initial conditions of axial moving rope model, the expression of initial traveling wave is derived. According to the boundary conditions of axial moving rope model, the traveling wave reflection equation is derived. According to the traveling wave reflection equation, the expression of each traveling wave is derived. According to the motion law of left traveling wave and right traveling wave in axial moving rope model, left traveling wave, right traveling wave and load wave are superimposed. According to the boundary optimal damping obtained by traveling wave reflection equation, the current size of coil in magneto-rheological damper is adjusted.

Description

Technical Field

[0001] This invention belongs to the field of mechanical system dynamics technology, specifically relating to a method for solving the lateral vibration of an axially moving rope under dual-sided magnetorheological damping control. Background Technology

[0002] Axial moving rope systems possess advantages such as high operating efficiency, strong adaptability, high load-bearing capacity, simple structure, and convenient control, and are widely used in various engineering fields, such as ropes for mining cranes, band saws, and flexible films in the adhesive printing industry. During operation, these devices inevitably generate noise and vibration. Lateral vibration not only affects the normal operation of these devices but can also significantly reduce their service life, thus limiting the application of axial moving rope systems. The lateral vibration problem of axial moving rope systems is a challenging topic that has been studied for many years and remains a focus of attention. Traditional research techniques are based on partial differential equations of motion established by Hamilton's principle and finite element dynamics equations established by Lagrange's equations. Numerical calculation methods, such as the Galerkin method, Runge-Kutta method, Newmark method, and time-varying state-space equations, are used to solve these equations to obtain the lateral vibration response of the axial moving rope system. However, the existing methods suffer from problems such as complex solution processes, low solution accuracy, and poor stability when solving the lateral vibration problem of moving rope systems under complex mixed boundary conditions. Furthermore, when the axial moving rope equipment moves at a high speed, approaching or reaching the critical speed, the vibration displacement amplitude of the equipment will increase abnormally, leading to increased errors. Summary of the Invention

[0003] The purpose of this application is to provide a method for solving the lateral vibration of an axially moving rope under dual-sided magnetorheological damping control, so as to solve the technical problems of complex solution process, low solution accuracy and poor stability in the prior art.

[0004] To achieve the above objectives, the technical solution adopted in this application is: to provide a method for solving the lateral vibration of an axially moving rope controlled by dual-sided magnetorheological damping, comprising at least the following steps:

[0005] S1: Install magnetorheological dampers at both ends of the axially moving rope model and form damping boundaries at both ends of the axially moving rope model, and uniformly distribute right harmonic loads on the axially moving rope model.

[0006] S2: Obtain the equations of motion and damping boundary constraint equations for the axially moving rope model based on Hamilton's principle;

[0007] S3: The solution to the motion equation of the axially moving rope model is expressed as a superposition of left-traveling wave, right-traveling wave and load wave;

[0008] S4: Solve for the expression of the load wave based on the form of the load;

[0009] S5: Derive the expression for the initial traveling wave based on the initial conditions of the axially moving rope model;

[0010] S6: Derive the traveling wave reflection equation based on the boundary conditions of the axially moving rope model, and derive the expressions for each traveling wave based on the traveling wave reflection equation.

[0011] S7: Based on the motion law of the left-traveling wave and the right-traveling wave in the axially moving rope model, the left-traveling wave and the right-traveling wave are superimposed with the load wave to obtain the lateral vibration response of the model;

[0012] S8: Obtain the optimal boundary damping based on the traveling wave reflection equation, and adjust the current in the coil of the magnetorheological damper based on the optimal boundary damping.

[0013] Optionally, in step S2, the equation of motion for the axially moving rope model is:

[0014] Optionally, in step S2, the damping boundary constraint equations are:

[0015]

[0016] Optionally, step S3 may include at least the following steps:

[0017] S31: The solution to the motion equation of the axially moving rope model is u(x,t);

[0018] S32: Represent u(x,t) as the superposition of a left-traveling wave, a right-traveling wave, and a load wave, as shown in the expression:

[0019] u(x,t)=F(xv r t)+G(x+v l t)+Q(t);

[0020] Optionally, in step S4, the expression for the load wave is:

[0021]

[0022] Optionally, step S5 may include at least the following steps:

[0023] S51: Initial condition expressions for establishing the axially moving rope model:

[0024]

[0025] S52: Substituting the general solution of the motion equations of the axially moving rope model into the initial condition expressions, we obtain the expressions for the initial left-traveling wave and the initial right-traveling wave:

[0026]

[0027] Optionally, step S6 includes at least the following steps:

[0028] S61: At the left damping boundary, based on the motion law of the traveling wave in the axially moving rope model and the boundary reflection law of the traveling wave at both ends of the axially moving rope model, the reflection period of the traveling wave is obtained. The expression for the reflection period is as follows:

[0029]

[0030] S62: Command Right-Wave Wave The initial left-traveling wave is Left-side damped boundary; based on the reflection and continuity laws of traveling waves in the nth period, the following expression can be obtained:

[0031] in

[0032] Solving this problem yields the following expression:

[0033]

[0034] S63: Based on the continuity of reflection, the expression for the continuity condition is obtained:

[0035] F2 n (-v r (n-1)T)=F1 n (-v r (n-1)T);

[0036] S64: In t b (t b =l0 / v l After that time, the right-bound wave is... Left-traveling wave Based on the reflection period of the traveling wave in the th period, the reflection law of the traveling wave, and the continuity law, the following expression can be obtained:

[0037]

[0038] Solving this problem yields the following expression:

[0039]

[0040] S65: Based on the continuity of reflection, the expression for the continuity condition is obtained:

[0041] F3 n (-v r (t b+(n-1)T))=F2 n (-v r (t b +(n-1)T));

[0042] S66: At the right-side damping boundary, the left traveling wave is... The initial right-traveling wave is F1 n Based on the reflection and continuity laws of traveling waves in the nth period, the following expression can be obtained:

[0043] in s = l0 + v l t;

[0044] Solving this problem yields the following expression:

[0045]

[0046] S67: Based on the continuity of reflection, the expression for the continuity condition is obtained:

[0047]

[0048] S68: In t a (t a =l0 / v r After a certain time, the left-bound wave is... Right-traveling wave Based on the reflection period of the traveling wave in the nth period, the reflection law of the traveling wave, and the continuity law, the following expression can be obtained:

[0049]

[0050] Optionally, in step S7, the axial vibration response u(x,t) of the axially moving rope model is obtained according to the expressions in steps S32, S62, S64, S66 and S67.

[0051] Optionally, in step S8, the right-side damping value is adjusted. Adjust the damping value on the left side

[0052] The beneficial effects of the method for solving the lateral vibration of an axially moving rope controlled by bilateral magnetorheological damping provided in this application are as follows:

[0053] The method for solving the lateral vibration of an axially moving rope controlled by dual-sided magnetorheological damping provided in this application solves the problem of difficulty in solving the forced vibration of an axially moving rope model using the traveling wave method. In related technologies, the traveling wave method is only applicable to solving the free vibration of a moving rope model. The method of this invention writes the traveling wave solution as a superposition of left and right traveling waves and load waves, thus solving the problem of difficulty in solving the vibration of an axially moving rope model using the traveling wave method.

[0054] Compared with commonly used numerical methods in engineering, the method of this invention has higher solution accuracy and better stability, and can obtain more accurate vibration response of axially moving rope equipment. It solves the problem of vibration response instability due to increased moving speed, improves computational efficiency, and greatly shortens the design cycle of vibration reduction for axially moving rope equipment.

[0055] Magnetorheological dampers are characterized by high reliability, continuous damping adjustment, low energy consumption, and relatively simple structure. The method of this invention involves installing magnetorheological dampers at both ends of an axially moving rope, forming damping boundaries at both ends. This adds no additional stiffness to the structure and does not alter its original dynamic characteristics. The power input to the dampers can be adjusted via a computer or other controller, thereby controlling the output force of the dampers and achieving superior control effects compared to passive dampers at a relatively low cost. This also extends the service life of axially moving rope equipment.

[0056] This invention provides a method applicable to damped boundary conditions, various speed conditions, and load conditions for moving rope equipment. The method yields accurate vibration displacement responses, meeting the needs of verifying the feasibility and effectiveness of various numerical calculation methods for lateral vibration of axially moving rope equipment. It allows adjustment of boundary conditions by changing the damping values ​​of the dual magnetorheological dampers, enabling rapid system stabilization and preventing resonance under load excitation. Attached Figure Description

[0057] To more clearly illustrate the technical solutions in the embodiments of this application, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0058] Figure 1 A flowchart of the method for solving the lateral vibration of an axially moving rope under dual-sided magnetorheological damping control provided in an embodiment of this application;

[0059] Figure 2 A plan view of the method for solving the lateral vibration of an axially moving rope under dual-sided magnetorheological damping control provided in the embodiments of this application. Detailed Implementation

[0060] To make the technical problems, technical solutions, and beneficial effects to be solved by this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and are not intended to limit the scope of this application.

[0061] It should be noted that when a component is referred to as being "fixed to" or "set on" another component, it can be directly on or indirectly on that other component. When a component is referred to as being "connected to" another component, it can be directly connected to or indirectly connected to that other component.

[0062] It should be understood that the terms "length", "width", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this application.

[0063] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this application, "multiple" means two or more, unless otherwise explicitly specified.

[0064] Based on this, the present invention provides a method for solving the transverse vibration of an axially moving rope controlled by dual-sided magnetorheological damping. This method greatly simplifies the solution process, has high solution accuracy, and good stability.

[0065] like Figure 1 and Figure 2 As shown, this application provides a method for solving the lateral vibration of an axially moving rope controlled by dual-sided magnetorheological damping, which includes at least the following steps:

[0066] S1: Magnetorheological dampers are installed at both ends of the axially moving rope model, and damping boundaries are formed at both ends of the axially moving rope model. Harmonic loads are evenly distributed on the axially moving rope model.

[0067] It should be noted that a fixed coordinate system is established with the axial movement direction of the axially moving rope model as the X direction and the lateral vibration direction as the U direction.

[0068] It should also be noted that, in this embodiment, the damping boundary closer to the origin is designated as the left damping boundary, and the boundary farther from the origin is designated as the right damping boundary.

[0069] S2: Obtain the equations of motion and damping boundary constraint equations for the axially moving rope model based on Hamilton's principle.

[0070] S3: The solution to the motion equation of the axially moving rope model is expressed as a superposition of left-traveling waves, right-traveling waves and load waves.

[0071] S4: Solve for the expression of the load wave based on the form of the load.

[0072] S5: Derive the expression for the initial traveling wave based on the initial conditions of the axially moving rope model.

[0073] S6: Derive the traveling wave reflection equation based on the boundary conditions of the axially moving rope model, and derive the expressions for each traveling wave based on the traveling wave reflection equation.

[0074] S7: Based on the motion law of the left-traveling wave and the right-traveling wave in the axially moving rope model, the left-traveling wave and the right-traveling wave are superimposed with the load wave to obtain the lateral vibration response of the model.

[0075] S8: Obtain the optimal boundary damping based on the traveling wave reflection equation, and adjust the current in the coil of the magnetorheological damper based on the optimal boundary damping.

[0076] This application provides a method for solving the lateral vibration of an axially moving rope controlled by dual-sided magnetorheological damping, which solves the problem of difficulty in solving the forced vibration of an axially moving rope model using the traveling wave method. In related technologies, the traveling wave method is only applicable to the free vibration of the moving rope model. This invention's method expresses the traveling wave method as a superposition of left and right traveling waves and load waves, thus solving the problem of difficulty in solving the vibration of an axially moving rope model using the traveling wave method. Compared with commonly used numerical methods in engineering, this invention's method has higher solution accuracy and better stability, and can obtain a more accurate vibration response of the axially moving rope equipment. It solves the problem of vibration response instability due to increased moving speed, improves computational efficiency, and greatly shortens the design cycle for vibration reduction of axially moving rope equipment. Magnetorheological dampers are characterized by high reliability, continuous damping adjustment, low energy consumption, and relatively simple structure. This invention's method installs magnetorheological dampers at both ends of the axially moving rope, forming damping boundaries at both ends, without adding additional stiffness to the structure and without changing the original dynamic characteristics of the structure. The power of the input damper can be adjusted by a computer or other controller, thereby controlling the output force of the damper and achieving a control effect superior to that of a passive damper at a relatively low cost; this can extend the service life of axially moving rope equipment. The method of this invention is applicable to damped boundary conditions, various speed conditions, and load conditions of moving rope equipment. The vibration displacement response obtained by this method is accurate, meeting the needs of verifying the feasibility and effectiveness of various numerical calculation methods for the lateral vibration of axially moving rope equipment. It can adjust the boundary conditions by changing the damping values ​​of the two-sided magnetorheological dampers, enabling the system to quickly stabilize and avoid resonance under load excitation.

[0077] In one embodiment of this application, in step S2, the equation of motion for the axially moving rope model is:

[0078]

[0079] The damped boundary constraint equations are as follows:

[0080]

[0081] It should be noted that in this embodiment, x represents the axial coordinate of the axially moving rope model, t represents time, v represents the axial velocity of the axially moving rope, and u represents the lateral displacement function of the axially moving rope model. u = u(x,t) represents the lateral displacement of the conveyor belt at x at time t. tt It is the second partial derivative of u with respect to t, u xx It is the second partial derivative of u with respect to x, u xt It is the second-order mixed partial derivative of u with respect to x and t. Let P represent the traveling wave velocity, ρ represent the tension in the axially moving rope model, ω represent the linear density of the axially moving rope model, and ω represent the angular frequency of the uniformly distributed harmonic load. Asin(ωt) characterizes the effect of the uniformly distributed harmonic load f(t), A represents the amplitude of the uniformly distributed harmonic load, and η represents the amplitude of the load. l For the left-side damping of the axially moving rope model, η r Damping on the right side of the rope model moving along the axis.

[0082] In one embodiment of this application, step S3 includes at least the following steps:

[0083] S31: The solution to the motion equation of the axially moving rope model is u(x,t).

[0084] S32: Represent u(x,t) as the superposition of a left-traveling wave, a right-traveling wave, and a load wave, as shown in the expression:

[0085] u(x,t)=F(xv r t)+G(x+v l t)+Q(t);

[0086] In one embodiment of this application, in step S4, the expression for the load wave is:

[0087]

[0088] It should be noted that in this embodiment, v r Let v be the velocity of the rightward-moving traveling wave in the axially moving rope model relative to a fixed coordinate system. l Let F(xv) be the velocity of the leftward-moving traveling wave in a rope model with axial movement relative to a fixed coordinate system. r t) represents a velocity of v r =c+v, a right-traveling wave, G(x+v) l t) represents a velocity of v l = The left-traveling wave of cv.

[0089] In one embodiment of this application, step S5 includes at least the following steps:

[0090] S51: Initial condition expressions for establishing the axially moving rope model:

[0091]

[0092] It should be noted that in this embodiment, the function φ(x) is the initial lateral displacement at different positions on the axially moving rope model in the fixed coordinate system, the function ψ(x) is the initial velocity at different positions on the axially moving rope model in the fixed coordinate system, and l0 is the distance between the two boundaries (that is, the distance between the left damping boundary and the right damping boundary).

[0093] S52: Substituting the general solution of the motion equations of the axially moving rope model into the initial condition expressions, we obtain the expressions for the initial left-traveling wave and the initial right-traveling wave:

[0094]

[0095] In one embodiment of this application, step S6 includes at least the following steps:

[0096] S61: At the left damping boundary, based on the motion law of the traveling wave in the axially moving rope model and the boundary reflection law of the traveling wave at both ends of the axially moving rope model, the reflection period of the traveling wave is obtained. The expression for the reflection period is as follows:

[0097]

[0098] S62: Command Right-Wave Wave The initial left-traveling wave is Left-side damping boundary: Based on the reflection and continuity laws of traveling waves in the nth period, the following expression can be obtained:

[0099] in r = -v r t;

[0100] Solving this problem yields the following expression:

[0101]

[0102] Specifically, the right traveling wave It is the initial left-traveling wave It is reflected from the damping boundary on the left.

[0103] S63: Based on the continuity of reflection, the expression for the continuity condition is obtained:

[0104] F2 n (-v r (n-1)T)=F1 n (-v r (n-1)T);

[0105] S64: In t b (t b =l0 / v l After that time, the right-bound wave is... Left-traveling wave The following expression can be obtained:

[0106]

[0107] Solving this problem yields the following expression:

[0108]

[0109] Specifically, the right traveling wave It is the initial left-traveling wave It is reflected from the damped boundary on the left. Where, t b The time it takes for the left-traveling wave to move from the right end to the left end of the axially moving rope model.

[0110] S65: Based on the continuity of reflection, the expression for the continuity condition is obtained:

[0111] F3 n (-v r (t b +(n-1)T))=F2 n (-v r (t b +(n-1)T));

[0112] S66: At the right-side damping boundary, the left traveling wave is... The initial right-traveling wave is F1 n Based on the reflection and continuity laws of traveling waves in the nth period, the following expression can be obtained:

[0113] in

[0114] Solving this problem yields the following expression:

[0115]

[0116] Specifically, the left traveling wave From the initial right-traveling wave F1 n It is reflected from the damping boundary on the right side.

[0117] S67: Based on the continuity of reflection, the expression for the continuity condition is obtained:

[0118]

[0119] S68: In t a (t a =l0 / v r After a certain time, the left-bound wave is... Right-traveling wave Based on the reflection period of the traveling wave in the nth period, the reflection law of the traveling wave, and the continuity law, the following expression can be obtained:

[0120]

[0121] In one embodiment of this application, in step S7, the axial vibration response u(x,t) of the axially moving rope model is obtained according to the expressions in steps S32, S62, S64, S66 and S67.

[0122] In one embodiment of this application, in step S8, the right-side damping value is adjusted. Adjust the damping value on the left side

[0123] Specifically, an external controller is used to control the magnetorheological damper. By changing the magnitude of the coil current in the magnetorheological damper, the damping value can be altered.

[0124] This configuration, utilizing the variable damping characteristic of the magnetorheological damper, can suppress the vibration of the system and prevent resonance caused by external excitation.

[0125] One or more embodiments in this application are intended to cover all such substitutions, modifications, and variations that fall within the broad scope of this application. Therefore, any omissions, modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of one or more embodiments in this application should be included within the protection scope of this application.

Claims

1. A method for solving the transverse vibration of an axially moving rope controlled by bilateral magnetorheological damping, characterized in that, At least the following steps are included: S1: Install magnetorheological dampers at both ends of the axially moving rope model, and form damping boundaries at both ends of the axially moving rope model, and uniformly distribute simple harmonic loads on the axially moving rope model. S2: Obtain the equations of motion and damping boundary constraint equations for the axially moving rope model based on Hamilton's principle; S3: The solution to the motion equation of the axially moving rope model is expressed as a superposition of left-traveling wave, right-traveling wave and load wave; S4: Solve for the expression of the load wave based on the form of the load; S5: Derive the expression for the initial traveling wave based on the initial conditions of the axially moving rope model; S6: Derive the traveling wave reflection equation based on the boundary conditions of the axially moving rope model, and derive the expressions for each traveling wave based on the traveling wave reflection equation. S7: Based on the motion law of the left-traveling wave and the right-traveling wave in the axially moving rope model, the left-traveling wave and the right-traveling wave are superimposed with the load wave to obtain the lateral vibration response of the model; S8: Obtain the optimal boundary damping based on the traveling wave reflection equation, and adjust the current in the coil of the magnetorheological damper based on the optimal boundary damping; In step S2, the equation of motion for the axially moving rope model is: ； The damped boundary constraint equations are as follows: ； in, The axial coordinates represent the axial movement of the rope model. Indicates time, This indicates the axial velocity of the rope as it moves axially. The lateral displacement function represents the axially moving rope model. Indicates that the conveyor belt is in hour Lateral displacement at that location, yes right The second-order partial derivative, yes right The second-order partial derivative, yes right and The second-order mixed partial derivative, Indicates the traveling wave velocity. This is represented as tension in an axially moving rope model. The linear density of the axially moving rope model. The angular frequency of a uniformly distributed harmonic load; Characterizing uniformly distributed harmonic loads The function, Let be the amplitude of the uniformly distributed harmonic load. For the left-side damping of the axially moving rope model, The right side of the rope model is resisted by the axis of movement; Step S3 includes at least the following steps: S31: Solve the motion equations of the axially moving rope model as follows ; S32: Will Represented as the superposition of left-traveling wave, right-traveling wave and load wave, the expression is: ； In step S4, the expression for the load wave is: ； in, Let be the velocity of the rightward-moving traveling wave in the axially moving rope model relative to a fixed coordinate system. Let be the velocity of the leftward-moving traveling wave in the axially moving rope model relative to a fixed coordinate system. Indicates speed as The right-traveling wave, Indicates speed as The left-facing wave.

2. The method for solving the transverse vibration of an axially moving rope controlled by double-sided magnetorheological damping as described in claim 1, characterized in that, Step S5 includes at least the following steps: S51: Initial condition expressions for establishing the axially moving rope model: ； S52: Substituting the general solution of the motion equations of the axially moving rope model into the initial condition expressions, we obtain the expressions for the initial left-traveling wave and the initial right-traveling wave: 。 3. The method for solving the transverse vibration of an axially moving rope controlled by double-sided magnetorheological damping as described in claim 2, characterized in that, Step S6 includes at least the following steps: S61: At the left damping boundary, based on the motion law of the traveling wave in the axially moving rope model and the boundary reflection law of the traveling wave at both ends of the axially moving rope model, the reflection period of the traveling wave is obtained. The expression for the reflection period is as follows: ； S62: Command Right-Wave Wave The initial left-traveling wave is According to the traveling wave in the first n The reflection and continuity patterns of the traveling wave within a period can be expressed as follows: ;in , ; Solving this problem yields the following expression: ； S63: Based on the continuity of reflection, the expression for the continuity condition is obtained: ； S64: In After a certain time, the right-bound wave is... Left-moving wave is This yields the following expression: ; Solving this problem yields the following expression: ； S65: Based on the continuity of reflection, the expression for the continuity condition is obtained: ； S66: At the right-side damping boundary, the left traveling wave is... The initial right-traveling wave is According to the traveling wave in the first n The reflection and continuity patterns of the traveling wave within a period can be expressed as follows: ;in , ; Solving this problem yields the following expression: ； S67: Based on the continuity of reflection, the expression for the continuity condition is obtained: ； S68: In After a certain time, the left-hand wave is ordered to... Right-traveling wave is According to the traveling wave in the first n The reflection and continuity patterns of the traveling wave within a period can be expressed as follows: 。 4. The method for solving the transverse vibration of an axially moving rope controlled by double-sided magnetorheological damping as described in claim 3, characterized in that, In step S7, the axial vibration response of the axially moving rope model is obtained according to the expressions in steps S32, S62, S64, S66, and S67. .

5. The method for solving the transverse vibration of an axially moving rope controlled by double-sided magnetorheological damping as described in claim 1, characterized in that, In step S8, the right-side damping value is adjusted. Adjust the damping value on the left side. .

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