CN113642127B - Multi-period calculation method for vibration and energy of axially moving rope equipment - Google Patents

Multi-period calculation method for vibration and energy of axially moving rope equipment Download PDF

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CN113642127B
CN113642127B CN202110930596.5A CN202110930596A CN113642127B CN 113642127 B CN113642127 B CN 113642127B CN 202110930596 A CN202110930596 A CN 202110930596A CN 113642127 B CN113642127 B CN 113642127B
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陈恩伟
贺钰腾
刘奕
任雪倩
陆益民
魏浩征
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Hefei University of Technology
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B66—HOISTING; LIFTING; HAULING
    • B66B—ELEVATORS; ESCALATORS OR MOVING WALKWAYS
    • B66B7/00—Other common features of elevators
    • B66B7/06—Arrangements of ropes or cables
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01H—MEASUREMENT OF MECHANICAL VIBRATIONS OR ULTRASONIC, SONIC OR INFRASONIC WAVES
    • G01H17/00—Measuring mechanical vibrations or ultrasonic, sonic or infrasonic waves, not provided for in the preceding groups
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

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Abstract

The invention discloses a multicycle calculation method of vibration and energy of an axial moving rope device, which comprises the following steps: 1. acquiring a traveling wave solution of a first period; 2. expanding the expression of each traveling wave solution to any period; 3. the line wave energy and its rate of change are calculated. The vibration damping method can accurately acquire the vibration displacement and the energy change of the rope equipment under the fixed-damping boundary condition, so that a proper damping value is selected to achieve the aim of damping the rope equipment, and the vibration damping method has the advantages of accurate calculation and high efficiency, and can greatly shorten the design period of the vibration damping design of the mobile rope equipment.

Description

Multi-period calculation method for vibration and energy of axially moving rope equipment
Technical Field
The invention belongs to the field of mechanical system dynamics modeling and vibration control, and particularly relates to a multicycle traveling wave solution method for axially moving rope vibration and energy.
Background
The axial moving rope type equipment can be applied to elevator ropes, sightseeing cable car ropeways, conveyor belts and even rope satellites, when the equipment is in overload operation, worn for a long time or excited by external factors, the moving rope easily generates transverse vibration, small-amplitude oscillation or noise is generated when the equipment is light, the use experience is affected, and safety accidents are caused when the equipment is heavy. Because of the wide application of numerical solutions, most of researches are to establish a finite element dynamics equation based on a Lagrange equation, select numerical algorithms such as a Newmark Bei Dafa, a Galerkin method, a Long Geku tower method and the like and a state space equation method to obtain the numerical solutions, and obtain the transverse vibration displacement of the axial moving rope device. However, when solving the partial differential equation problem of the complex boundary condition, the numerical solution has the problems of low solving precision, difficult solving process, poor stability and the like.
Compared with infinite and semi-infinite axial movement rope lock equipment, the finite axial movement rope lock equipment is more in line with engineering practical application. When the motion equation is solved by the traveling wave method, traveling waves do not occur or only reflect once, so that the solving process is simpler, but in the limited-length axial movement rope lock device with damping boundary types and the like, vibration energy is not always completely dissipated after one period, the rope still can continue vibrating, so that the reflection superposition and energy transmission of the traveling waves have multiple periodicity in the limited-length axial movement rope lock device, and the solving process is more complex due to the fact that the traveling waves reflect for multiple times at the boundary.
It can be seen that the displacement of the moving rope cannot be accurately obtained by the numerical solution obtained by the prior art, and the calculation efficiency is low, and the traveling wave solution related to the infinite and semi-infinite length axial moving rope lock device is not suitable for being applied to engineering practice.
Disclosure of Invention
The invention aims to solve the defects of the prior art, and provides a multicycle calculation method of vibration and energy of an axial moving rope device, so as to accurately acquire displacement of the axial moving rope lock device under a fixed-damping boundary condition, reveal energy change and acquire an optimal damping value, thereby realizing the effect of rope vibration reduction, achieving the purposes of high calculation efficiency and shortening the design period of vibration reduction design of the device.
The invention adopts the following technical scheme for solving the technical problems:
the invention relates to a multicycle calculation method of vibration and energy of an axial moving rope device, which is characterized in that: calculating multicycle transverse vibration and energy of the axial moving rope device under the boundary conditions of two ends of the axial moving rope device, wherein one end is a fixed boundary, and the other end is a damping boundary; the multicycle computing method comprises the following steps:
step 1: acquiring a traveling wave solution of a first period;
step 1.1: collecting parameters of the axially moving rope device, including: axial displacement velocity v, tension P of rope arrangement, damping coefficient eta at the nylon boundary, length l of rope 0 The linear density ρ of the rope;
establishing a fixed coordinate system by taking a fixed boundary as a coordinate origin, taking the axial moving direction of the axial moving rope equipment as an x direction and taking the transverse vibration direction as a u direction;
step 1.2: calculating the wave velocity c using formula (1):
step 1.3: intermediate parameters of the orderWherein v is r =c+v is the speed of the right-travelling wave in the rope relative to the fixed coordinate system; v l C-v is the velocity of the left-travelling wave in the rope relative to the fixed coordinate system;
step 1.4: when β=0, the optimal damping value η after one period is calculated using the equation opt ：
Step 1.5: the lateral vibration displacement u (x, t) is obtained by using:
u(x,t)＝F(x-v r t)+G(x+v l t) (3)
wherein F (·) represents a right traveling wave function and G (·) represents a left traveling wave function; x is a axial coordinate variable in the fixed coordinate system, and t is a time variable;
step 1.6: the expression of each traveling wave solution in the first period is obtained by using the formula:
in the formula, G 1 (. Cndot.) is a velocity v l Is a first left-shifted traveling wave; f (F) 1 (. Cndot.) is a velocity v r Is a first right traveling wave; g 2 (. Cndot.) is a velocity v l Is a second left-shifted traveling wave; f (F) 2 (. Cndot.) is a velocity v r Is a second right traveling wave; g 3 (. Cndot.) is a velocity v l Is a third leftwards traveling wave; f (F) 3 (. Cndot.) is a velocity v r Is a third right traveling wave; the function phi (·) is the initial lateral displacement of the rope arrangement at different positions in the fixed coordinate system; the function ψ (·) is the initial velocity of the different positions on the rope arrangement in the fixed coordinate system; a is a parameter;
when G 2 (l 0 )＝G 1 (l 0 )，G 3 (l 0 +v l t a )＝G 2 (l 0 +v l t a ) When the transverse vibration displacement u (x, t) satisfies the continuity condition;
step 2: expanding the expression of each traveling wave solution to any nth period by using a formula-formula, wherein n is more than or equal to 2:
in the formula, T is a cycle time, andF 1 n a first right traveling wave representing an nth period; f (F) 2 n A second right traveling wave representing an nth period; f (F) 3 n A third right traveling wave representing an nth period; g 1 n A first left traveling wave representing an nth period; g 2 n A second left traveling wave representing an nth period; g 3 n A third left traveling wave representing an nth period; f (F) 1 j A first right traveling wave representing a j-th period; f (F) 2 j A second right traveling wave representing a j-th period; g 1 j A first left traveling wave representing a j-th period; g 2 j A second left representing a j-th periodTraveling wave; g 3 j A second left traveling wave representing a j-th period;
step 3: calculating the wave energy and the change rate thereof;
step 3.1: dividing the energy change rate into the energy change rate of the system and the energy change rate of the control volume according to fluid mechanics;
establishing energy E of t-moment system sys (t) energy E with control volume CV The equation relationship between (t) is as follows:
wherein F 'represents the derivative of the right traveling wave function F (-), and G' represents the derivative of the left traveling wave function G (-);
establishing energy change rate of system by utilizing formula and formula respectivelyEnergy rate of change with control volume +.>Is represented by the expression:
and 3.2, obtaining the energy expression of each traveling wave by using the formula:
in the formula, x 1 ，x 2 Is two coordinates in the axial moving direction of the fixed coordinate system, and 0<x 1 <x 2 <l 0 ；F 1 1′ Representing a first period of a first right traveling wave F 1 1 Derivative of (-), G 1 1′ Representing a first period of a first left traveling wave G 1 1 Derivative of (-);the first right traveling wave F is the nth period at the t moment 1 n In the coordinates x 1 To the coordinate x 2 Energy on->The second right traveling wave F is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Energy on->The first right traveling wave F is the nth period at the t moment 3 n In the coordinates x 1 To the coordinate x 2 Energy on->First left traveling wave G of nth period at t time 1 n In the coordinates x 1 To the coordinate x 2 Energy on; />The second left traveling wave G is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Energy on; />Third left traveling wave G in nth period at t time 3 n In the coordinates x 1 To the coordinate x 2 Energy on;
obtaining the coordinate x by using the formula 1 To the coordinate x 2 The energy change rate of each traveling wave:
in the formula (I) and the formula (II),the first right traveling wave F is the nth period at the t moment 1 n In the coordinates x 1 To the coordinate x 2 Rate of energy change in>The second right traveling wave F is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Rate of energy change in>The first right traveling wave F is the nth period at the t moment 3 n In the coordinates x 1 To the coordinate x 2 The rate of change of the energy in the above,first left traveling wave G of nth period at t time 1 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;the second left traveling wave G is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;third left traveling wave G in nth period at t time 3 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;
step 3.3: obtaining energy and the change rate of the energy according to time intervals;
defining two time parameters
When (n-1) T<t<(n-1)T+t a When the energy E of the rope equipment at the time t under the nth period is obtained by using the formula and the formula respectively n (t) and controlling energy variation of volumeRate of
When (n-1) T+t a <t<(n-1)T+t b At the time, the energy E of the rope equipment in the nth period of the moment t is obtained by using the formula and the formula respectively n (t) and controlling the rate of energy change of the volume
When (n-1) T+t b <t<At nT, the energy E of the rope device at the nth period at time t is obtained by using the formulas (32) and (33) n (t) and controlling the rate of energy change of the volume
Compared with the prior art, the invention has the beneficial effects that:
1. the method solves the problem that the vibration of the limited-length mobile rope equipment is difficult to solve by using the traveling wave method, has wide application range, and is suitable for various complex boundaries, such as a spring boundary, a free boundary, a mass boundary and the like.
2. The method expands the single-period traveling wave solution to multiple periods, solves the problem of solving the transverse vibration displacement response of the axial moving rope equipment at any moment, and has simple process and easy understanding.
3. Compared with a numerical method which is frequently used, the method has the advantages of accuracy, good stability, high calculation efficiency and the like, and the design period of vibration reduction design of the mobile rope equipment is greatly shortened.
4. The method of the invention displays the energy and the change rate thereof in a traveling wave mode through quantitative calculation, and reveals the physical significance of the system and the control volume energy change rate, wherein the energy change rate of the system corresponds to the boundary working power, and the energy change rate of the system is the superposition of the boundary working power and the energy change caused by the inflow and outflow of the rope into the boundary.
Drawings
FIG. 1 is a fixed-damped boundary moving rope model;
FIG. 2a is a schematic diagram of the superposition of traveling wave reflections at time 0 in the method of the present invention;
FIG. 2b shows a first stage [0, t ] of the process according to the invention a ]A traveling wave reflection superposition schematic diagram;
FIG. 2c shows t in the method of the invention a A schematic diagram of the reflection superposition of the traveling wave at the moment;
FIG. 2d shows a second stage [ t ] of the method of the invention a ,t b ]A traveling wave reflection superposition schematic diagram;
FIG. 2e shows t in the method of the present invention b A schematic diagram of the reflection superposition of the traveling wave at the moment;
FIG. 2f shows a third stage [ t ] in the process according to the invention a ,T 0 ]A traveling wave reflection superposition schematic diagram.
Detailed Description
In the embodiment, the multicycle calculation method of the vibration and the energy of the axial moving rope device solves the problem of the boundary value of the rope with the limited length by using the darebel formula, expands the traveling wave solution to multicycle by recursion according to the obtained traveling wave boundary reflection relation and the continuity condition, is suitable for calculating the vibration and the energy multicycle under the fixed-damping boundary condition of the moving rope device under any cycle, and can also test the feasibility and the effectiveness of the transverse vibration numerical solution of the axial moving rope. Specifically, as shown in fig. 1, a fixed-damped boundary moving rope model is provided, one end is a typical fixed boundary, and the other end is a damped boundary, and the method is carried out according to the following steps:
step 1: as shown in fig. 1, a fixed coordinate system is established by taking a fixed boundary as a coordinate origin, taking the axial moving direction of the axial moving rope device as an x direction and taking the transverse vibration direction as a u direction;
deducing a motion control equation and boundary conditions of the motion control equation according to the modified Hamiltonian principle, wherein the motion control equation and the boundary conditions are as shown in the formula:
u tt +2vu xt +(v 2 -c 2 )u xx ＝0 (1)
x is a axial coordinate variable in a fixed coordinate system, and t is a time variable; p represents the tension of the rope arrangement; v denotes the axial movement speed of the rope arrangement,representing the velocity of the traveling wave, ρ representing the linear density of the rope; the rope lateral vibration displacement u is a function of the coordinate variable x and the time variable t, u=u (x, t); u (u) tt Is the second partial derivative of u to t; u (u) xx Is the second partial derivative of u to x; u (u) xt Is the first partial derivative of u for x and t, respectively; for both end boundaries of the axially moving rope device, a left end boundary is defined as x=0 and a right end boundary is defined as x=l 0 ，l 0 Is the rope length; u (0, t) represents the vibration displacement of the rope arrangement at x=0; η represents a damping coefficient; u (u) t (l 0 T) is shown inx＝l 0 First partial derivative of u to t; u (u) x (l 0 T) is expressed in x=l 0 First order partial derivative of u to x;
step 2: the lateral vibration displacement u (x, t) is obtained by using:
u(x,t)＝F(x-v r t)+G(x+v l t) (3)
in the formula, v r =c+v is the speed of the right-travelling wave in the rope relative to the fixed coordinate system; v l C-v is the velocity of the left-travelling wave in the rope relative to the fixed coordinate system; f (·) represents a right traveling wave function, and G (·) represents a left traveling wave function;
knowing t=0, the initial conditions of the movement of the rope device are as follows:
in formula (4), u (x, 0) represents an initial lateral displacement response of the rope arrangement at time t=0; u (u) t (x, 0) represents the initial lateral velocity response of the rope arrangement at time t=0; the function phi (·) is the initial lateral displacement of the rope arrangement at different positions in the fixed coordinate system; the function ψ (·) is the initial velocity of the different positions on the rope arrangement in the fixed coordinate system;
bringing the general solution into a boundary condition to obtain the boundary reflection relation of the traveling wave as the following formula and expression:
F(-v r t)＝-G(v l t) (5)
G′(l 0 +v l t)＝βF′(l 0 -v r t) (6)
intermediate parameters of the order
By changing the element, let r= -v r t,s＝l 0 +v l t, obtainable by the formula:
as shown in fig. 2 a-2 f, according to the motion law of the traveling wave in the rope device and the reflection superposition law of the traveling wave at the boundaries of two ends of the rope device, the vibration period T is divided into three phases and three key moments, wherein the three phases refer to a first phase [0, T respectively a ]Second stage [ t ] a ,t b ]And third stage [ t ] b ,T]Wherein 0 is<t a <t b <T, the three key moments refer to moment t=0, moment t=t, respectively a And time t=t b ；
The traveling wave reflection period T can be obtained by the formula:
time node t dividing one cycle into three phases a And t b The method can be calculated by the formula:
bringing the general solution into the initial condition to obtain a traveling wave F 1 And G 1 Is represented by the expression:
wherein a is an arbitrary constant.
For pairs from x to l respectively 0 And think of I 0 +v l t a Integrating to obtain G 2 And G 3 Is represented by the expression:
similarly, F 2 And F 3 The expression of (2) can be obtained by the formula:
finally, each traveling wave expression in the first period is:
in the formula, G is as shown in FIGS. 2a to 2f 1 (. Cndot.) is a velocity v l Is a first left-shifted traveling wave; f (F) 1 (. Cndot.) is a velocity v r Is a first right traveling wave; g 2 (. Cndot.) is a velocity v l Is a second left-shifted traveling wave; f (F) 2 (. Cndot.) is a velocity v r Is a second right traveling wave; g 3 (. Cndot.) is a velocity v l Is a third leftwards traveling wave; f (F) 3 (. Cndot.) is a velocity v r Is a third right traveling wave; as shown in FIGS. 2a and 2c, when G 2 (l 0 )＝G 1 (l 0 )，G 3 (l 0 +v l t a )＝G 2 (l 0 +v l t a ) When the transverse vibration displacement u (x, t) satisfies the continuity condition;
the same principle expands the traveling wave solution to any nth week:
in the formula, the first period, F 1 n A first right traveling wave representing an nth period; f (F) 2 n A second right traveling wave representing an nth period; f (F) 3 n A third right traveling wave representing an nth period; g 1 n A first left traveling wave representing an nth period; g 2 n A second left traveling wave representing an nth period; g 3 n A third left traveling wave representing an nth period;
carrying the formula into the formula to obtain an initial traveling wave (G) 1 n ,F 1 n ) And end wave (G) 3 n ,F 3 n ) Is the relation of:
the final traveling wave of the nth cycle is equal to the initial traveling wave of the (n+1) th cycle, and the continuity of the traveling wave among the cycles is easy to obtain:
F 3 n (x-nv r T)＝F 1 n+1 (x-nv r T)or F 3 n ＝F 1 n+1 (28)
G 3 n (x+nv l T)＝G 1 n+1 (x+nv l T)or G 3 n ＝G 1 n+1 (29)
the initial traveling wave of the nth period can be obtained through recursion:
analogically to the first period, based on the boundary reflection relationship, we can obtain:
in the formula, F 1 j A first right traveling wave representing a j-th period; f (F) 2 j A second right traveling wave representing a j-th period; g 1 j A first left traveling wave representing a j-th period; g 2 j A second left traveling wave representing a j-th period; g 3 j A second left traveling wave representing a j-th period;
step 3: obtaining an optimal damping value after one period;
calculating an optimal damping value when β=0:
step 4: calculating the wave energy and the change rate thereof;
the energy density of the system is:
according to the formula, the lateral displacement u is derived by respectively calculating the bias of x and t:
wherein F 'represents the derivative of the right traveling wave function F (-), and G' represents the derivative of the left traveling wave function G (-); u (u) t Representing the first partial derivative of u with respect to t; u (u) x Representing the first partial derivative of u with respect to x;
the formula of the formula is available:
establishing energy E of t-moment system sys (t) energy E with control volume CV The equation relationship between (t) is as follows:
the method is characterized by comprising the following steps of:
E CV (t) deriving t to obtain the rate of change of energy of the control volume
Since the energy change rate of the system is instantaneously defined, it is necessary to determine the energy change rate of the system according to the Reynolds transmission theorem/>
Step 4.1: calculating the derivatives (G 'and F') of each travelling wave function and applying to the energy and its rate of change;
for x-v r t is derived:
wherein F is 1 1′ Representing a first period of a first right traveling wave F 1 1 Derivative of (-);
carrying in to obtain F 1 n The energy expression of the traveling wave:
wherein x is 1 ，x 2 Is two coordinates in the axial moving direction of the fixed coordinate system, and 0<x 1 <x 2 <l 0 ，x 1 ,x 2 The value of (2) is determined according to three phases corresponding to the period;the first right traveling wave F is the nth period at the t moment 1 n In the coordinates x 1 To the coordinate x 2 Energy on the surface.
Other traveling wave energies may be obtained by similar methods:
in the formula, G 1 1 ' first left traveling wave G representing first period 1 1 Derivative of (-);the second right traveling wave F is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Energy on->The first right traveling wave F is the nth period at the t moment 3 n In the coordinates x 1 To the coordinate x 2 Energy on->First left traveling wave G of nth period at t time 1 n In the coordinates x 1 To the coordinate x 2 Energy on; />The second left traveling wave G is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Energy on;third left traveling wave G in nth period at t time 3 n In the coordinates x 1 To the coordinate x 2 Energy on;
the derivation of formula-formula can be obtained:
in the formula (I), in the formula (II),the first right traveling wave F is the nth period at the t moment 1 n In the coordinates x 1 To the coordinate x 2 Rate of energy change in>The second right traveling wave F is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Rate of energy change in>The first right traveling wave F is the nth period at the t moment 3 n In the coordinates x 1 To the coordinate x 2 The rate of change of the energy in the above,first left traveling wave G of nth period at t time 1 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;the second left traveling wave G is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;third left traveling wave G in nth period at t time 3 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;
step 4.2: solving energy and change rate thereof in time intervals:
in the nth cycle, FIG. 2b is simulated when (n-1) T<t<(n-1)T+t a When, as shown in FIG. 2b, this stage includes F 1 n ,F 2 n ,G 1 n ,G 2 n Four travelling waves, so energy E of rope arrangement at time t n (t) and controlling the rate of energy change of the volumeIs of the formula: />
In the nth cycle, FIG. 2d is simulated when (n-1) T+t a <t<(n-1)T+t b When, as in FIG. 2d, this stage includes F 2 n ,G 1 n ,G 3 n Three traveling waves, so energy E of rope arrangement at time t n (t) and controlling the rate of energy change of the volumeIs of the formula:
in the nth period, the analog diagram2f, when (n-1) T+t b <t<At nT, this stage includes F as in FIG. 2F 2 n ,F 3 n ,G 2 n ,G 3 n Four travelling waves, so energy E of rope arrangement at time t n (t) and controlling the rate of energy change of the volumeIs of the formula and formula:
/>

Claims (1)

1. a multicycle calculation method of vibration and energy of an axial moving rope device is characterized in that: calculating multicycle transverse vibration and energy of the axial moving rope device under the boundary conditions of two ends of the axial moving rope device, wherein one end is a fixed boundary, and the other end is a damping boundary; the multicycle computing method comprises the following steps:
step 1: acquiring a traveling wave solution of a first period;
step 1.1: collecting parameters of the axially moving rope device, including: axial displacement velocity v, tension P of rope arrangement, damping coefficient eta at the nylon boundary, length l of rope 0 The linear density ρ of the rope;
establishing a fixed coordinate system by taking a fixed boundary as a coordinate origin, taking the axial moving direction of the axial moving rope equipment as an x direction and taking the transverse vibration direction as a u direction;
step 1.2: calculating the wave velocity c using formula (1):
step 1.3: intermediate parameters of the orderWherein v is r =c+v is the speed of the right-travelling wave in the rope relative to the fixed coordinate system; v l C-v is the velocity of the left-travelling wave in the rope relative to the fixed coordinate system;
step 1.4: when β=0, the optimal damping value η after one cycle is calculated using the equation (2) opt ：
Step 1.5: the lateral vibration displacement u (x, t) is obtained using equation (3):
u(x,t)＝F(x-v r t)+G(x+v l t) (3)
in the formula (3), F (·) represents a right traveling wave function, and G (·) represents a left traveling wave function; x is a axial coordinate variable in the fixed coordinate system, and t is a time variable;
step 1.6: using the formulas (4) - (9), the expression of each traveling wave solution in the first period is obtained:
in the formulae (4) - (9), G 1 (. Cndot.) is a velocity v l Is a first left-shifted traveling wave; f (F) 1 (. Cndot.) is a velocity v r Is a first right traveling wave; g 2 (. Cndot.) is a velocity v l Is a second left-shifted traveling wave; f (F) 2 (. Cndot.) is a velocity v r Is a second right traveling wave; g 3 (. Cndot.) is a velocity v l Is a third leftwards traveling wave; f (F) 3 (. Cndot.) is a velocity v r Is a third right traveling wave; the function phi (·) is the initial lateral displacement of the rope arrangement at different positions in the fixed coordinate system; the function ψ (·) is the initial velocity of the different positions on the rope arrangement in the fixed coordinate system; a is a parameter;
when G 2 (l 0 )＝G 1 (l 0 )，G 3 (l 0 +v l t a )＝G 2 (l 0 +v l t a ) When the transverse vibration displacement u (x, t) satisfies the continuity condition;
step 2: expanding the expression of each traveling wave solution to any nth period by using the formulas (10) - (15), wherein n is more than or equal to 2:
in the formulae (10) to (15), T is a cycle time, andF 1 n a first right traveling wave representing an nth period; f (F) 2 n A second right traveling wave representing an nth period; f (F) 3 n A third right traveling wave representing an nth period; g 1 n A first left traveling wave representing an nth period; g 2 n A second left traveling wave representing an nth period; g 3 n A third left traveling wave representing an nth period; f (F) 1 j A first right traveling wave representing a j-th period; f (F) 2 j A second right traveling wave representing a j-th period; g 1 j A first left traveling wave representing a j-th period; g 2 j A second left traveling wave representing a j-th period; g 3 j A second left traveling wave representing a j-th period;
step 3: calculating the wave energy and the change rate thereof;
step 3.1: dividing the energy change rate into the energy change rate of the system and the energy change rate of the control volume according to fluid mechanics;
establishing energy E of t-moment system sys (t) energy E with control volume CV The equation relationship between (t) is shown in equation (16):
in the formula (16), F 'represents the derivative of the right traveling wave function F (·) and G' represents the derivative of the left traveling wave function G (·);
establishing the energy change rate of the system by using the formula (17) and the formula (18), respectivelyEnergy rate of change with control volume +.>Is represented by the expression:
step 3.2, obtaining the energy expression of each traveling wave by using the formulas (19) - (24):
in the formulae (19) - (24), x 1 ，x 2 Is two coordinates in the axial moving direction of the fixed coordinate system, and 0<x 1 <x 2 <l 0 ；F 1 1′ Representing a first period of a first right traveling wave F 1 1 Derivative of (-), G 1 1′ Representing a first period of a first left traveling wave G 1 1 Derivative of (-);the first right traveling wave F is the nth period at the t moment 1 n In the coordinates x 1 To the coordinate x 2 Energy on->The second right traveling wave F is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Energy on->The first right traveling wave F is the nth period at the t moment 3 n In the coordinates x 1 To the coordinate x 2 Energy on->First left traveling wave G of nth period at t time 1 n In the coordinates x 1 To the coordinate x 2 Energy on; />The second left traveling wave G is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Energy on; />Third left traveling wave G in nth period at t time 3 n In the coordinates x 1 To the coordinate x 2 Energy on;
obtaining the coordinate x by using the formulas (25) - (30) 1 To the coordinate x 2 The energy change rate of each traveling wave:
in the formulas (25) and (30),the first right traveling wave F is the nth period at the t moment 1 n In the coordinates x 1 To the coordinate x 2 Rate of energy change in>The second right traveling wave F is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 Rate of energy change in>The first right traveling wave F is the nth period at the t moment 3 n In the coordinates x 1 To the coordinate x 2 The rate of change of the energy in the above,first left traveling wave G of nth period at t time 1 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;the second left traveling wave G is the nth period at the t moment 2 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;third left traveling wave G in nth period at t time 3 n In the coordinates x 1 To the coordinate x 2 The rate of change of energy in the above;
step 3.3: obtaining energy and the change rate of the energy according to time intervals;
defining two time parameters
When (n-1) T<t<(n-1)T+t a In this case, the energy E of the rope device at time t in the nth cycle is obtained by using the equations (31) and (32), respectively n (t) and controlling the rate of energy change of the volume
When (n-1) T+t a <t<(n-1)T+t b In this case, the energy E of the rope device at the nth cycle at time t is obtained by using the equations (33) and (34) n (t) and controlling the rate of energy change of the volume
When (n-1) T+t b <t<At nT, the energy E of the rope device at the nth period at time t is obtained by using the formulas (35) and (36) n (t) and controlling the rate of energy change of the volume
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