CN112347576B - Method for calculating vibration energy of axially moving rope device under mixed boundary condition - Google Patents
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Abstract
The invention discloses a vibration energy calculating method of an axial moving rope device under a mixed boundary condition, which is to obtain a motion equation of the axial moving rope device according to a Hamiltonian principle, and represent displacement response of the motion equation as superposition of two traveling waves; deducing expressions of two initial traveling waves according to initial motion conditions of the axially moving rope equipment, obtaining constraint equations of a mixed boundary, and then combining the constraint equations of the mixed boundary with the motion equations to obtain reflection wave response equations of all stages; and superposing the incident wave and the reflected wave of the mobile rope equipment to obtain transverse displacement vibration, and finally, characterizing the energy expression of each traveling wave through each traveling wave expression in a vibration period, thereby calculating the vibration energy of the rope equipment. The method is suitable for energy calculation of vibration displacement response obtained by the mixed boundary condition and various speed working conditions of the mobile rope equipment, and can analyze energy change of the rope equipment under various working conditions.
Description
Technical Field
The invention belongs to the field of mechanical system dynamics modeling and vibration control, and particularly relates to a method for calculating transverse vibration energy of axial movement rope equipment under a mixed boundary condition.
Background
The axial moving rope device has the advantages of high operation efficiency, strong self-adaption, large bearing capacity, simple structure, flexibility, controllability and the like, and has very important application values in a plurality of engineering fields, such as a rope satellite cable, a power transmission belt, an elevator cable, a passenger-cargo ropeway and the like. Noise and vibration accompany the operation of these devices, and in particular lateral vibrations have a great influence on the functioning and safety of these devices. The problem of lateral vibrations of axially moving rope arrangements is a challenging task that has been studied for many years and has remained of great interest to date. The traditional research technology is to solve partial differential motion equations established based on Hamiltonian principle and finite element dynamics equations established based on Lagrange equation by using numerical calculation methods such as Galerkin method, dragon-Gerdostat method, newmark method and time-varying state space equation to obtain transverse vibration response of the axial moving rope equipment. However, the conventional method has problems of complex solving process, low solving precision and poor stability when solving the transverse vibration problem of the mobile rope equipment under complex mixed boundary conditions. Moreover, when the speed of the axially moving rope device is high and approaches or reaches a critical speed, the vibration displacement amplitude of the device is abnormally increased, and the error is increased.
The darebel principle indicates that the transverse vibration of an infinitely long uniform string can be expressed as superposition of two traveling waves in opposite directions, which lays a theoretical foundation for obtaining the transverse vibration of the axially movable rope equipment by utilizing the wave superposition theory. However, the darebel principle is directed to vibration and energy variation characteristics of a traveling wave reflected once at different boundaries of a semi-infinitely long chord line. In the engineering application of the actual axial moving rope device, under the condition of a complex mixed boundary, traveling waves in different directions can be reflected for multiple times at the boundary of the limited-length moving rope device and overlapped with the incident wave to form the transverse vibration of the moving rope device, so that the problem of accurately acquiring the transverse vibration formed by repeated reflection and overlapping of the traveling waves in the fixed-length moving rope device under the constraint condition of the mixed boundary cannot be solved by using the method of the Darby principle.
Disclosure of Invention
The invention aims to avoid the defects of the prior art, and provides a vibration energy calculating method of axial moving rope equipment under a mixed boundary condition, so as to solve the problems of low solving precision and poor stability of transverse vibration displacement response of the axial moving rope equipment and the problem of instability of the vibration displacement response due to the increase of the moving speed; the method is suitable for energy calculation of vibration displacement response obtained by the mixed boundary condition and various speed working conditions of the mobile rope equipment, and can analyze energy change of the rope equipment under various working conditions.
The invention adopts the following technical scheme for solving the technical problems:
the invention relates to a vibration energy calculating method of an axial moving rope device under a mixed boundary condition, wherein the mixed boundary condition is that one end boundary of two end boundaries of the axial moving rope device is an atypical boundary, the other end boundary is a typical boundary, the atypical boundary is taken as a coordinate origin, the axial moving direction of the axial rope device is taken as an x direction, and the transverse vibration direction is taken as a u direction, so as to establish a fixed coordinate system; the method is characterized in that: the vibration energy calculating method comprises the following steps:
step 1, according to the Hamiltonian principle, obtaining a motion equation of the axial moving rope device by using a formula (1), and expressing a solution u (x, t) of the motion equation of the axial moving rope device as a superposition of two traveling waves shown in a formula (2), wherein the superposition of the two traveling waves refers to the superposition of a left traveling wave and a right traveling wave;
u tt +2vu xt -(c 2 -v 2 )u xx ＝0 (1)
in the formula (1), u tt Is the second partial derivative of the transverse vibration displacement u of the rope with respect to time t; v denotes the axially moving rope deviceIs fixed, the axial movement speed of the device; u (u) xt The first partial derivative of the transverse vibration displacement u of the rope on the axial coordinate x of the axial moving rope device in a fixed coordinate system and on the time t respectively; c represents the velocity of the traveling wave; u (u) xx Is the second partial derivative of the axial coordinate x of the rope transverse vibration displacement u to the axially moving rope device;
u(x,t)＝F(x-v r t)+G(x+v l t) (2)
in the formula (2), v r The velocity of the right traveling wave relative to the fixed coordinate system; v l The velocity of the traveling wave is shifted to the left relative to a fixed coordinate system; f (x-v) r t) represents a velocity v r Right-shifting traveling wave of (a); g (x+v) l t) represents a velocity v l Is a left-shifted traveling wave;
step 2, let the atypical boundary at x=0 be the spring-damper boundary, where x=l 0 The typical boundary at the point is a fixed boundary, so that a mixed boundary constraint equation set formula is obtained by using the formula (3);
in the formula (3), u (0, t) represents a vibration displacement of the axially moving rope device at x=0; u (u) t (l 0 T) is expressed in x=l 0 First partial derivative of the transverse vibration displacement u of the rope with respect to time t; u (u) x (l 0 T) is expressed in x=l 0 A first partial derivative of the transverse vibration displacement u of the rope to the axial coordinate x; η represents an atypical boundary damping coefficient of the axially moving rope arrangement; ρ represents the linear density of the rope; p represents the tension of the axially moving rope arrangement; k represents the stiffness coefficient of the spring;
and 3, establishing a relational expression of the two traveling waves by using the expression (4) and the expression (5):
G(l 0 +v l t)＝-F(l 0 -v r t) (4)
in the formula (5), F 'and G' respectively represent derivatives of two traveling waves with respect to time t;
two intermediate variables α, β are obtained using formula (6):
obtaining a simplified traveling wave relational expression by using the formula (7):
in the formula (7), s represents the displacement of the right traveling wave, and s=v r t；
Obtaining a general traveling wave relation expression by using the formula (8):
in the formula (8), F (x) represents the traveling wave relational expression of the right traveling wave relative to the axial coordinate x; e, e -αs Representing an integral factor;
step 4, giving an initial condition for movement of the axially moving rope device by using the formula (9):
in equation (9), the function phi (x) is the initial lateral displacement of the different locations on the axially moving rope device in a fixed coordinate system; the function ψ (x) is the initial velocity of the different positions on the axially moving rope arrangement in the fixed coordinate system;
according to the motion initial conditions, an initial expression of two traveling waves is obtained by using the formula (10):
in the formula (10), ζ is an integral variable; g (x) represents the traveling wave relational expression of the left traveling wave relative to the axial coordinate x; c is an integration constant;
step 5, determining the vibration period T of the axial moving rope device by using the formula (11) according to the motion rule of the two traveling waves in the axial moving rope device and the reflection superposition rule of the two traveling waves at the boundaries of the two ends of the axial moving rope device 0 ；
Step 6, combining the motion equation, motion initiation condition and solution u (x, T) of the motion equation of the axial moving rope device with the mixed boundary constraint equation set, respectively, and for the vibration period T 0 The traveling waves at different stages are overlapped, so that a transverse vibration displacement response type is obtained;
in travelling wave displacement from axial coordinate x 1 To the axial coordinate x 2 The left traveling wave G is divided into a first left traveling wave G 1 And a second leftwards traveling wave G 2 The right traveling wave F is divided into a first right traveling wave F 1 And a second right traveling wave F 2 The method comprises the steps of carrying out a first treatment on the surface of the Wherein the second left traveling wave G 2 Is the first right traveling wave F 1 At the right boundary x=l 0 Reflected wave at the position, the second right-shift traveling wave F 2 Is the first left traveling wave G 1 Reflected waves at the left boundary x=0;
obtaining a first right traveling wave F by using (12) 1 And a first left traveling wave G 1 Is represented by the expression:
in the formula (12), F 1 (x-v r t) represents a velocity v r Is a first right traveling wave; g 1 (x+v l t) is the velocity v l Is shifted left by the first left-shift traveling wave; c is an integration constant;
the continuity condition of two traveling waves on the atypical boundary at x=0 is constructed using equation (13):
according to the formulas (3) and (13), the second leftwards traveling wave G is obtained by the formulas (14) and (15), respectively 2 And a second right traveling wave F 2 ：
In the formula (14) and the formula (15), G 2 (x+v l t) is the velocity v l Is a second left-shifted traveling wave; f (F) 2 (x-v r t) is the velocity v r Is a second right traveling wave;
step 7, according to the first phase [0, t a ]The travelling wave expression in (a) is obtained in the first stage [0, t a ]A mid-range traveling wave energy expression to calculate a total vibration energy of the axially moving rope apparatus:
step 7.1 obtaining the vibration energy of the second right traveling wave F2 by using the method (16)
Step 7.2 obtaining the vibration energy of the first right traveling wave F1 by using the method (17)
Step 7.3, first left-shift traveling wave G using (18) 1 Vibration energy of (a)
Step 7.4 obtaining a second left traveling wave G by using the method (19) 2 Vibration energy of (a)
Step 7.5, obtaining total vibration energy E (t) by using the formula (20):
in the formula (20), E k (t) is the potential energy of the spring, an
Compared with the prior art, the invention has the beneficial effects that:
1. the method has high solving precision and good stability. Compared with a numerical solution method, the method provided by the invention has the advantages that the accurate analytical expression of the vibration displacement is obtained by the analytical solution, the method has the characteristics of high solving precision and good stability of the transverse vibration response of the axial moving rope equipment, and the problem of instability of the vibration response due to the increase of the moving speed can be solved.
2. The method solves the problem of accurately acquiring transverse vibration formed by repeated reflection and superposition of the traveling wave in the fixed-length mobile rope equipment under the constraint condition of the mixed boundary, and has simple process.
3. The method is suitable for the mixed boundary conditions and various speed working conditions of the mobile rope equipment. The method can adjust the boundary conditions according to different boundaries, and is suitable for various mixed boundary constraint conditions. The obtained vibration displacement response is accurate, and the energy expression of each traveling wave is represented through each traveling wave expression in three stages of a vibration period, so that the total vibration energy of the rope equipment can be calculated. The method is suitable for calculating the energy of vibration displacement response obtained by the mixed boundary condition of the mobile rope equipment and the various speed working conditions, and the method for analyzing the energy change of the rope equipment under the various working conditions can meet the requirements of feasibility and effectiveness of various numerical calculation methods for checking the transverse vibration of the axial mobile rope equipment.
Drawings
FIG. 1 is a spring-damper-fixed hybrid boundary model in accordance with the present invention;
FIG. 1a shows a first stage [0, t ] of the invention a ]A traveling wave reflection superposition schematic diagram;
FIG. 1b shows a second stage [ t ] of the present invention a ，t b ]A traveling wave reflection superposition schematic diagram;
FIG. 1c shows a first stage [ t ] of the present invention b ，T 0 ]A traveling wave reflection superposition schematic diagram.
Detailed Description
In this embodiment, a method of vibration energy calculation for an axially moving rope device under mixed boundary conditions. As shown in fig. 1, the mixed boundary condition refers to that one end boundary of two end boundaries of the axial moving rope device is an atypical boundary, the other end boundary is a typical boundary, the atypical boundary is taken as a coordinate origin, the axial moving direction of the axial rope device is taken as an x direction, and the transverse vibration direction is taken as a u direction, and a fixed coordinate system is established; the vibration energy calculating method comprises the following steps:
step 1, according to the Hamiltonian principle, obtaining a motion equation of the axial moving rope device by using a formula (1), and expressing a solution u (x, t) of the motion equation of the axial moving rope device as superposition of two traveling waves shown in a formula (2), wherein the superposition of the two traveling waves refers to superposition of a left traveling wave and a right traveling wave;
u tt +2vu xt -(c 2 -v 2 )u xx ＝0 (1)
in the formula (1), u tt Is the second partial derivative of the transverse vibration displacement u of the rope with respect to time t; v denotes the axial movement speed of the axially moving rope device; u (u) xt The first partial derivative of the transverse vibration displacement u of the rope on the axial coordinate x and the time t of the axial moving rope equipment in a fixed coordinate system respectively; c represents the velocity of the traveling wave; u (u) xx Is the second partial derivative of the axial coordinate x of the rope transverse vibration displacement u to the axially moving rope device;
u(x,t)＝F(x-v r t)+G(x+v l t) (2)
in the formula (2), v r The velocity of the right traveling wave relative to the fixed coordinate system; v l The velocity of the traveling wave is shifted to the left relative to a fixed coordinate system; f (x-v) r t) represents a velocity v r Right-shifting traveling wave of (a); g (x+v) l t) represents a velocity v l Is a left-shifted traveling wave;
step 2, let the atypical boundary at x=0 be the spring-damper boundary, where x=l 0 The typical boundary at the point is a fixed boundary, so that a mixed boundary constraint equation set formula is obtained by using the formula (3);
in the formula (3), u (0, t) represents a vibration displacement of the axially moving rope device at x=0; u (u) t (l 0 T) is expressed in x=l 0 First partial derivative of the transverse vibration displacement u of the rope with respect to time t; u (u) x (l 0 T) is expressed in x=l 0 A first partial derivative of the transverse vibration displacement u of the rope to the axial coordinate x; η represents an atypical boundary damping coefficient of the axially moving rope arrangement; ρ represents the linear density of the rope; p represents the tension of the axially moving rope arrangement; k represents the stiffness coefficient of the spring;
and 3, establishing a relational expression of the two traveling waves by using the expression (4) and the expression (5):
G(l 0 +v l t)＝-F(l 0 -v r t) (4)
in the formula (5), F 'and G' respectively represent derivatives of two traveling waves with respect to time t;
two intermediate variables α, β are obtained using formula (6):
obtaining a simplified traveling wave relational expression by using the formula (7):
in the formula (7), s represents the displacement of the right traveling wave, and s=v r t；
Using integral factor e -αs Obtaining the formula (8):
obtaining a general traveling wave relation expression by using the formula (9):
in the formula (9), F (x) represents the traveling wave relational expression of the right traveling wave relative to the axial coordinate x; e, e - α s Representing an integral factor;
step 4, giving an initial condition for movement of the axial moving rope device by using the formula (10):
in equation (10), the function phi (x) is the initial lateral displacement of different positions on the axially moving rope device in the fixed coordinate system; the function ψ (x) is the initial velocity of the different positions on the axially moving rope arrangement in the fixed coordinate system;
according to the motion initial conditions, an initial expression of two traveling waves is obtained by using the expression (11):
in the formula (11), ζ is an integral variable; g (x) represents the traveling wave relational expression of the left traveling wave relative to the axial coordinate x; c is an integration constant;
step 5, determining the vibration period T of the axial moving rope device by using the (12) according to the motion rule of the two traveling waves in the axial moving rope device and the reflection superposition rule of the two traveling waves at the boundaries of the two ends of the axial moving rope device 0 ；
According to the motion rule of the traveling wave in the rope device and the reflection superposition rule of the traveling wave at the boundary of two ends of the rope device, the vibration period T is divided into three stages, namely a first stage [0, T a ]Second stage [ t ] a ,t b ]And third stage [ t ] b ,T 0 ]Wherein 0 is<t a <t b <T 0 ；
Step 6, combining the motion equation, the motion initial condition and the solution u (x, T) of the motion equation of the axially moving rope device with the mixed boundary constraint equation set respectively, and for the vibration period T 0 The traveling waves at different stages are overlapped, so that a transverse vibration displacement response type is obtained;
step 6.1, as shown in FIG. 1a, first stage [0, t a ]In which the left traveling wave G is divided into a first left traveling wave G 1 And a second leftwards traveling wave G 2 The right traveling wave F is divided into a first right traveling wave F 1 And a second right traveling wave F 2 The method comprises the steps of carrying out a first treatment on the surface of the Wherein the second left traveling wave G 2 Is the first right traveling wave F 1 At the right boundary x=l 0 Reflected wave at the position, the second right-shift traveling wave F 2 Is the first left traveling wave G 1 Reflected waves at the left boundary x=0;
obtaining a first right traveling wave F by using (13) 1 And a first left traveling wave G 1 Is represented by the expression:
in the formula (13), F 1 (x-v r t) represents a velocity v r Is a first right traveling wave; g 1 (x+v l t) is the velocity v l Is shifted left by the first left-shift traveling wave; c is an integration constant;
the continuity condition of two traveling waves on the atypical boundary at x=0 is constructed using equation (14):
according to the formulas (3) and (14), the second leftwards traveling wave G is obtained by the formulas (15) and (16), respectively 2 And a second right traveling wave F 2 ：
In the formula (15) and the formula (16), G 2 (x+v l t) is the velocity v l Is a second left-shifted traveling wave; f (F) 2 (x-v r t) is the velocity v r Is a second right traveling wave;
step 6.2, as shown in FIG. 1b, in the second stage [ t a ,t b ]In which the left traveling wave G is divided into a first left traveling wave G 1 Second leftwards traveling wave G 2 And a third leftwards traveling wave G 3 The right traveling wave F is a second right traveling wave F 2 The method comprises the steps of carrying out a first treatment on the surface of the Second right traveling wave F 2 To the third left shift G 3 Is the second right shift F 2 At the right boundary x=l 0 Reflected waves at;
obtaining a third left shift G by using the formula (17) 3 And a second right shift F 2 Is a relational expression of:
in the formula (17), G 3 (x+v l t) is the velocity v l Is a third leftwards traveling wave;
obtaining a third left shift G by using (18) 3 And a first right shift F 1 Is a relational expression of:
step 6.3, as shown in FIG. 1c, at a third stage [ t b ,T 0 ]In which the left traveling wave G is divided into a second left traveling wave G 2 And a third leftwards traveling wave G 3 The right traveling wave F is divided into a second right traveling wave F 2 And a third right traveling wave F 3 The method comprises the steps of carrying out a first treatment on the surface of the Travelling wave G 3 Is the second right traveling wave F 2 At the right boundary x=l 0 Reflected waves at; third right traveling wave F 3 Is the second left traveling wave G 2 Reflected waves at left boundary x=0;
obtaining a third right traveling wave F by using (19) 3 The expression:
obtaining a second left traveling wave G by using the method (20) 2 At the left side edge expression:
obtaining a traveling wave F by using (21) 2 And F 3 The continuity conditions of (2) are:
F 3 (0,t b )＝F 2 (0,t b ) (21)
obtaining a third right traveling wave F by using (22) 3 The expression:
in the formula (22), F 3 (x-v r t) represents a velocity v r Is a third right traveling wave;
step 7, according to the first phase [0, t a ]The travelling wave expression in (a) is obtained in the first stage [0, t a ]The mid traveling wave energy expression, thereby calculating the total vibration energy of the axially moving rope apparatus:
step 7.1 potential energy of spring when spring-damper securing System is on atypical boundaryThe total system energy E (t) of the rope arrangement is obtained using equation (23):
and (3) deriving x and t from the travelling wave superposition formula in the formula (2) to obtain a formula (24):
in formula (24): u (u) t Is the first order partial derivative of u to t; u (u) x Is the first order partial derivative of u to x;
from formulas (23) and (24), the energy characterization of the rope as shown in formula (25) can be obtained:
from the derivation of equation (16), equation (26) can be obtained:
step 7.2 obtaining the vibration energy of the second right traveling wave F2 by using the formula (27)
Step 7.3 obtaining the vibration energy of the first right traveling wave F1 by using the formula (28)
Step 7.4, first left-shift traveling wave G using (29) 1 Vibration energy of (a)
Step 7.5 obtaining a second left traveling wave G by using the method (30) 2 Vibration energy of (a)
Step 7.6 obtaining a second left traveling wave G by using the method (31) 2 Vibration energy of (a)
Step 7.7, obtaining a second left traveling wave G by using the method (32) 2 Vibration energy of (a)
Step 7.8, obtaining three stages of total vibration energy E (t) by using the formula (33):
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Claims (1)
1. the vibration energy calculating method of the axial moving rope device under the mixed boundary condition is that one end boundary of two end boundaries of the axial moving rope device is an atypical boundary, the other end boundary is a typical boundary, the atypical boundary is taken as a coordinate origin, the axial moving direction of the axial rope device is taken as an x direction, and the transverse vibration direction is taken as a u direction, and a fixed coordinate system is established; the method is characterized in that: the vibration energy calculating method comprises the following steps:
step 1, according to the Hamiltonian principle, obtaining a motion equation of the axial moving rope device by using a formula (1), and expressing a solution u (x, t) of the motion equation of the axial moving rope device as a superposition of two traveling waves shown in a formula (2), wherein the superposition of the two traveling waves refers to the superposition of a left traveling wave and a right traveling wave;
u tt +2vu xt -(c 2 -v 2 )u xx ＝0 (1)
in the formula (1), u tt Is the second partial derivative of the transverse vibration displacement u of the rope with respect to time t; v represents the axial movement speed of the axially moving rope device; u (u) xt The first partial derivative of the transverse vibration displacement u of the rope on the axial coordinate x of the axial moving rope device in a fixed coordinate system and on the time t respectively; c represents the velocity of the traveling wave; u (u) xx Is the second partial derivative of the axial coordinate x of the rope transverse vibration displacement u to the axially moving rope device;
u(x,t)＝F(x-v r t)+G(x+v l t) (2)
in the formula (2), v r The velocity of the right traveling wave relative to the fixed coordinate system; v l The velocity of the traveling wave is shifted to the left relative to a fixed coordinate system; f (x-v) r t) represents a velocity v r Right-shifting traveling wave of (a); g (x+v) l t) represents a velocity v l Is a left-shifted traveling wave;
step 2, let the atypical boundary at x=0 be the spring-damper boundary, where x=l 0 The typical boundary at the point is a fixed boundary, so that a mixed boundary constraint equation set formula is obtained by using the formula (3);
in the formula (3), u (0, t) represents a vibration displacement of the axially moving rope device at x=0; u (u) t (l 0 T) is expressed in x=l 0 First partial derivative of the transverse vibration displacement u of the rope with respect to time t; u (u) x (l 0 T) is expressed in x=l 0 A first partial derivative of the transverse vibration displacement u of the rope to the axial coordinate x; eta represents atypical of axially moving rope arrangementsBoundary damping coefficient; ρ represents the linear density of the rope; p represents the tension of the axially moving rope arrangement; k represents the stiffness coefficient of the spring;
and 3, establishing a relational expression of the two traveling waves by using the expression (4) and the expression (5):
G(l 0 +v l t)＝-F(l 0 -v r t) (4)
in the formula (5), F 'and G' respectively represent derivatives of two traveling waves with respect to time t;
two intermediate variables α, β are obtained using formula (6):
obtaining a simplified traveling wave relational expression by using the formula (7):
in the formula (7), s represents the displacement of the right traveling wave, and s=v r t；
Obtaining a general traveling wave relation expression by using the formula (8):
in the formula (8), F (x) represents the traveling wave relational expression of the right traveling wave relative to the axial coordinate x; e, e -αs Representing an integral factor;
step 4, giving an initial condition for movement of the axially moving rope device by using the formula (9):
in equation (9), the function phi (x) is the initial lateral displacement of the different locations on the axially moving rope device in a fixed coordinate system; the function ψ (x) is the initial velocity of the different positions on the axially moving rope arrangement in the fixed coordinate system;
according to the motion initial conditions, an initial expression of two traveling waves is obtained by using the formula (10):
in the formula (10), ζ is an integral variable; g (x) represents the traveling wave relational expression of the left traveling wave relative to the axial coordinate x; c is an integration constant;
step 5, determining the vibration period T of the axial moving rope device by using the formula (11) according to the motion rule of the two traveling waves in the axial moving rope device and the reflection superposition rule of the two traveling waves at the boundaries of the two ends of the axial moving rope device 0 ；
Step 6, combining the motion equation, motion initiation condition and solution u (x, T) of the motion equation of the axial moving rope device with the mixed boundary constraint equation set, respectively, and for the vibration period T 0 The traveling waves at different stages are overlapped, so that a transverse vibration displacement response type is obtained;
in travelling wave displacement from axial coordinate x 1 To the axial coordinate x 2 The left traveling wave G is divided into a first left traveling wave G 1 And a second leftwards traveling wave G 2 The right traveling wave F is divided into a first right traveling wave F 1 And a second right traveling wave F 2 The method comprises the steps of carrying out a first treatment on the surface of the Wherein the second left traveling wave G 2 Is the first right traveling wave F 1 At the right boundary x=l 0 Reflected wave at the position, the second right-shift traveling wave F 2 Is the first left traveling wave G 1 Reflected waves at the left boundary x=0;
obtaining a first right traveling wave F by using (12) 1 And a first left traveling wave G 1 Is represented by the expression:
in the formula (12), F 1 (x-v r t) represents a velocity v r Is a first right traveling wave; g 1 (x+v l t) is the velocity v l Is shifted left by the first left-shift traveling wave; c is an integration constant;
the continuity condition of two traveling waves on the atypical boundary at x=0 is constructed using equation (13):
according to the formulas (3) and (13), the second leftwards traveling wave G is obtained by the formulas (14) and (15), respectively 2 And a second right traveling wave F 2 ：
In the formula (14) and the formula (15), G 2 (x+v l t) is the velocity v l Is a second left-shifted traveling wave; f (F) 2 (x-v r t) is the velocity v r Is a second right traveling wave;
step 7, according to the first phase [0, t a ]The travelling wave expression in (a) is obtained in the first stage [0, t a ]A mid-range traveling wave energy expression to calculate a total vibration energy of the axially moving rope apparatus:
step 7.1 obtaining the vibration energy of the second right traveling wave F2 by using the method (16)
Step 7.2 obtaining the vibration energy of the first right traveling wave F1 by using the method (17)
Step 7.3, first left-shift traveling wave G using (18) 1 Vibration energy of (a)
Step 7.4 obtaining a second left traveling wave G by using the method (19) 2 Vibration energy of (a)
Step 7.5, obtaining total vibration energy E (t) by using the formula (20):
in the formula (20), E k (t) is the potential energy of the spring, an
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