CN109614745B - Method for acquiring transverse vibration of axial moving rope equipment under mixed boundary condition - Google Patents
Method for acquiring transverse vibration of axial moving rope equipment under mixed boundary condition Download PDFInfo
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Abstract
The invention discloses a method for acquiring transverse vibration of axial moving rope equipment under a mixed boundary condition, which is characterized by acquiring a motion equation of the axial moving rope equipment according to a Hamilton principle, and regarding the displacement response of the motion equation as the superposition of two traveling waves; deriving expressions of two initial traveling waves according to initial motion conditions of the axial movement rope equipment, and dividing a vibration cycle into three stages according to a traveling wave motion rule in the rope equipment; obtaining a constraint equation of a mixed boundary, combining the constraint equation of the mixed boundary with a motion equation, and obtaining a reflected wave response equation of each stage; and respectively superposing incident waves and reflected waves of the movable rope equipment at the boundary in three different time stages in a vibration period to obtain the transverse displacement vibration of the movable rope equipment. The invention is applicable to mixed boundary conditions and multiple speed conditions of moving rope equipment. The obtained vibration displacement response is accurate, and the requirements for testing the feasibility and the effectiveness of various numerical calculation methods for the transverse vibration of the axial moving rope can be met.
Description
Technical Field
The invention belongs to the field of mechanical system dynamics modeling and vibration control, and particularly relates to a method for acquiring transverse vibration of axial movement rope equipment under a mixed boundary condition.
Background
The axial movement rope device has the advantages of high running efficiency, strong self-adaptation, large bearing capacity, simple structure, flexibility, controllability and the like, and has very important application values in various engineering fields, such as tethered satellite cables, power transmission belts, elevator cables, passenger and freight ropeways and the like. Noise and vibration accompany the operation of these devices, and lateral vibration in particular has a great influence on the function and safety of these devices. The problem of lateral vibration of axially moving rope devices is a challenging problem that has been studied for many years and has received much attention to date. The traditional research technology is that a partial differential motion equation established based on the Hamilton principle and a finite element kinetic equation established based on a Lagrange equation are utilized to solve the equations by using a numerical calculation method, such as a Galerkin method, a Runge-Kutta method, a Newmark method, a time-varying state space equation and the like, so as to obtain the transverse vibration response of the axial movement rope device. However, when solving the problem of the lateral vibration of the moving rope device under the complicated mixed boundary condition, the conventional method has the problems of complicated solving process, low solving precision and poor stability. Also, when the axially moving rope device is at a high speed, approaching or reaching a critical speed, it can cause the device to vibrate with an abnormally increased displacement amplitude, resulting in increased error.
The darnobel principle indicates that the infinite-length uniform string transverse vibration can be expressed as the superposition of two traveling waves in opposite directions, lays a theoretical foundation for acquiring the transverse vibration of the axial movement rope equipment by using a wave superposition theory, and has the advantage that the vibration response is not unstable due to the increase of the movement speed. However, the darnobel principle is directed to the vibration and energy variation characteristics of a single reflection of a traveling wave at different boundaries of a semi-infinite chord. In the practical engineering application of the axial moving rope equipment, 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 equipment and are superposed with incident waves to form transverse vibration of the moving rope equipment, so that the problem of accurately acquiring the transverse vibration formed by the repeated reflection superposition of the traveling waves in the fixed-length moving rope equipment under the constraint condition of the mixed boundary cannot be solved by using the method of the Dalabel principle.
Disclosure of Invention
The invention aims to avoid the defects of the prior art and provides a method for acquiring the transverse vibration of the axial moving rope equipment under the mixed boundary condition, thereby acquiring an accurate analytical expression of the transverse vibration displacement response of the axial moving rope equipment under the mixed boundary condition and solving the problems of low solving precision and poor stability of the transverse vibration displacement response of the axial moving rope equipment; the problem of instability of vibration displacement response caused by the increase of the moving speed is solved; under the constraint condition of a mixed boundary, the problem of accurately acquiring the transverse vibration formed by the repeated reflection and superposition of the traveling wave in the fixed-length moving rope equipment is solved.
The invention adopts the following technical scheme for solving the technical problems:
the invention obtains the method of the axial movement rope equipment transverse vibration under the mixed boundary condition, the mixed boundary condition means that in the two end boundaries of the axial movement rope equipment, one end boundary is a typical boundary, and the other end boundary is a non-typical boundary; the method is characterized in that: the method for acquiring the transverse vibration of the axial moving rope device under the mixed boundary condition comprises the following steps:
obtaining an axial movement rope equipment motion equation according to a Hamilton principle, and expressing the solution of the axial movement rope equipment motion equation as the superposition of two traveling waves, wherein the superposition of the two traveling waves is the superposition of a left traveling wave and a right traveling wave; and obtaining a mixed boundary constraint equation set;
deriving an initial representation of the two travelling waves from initial conditions of motion of the axially moving rope device;
determining the vibration period T of the rope equipment according to the motion rule of the traveling wave in the rope equipment and the reflection superposition rule of the traveling wave on the boundaries of the two ends of the rope equipment;
dividing a vibration period T into three stages, and combining a traveling wave superposition formula of the motion equation, the motion initial condition and the motion equation solution of the axial movement rope equipment with a mixed boundary constraint equation respectively to obtain a reflected wave response equation of each stage in the vibration period T; and aiming at three different stages in a vibration cycle, respectively superposing incident waves and reflected waves on the boundary of the axial movement rope equipment to obtain transverse vibration displacement response.
The method for acquiring the transverse vibration of the axial movement rope device under the mixed boundary condition is characterized by comprising the following steps of:
step 1: obtaining an axial movement rope device motion equation according to the Hamilton principle as shown in the formula (1):
u tt +2vu xt +(v 2 -c 2 )u xx ＝0 (1)
x represents the axial coordinate of the rope device, t represents time, v represents the axial moving speed of the rope device;
u represents the rope lateral vibration displacement, u is a function of x and t, u = u (x, t);
u tt is the second partial derivative of u over t; u. u xx Is the second partial derivative of u over x; u. of xt Is the first partial derivative of u with respect to x and t, respectively; c represents the velocity of the traveling wave;
the solution u (x, t) of equation (1) is expressed in the form of superposition of two traveling waves, as equation (2):
u(x,t)＝F(x-v r t)+G(x+v l t) (2)
for the two-end boundary of the axially moving rope device, the left-end boundary is defined as x =0, and the right-end boundary is defined as x = l 0 ，l 0 Is the length of the rope;
v r is the velocity of the right traveling wave in the rope relative to a fixed coordinate system;
v l is the velocity of the left traveling wave in the rope relative to a fixed coordinate system;
F(x-v r t) represents a velocity v r The right traveling wave of (a) is marked as F;
G(x+v l t) represents a velocity v l The left traveling wave of (a) is marked as G;
and 2, step: determining a motion initiation condition and a mixed boundary condition constraint equation set of the rope device:
when t =0 is set, the initial condition of the movement of the rope apparatus is as follows (3):
in formula (3):
u (x, 0) represents the lateral vibration displacement of the rope at time t =0, u t (x, 0) represents the first partial derivative of u (x, 0) with respect to t;
the function phi (x) is the initial transverse displacement at different positions on the rope device in a fixed coordinate system;
the function ψ (x) is the initial velocity at different positions on the rope arrangement in a fixed coordinate system;
the typical boundary of the mixed boundary condition constraint equation system at x =0 is a fixed boundary at x = l 0 The atypical boundary at (a) is a damped boundary, characterized by equation (4):
in formula (4):
u (0, t) represents the vibrational displacement of the rope arrangement at x = 0; η represents the damping coefficient;
u t (l 0 t) is expressed at x = l 0 The first partial derivative of position u with respect to t; ρ represents the linear density of the rope;
u x (l 0 t) is expressed at x = l 0 The first partial derivative of position u with respect to x; p represents the tension of the rope arrangement;
and step 3: the vibration period T is determined by equation (5):
dividing the vibration period T into three stages according to the motion rule of the traveling wave in the rope equipment and the reflection superposition rule of the traveling wave at the boundaries of two ends of the rope equipment, wherein the three stages are respectively a first stage [0, T ] a ]Second stage [ t ] a ,t b ]And a third stage [ t ] a , T]Wherein 0 is<t a <t b <T；
And 4, step 4: obtaining the first stage [0, t ] of the rope equipment by combining the motion initial condition and the mixed boundary constraint equation a ]The lateral vibration displacement of (2):
first stage [0, t a ]In, the left traveling wave G is divided into traveling waves G 1 Sum traveling wave G 2 The right traveling wave F is divided into traveling waves F 1 Sum traveling wave F 2 (ii) a Wherein the traveling wave G 2 Is a travelling wave F 1 At right border x = l 0 Reflected wave, travelling wave F of 2 Is a travelling wave G 1 Reflected waves at the left border x = 0;
respectively obtaining traveling waves F by combining initial motion conditions (3) and formula (2) 1 Sum traveling wave G 1 Is represented by the formula (6):
in formula (6):
F 1 (x-v r t) represents a velocity v r Is denoted as F 1 ；
G 1 (x+v l t) is a velocity v l Left traveling wave of (1), denoted as G 1 (ii) a Xi is an integral variable; d is an integral constant;
combining the mixed boundary condition constraint equation system characterized by the formula (4) and the continuity condition characterized by the formula (7), respectively, the traveling wave G characterized by the formula (8) is obtained 2 And a traveling wave F characterized by the formula (9) 2
G 2 (l 0 )＝G 1 (l 0 ) (7)
Wherein:
G 2 (x+v l t) is a velocity v l Left traveling wave of (1), denoted as G 2 ；
F 2 (x-v r t) is velocity v r Is denoted as F 2 ；
The left traveling wave and the right traveling wave of the first stage are obtained from the formula (6), the formula (8) and the formula (9) and expressed as the formula (10) and the formula (11):
the vibration displacement response of the rope equipment in the first stage is obtained by superposing the left traveling wave and the right traveling wave according to the formula (10) and the formula (11) as shown in the formula (12):
u(x,t)＝F(x-v r t)+G(x+v l t),0＜t＜t a (12)
and 5: the rope equipment is obtained in the second stage [ t ] by combining the motion equation, the traveling wave superposition formula (2) of the motion equation solution and the mixed boundary constraint equation a ,t b ]The lateral vibration displacement of (2):
in the second stage [ t ] a ,t b ]In respect of v>0, the left traveling wave G is divided into traveling waves G 1 Traveling wave G 2 Sum traveling wave G 3 The right traveling wave F is a traveling wave F 2 (ii) a Travelling wave F 2 Is an incident wave, G 3 Is F 2 At right border x = l 0 A reflected wave of (c);
from the hybrid boundary constraint equation and equation (2) to obtain G 3 And F 2 Is characterized by the relationship of equation (13):
the continuity conditions are characterized by formula (14) and formula (15):
substituting formula (14) and formula (15) into formula (13) to obtain G characterized by formula (16) 3 ：
In the second stage [ t ] a ,t b ]In respect of v<0, the left moving wave G is the traveling wave G 2 The right traveling wave F is divided into traveling waves F 1 Traveling wave F 2 And a traveling wave F 3 (ii) a Travelling wave F 3 Is a travelling wave G 2 Reflected waves at the left border x = 0; obtaining F using the hybrid boundary constraint equation and equation (2) 3 And G 2 Is as in formula (17):
in the formula (17), G 2 (l 0 ) And F 1 (l 0 ) Satisfy the continuity condition, respectively, characterized by formula (18) and formula (19):
G 2 (l 0 )＝G 2 (l 0 +v l t b ) (18)
F 1 (l 0 )＝F 1 (l 0 -v r t b ) (19)
let x = l 0 ,t＝t b G is obtained by substituting formulae (6) and (8) with the combination of formulae (18) and (19) 2 (l 0 )、F 1 (l 0 ) Characterized by formula (20) and formula (21):
substituting formula (20) and formula (21) into formula (17) to obtain formula (22):
the second stage [ t ] is obtained from the formula (6), the formula (8), the formula (9), the formula (16) and the formula (22) a ,t b ]The left traveling wave and the right traveling wave in (1) are expressed as formula (23) and formula (24):
the second stage [ t ] is obtained by superposing the left traveling wave and the right traveling wave according to the equations (23) and (24) a ,t b ]The vibration displacement response of the rope arrangement of (1) is characterized by equation (25):
u(x,t)＝F(x-v r t)+G(x+v l t),t a ＜t＜t b (25)
step 6: according to a first phase [0, t a ]And a second stage [ t ] a ,t b ]Obtaining the moving rope device in the third stage t b ,T]The lateral vibration displacement of (2):
in a third stage [ t ] b ,T]In, the left traveling wave G is divided into traveling waves G 2 Sum traveling wave G 3 The right traveling wave F is divided into traveling waves F 2 And a traveling wave F 3 (ii) a Travelling wave G 3 Is a travelling wave F 2 At right border x = l 0 A reflected wave of (d); travelling wave F 3 Is a travelling wave G 2 Reflected waves at the left border x = 0;
the third stage [ t ] is obtained from the formula (8), the formula (9), the formula (16) and the formula (22) b ,T]The left traveling wave and the right traveling wave in (1) are represented by the following expressions (26) and (27):
the left traveling wave and the right traveling wave are superposed by the equations (26) and (27) to obtain a third stage [ t ] b ,T]The vibrational displacement response of the rope arrangement of (a), as characterized by equation (28):
u(x,t)＝F(x-v r t)+G(x+v l t),t b ＜t＜T (28)
the expression (12), the expression (25) and the expression (28) are piecewise analytical expressions of the transverse vibration displacement of the rope device in one vibration period T.
Compared with the prior art, the invention has the beneficial effects that:
1. the method has high solving precision and good stability. The method adopts an analytic solution method, compared with a numerical solution method, the analytic solution method obtains an accurate analytic expression of the vibration displacement, has the characteristics of high solving precision and good stability for solving the transverse vibration response of the axial movement rope device, and can solve the problem of instability of the vibration response due to the increase of the movement speed.
2. The method solves the problem of accurately acquiring the transverse vibration formed by repeated reflection and superposition of traveling waves in the fixed-length moving rope equipment under the mixed boundary constraint condition, and has a simple process. According to the method, according to the Dalabel principle, the vibration cycle is divided into three stages according to the motion rule of the traveling waves in the rope equipment, and the traveling waves in different directions in each stage are reflected on the boundaries of two ends; then boundary incident waves and boundary reflected waves of the rope equipment in three stages are respectively obtained by combining a traveling wave movement rule and boundary conditions in the rope equipment; and finally, overlapping the boundary incident wave and the boundary reflected wave to respectively obtain the transverse displacement of the rope equipment in three stages, thereby solving the problem of accurately obtaining the transverse vibration formed by repeated reflection and superposition of traveling waves in the fixed-length moving rope equipment under the mixed boundary constraint condition.
3. The method of the invention is suitable for mixed boundary conditions and various speed working conditions of moving 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 requirements for testing the feasibility and the effectiveness of various numerical calculation methods for the transverse vibration of the axial movement rope device can be met.
Drawings
FIG. 1 is a fixed _ damped hybrid boundary model;
FIG. 1a is a fixed _ spring-damping boundary model;
FIG. 1b is a fixed _ mass-spring-damping boundary model;
FIG. 2a shows the first stage [0, t ] of the process of the present invention a ]A traveling wave reflection superposition schematic diagram;
FIG. 2b shows the second stage [ t ] of the process of the present invention a ,t b ]、v>A 0-time traveling wave reflection superposition diagram;
FIG. 2c shows a second stage [ t ] of the process according to the invention a ,t b ]、v<A 0-time traveling wave reflection superposition schematic diagram;
FIG. 2d shows the third stage [ t ] of the process according to the invention a ,T]A traveling wave reflection superposition diagram;
Detailed Description
The reflected wave superposition technology is applied to solving the transverse vibration response of the axial moving rope system under the mixed boundary condition, and comprises a fixed _ damping boundary condition shown in figure 1, a fixed _ spring-damping boundary condition shown in figure 1a and a fixed _ mass-spring-damping boundary condition shown in figure 1 b; the vibration response obtained by the reflection wave superposition technology is high in precision, and the requirements for testing the feasibility and the effectiveness of the axial movement rope transverse vibration numerical calculation method are met.
In this embodiment, the fixed _ damping boundary condition shown in fig. 1 is taken as an example for explanation, and the hybrid boundary condition means that, in the boundaries of the two ends of the axially moving rope device, one end boundary is a typical boundary, and the other end boundary is an atypical boundary; in this embodiment, the method of obtaining the lateral vibration of the axial movement rope device under the mixed boundary condition includes:
obtaining a motion equation of the axial moving rope equipment according to a Hamilton principle, and expressing the solution of the motion equation of the axial moving rope equipment as the superposition of two traveling waves, wherein the superposition of the two traveling waves is the superposition of a left traveling wave and a right traveling wave; and a system of hybrid boundary constraint equations is obtained.
An initial representation of the two travelling waves is derived from the initial conditions of the movement of the axially moving rope arrangement.
Determining the vibration period T of the rope equipment according to the motion rule of the traveling wave in the rope equipment and the reflection superposition rule of the traveling wave on the boundaries of the two ends of the rope equipment; dividing a vibration period T into three stages, and combining a traveling wave superposition formula of an axial movement rope device motion equation, a motion initial condition and a motion equation solution with a mixed boundary constraint equation to obtain a reflected wave response equation of each stage in the vibration period T; and aiming at three different stages in a vibration cycle, respectively superposing incident waves and reflected waves on the boundary of the axial movement rope equipment to obtain transverse vibration displacement response.
The method for acquiring the transverse vibration of the axial moving rope device under the mixed boundary condition in the embodiment is carried out according to the following steps:
step 1: obtaining an axial movement rope device motion equation according to the Hamilton principle as shown in the formula (1):
u tt +2vu xt +(v 2 -c 2 )u xx ＝0 (1)
x represents the axial coordinate of the rope device, t represents time, and v represents the axial moving speed of the rope device;
u represents the rope lateral vibration displacement, u is a function of x and t, u = u (x, t);
u tt is the second partial derivative of u over t; u. of xx Is the second partial derivative of u over x; u. of xt Is the first partial derivative of u with respect to x and t, respectively; c represents the velocity of the traveling wave; c = (P/ρ) 0.5 P is the tension of the rope, ρ is the linear density of the rope;
the solution u (x, t) of equation (1) is expressed in the form of superposition of two travelling waves, as in equation (2):
u(x,t)＝F(x-v r t)+G(x+v l t) (2)
for the two end boundaries of the axially moving rope device, the left end boundary is defined as x =0 and the right end boundary is defined as x = l 0 ，l 0 Is the length of the rope;
v r velocity of the right traveling wave in the rope with respect to a fixed coordinate system, v r ＝c+v；
v l Is the velocity, v, of the left traveling wave in the rope relative to a fixed coordinate system l ＝c-v；
F(x-v r t) represents a velocity v r The right traveling wave of (a) is marked as F;
G(x+v l t) represents a velocity v l The left traveling wave of (a) is marked as G; f and G are both arbitrary quadratic continuous differentiable functions.
Step 2: determining a motion initiation condition and a mixed boundary condition constraint equation set of the rope device:
when t =0 is set, the initial condition of the movement of the rope apparatus is as follows (3):
in formula (3):
u (x, 0) represents the lateral vibration displacement of the rope at time t =0, u t (x, 0) represents the first partial derivative of u (x, 0) with respect to t;
the function phi (x) is the initial transverse displacement at different positions on the rope device in a fixed coordinate system;
the function ψ (x) is the initial velocity at different positions on the rope arrangement in a fixed coordinate system;
the typical boundary of the mixed boundary condition constraint equation system at x =0 is a fixed boundary at x = l 0 The atypical boundary at (a) is a damped boundary, characterized by equation (4):
in formula (4):
u (0, t) represents the vibrational displacement of the rope arrangement at x = 0; η represents the damping coefficient;
u t (l 0 t) is expressed at x = l 0 The first partial derivative of position u with respect to t; ρ represents the linear density of the rope;
u x (l 0 t) is expressed at x = l 0 The first partial derivative of position u with respect to x; p denotes the tension of the rope arrangement.
And step 3: determining a vibration period T by the formula (5), wherein the vibration period T is the time of the incident wave after being reflected back to the initial state;
dividing the vibration period T into three stages according to the motion rule of the traveling wave in the rope equipment and the reflection superposition rule of the traveling wave on the boundary of two ends of the rope equipment, wherein the three stages are respectively a first stage [0, T ] a ]Second stage [ t ] a ,t b ]And a third stage [ t ] a ,T]Wherein 0 is<t a <t b <T; the motion state of the traveling wave is changed along with time, and analysis in three stages is more favorable for solving the transverse vibration displacement; when v is>0, i.e. the rope moves axially from left to right, t a ＝l 0 /v r ，t b ＝l 0 /v l (ii) a When v is>At 0, namely the rope moves axially from right to left, then: t is t a ＝l 0 /v l ，t b ＝l 0 /v r 。
And 4, step 4: obtaining the first stage [0, t ] of the rope equipment by combining the motion initial condition and the mixed boundary constraint equation a ]The lateral vibration displacement of (2):
first stage [0, t ] a ]In (2 a), the left-hand traveling wave G is divided into traveling waves G 1 Sum traveling wave G 2 The right traveling wave F is divided into traveling waves F 1 Sum traveling wave F 2 (ii) a Wherein the traveling wave G 2 Is a travelling wave F 1 At right border x = l 0 Reflected wave of (F), travelling wave 2 Is a travelling wave G 1 Reflected wave at left border x = 0.
Respectively obtaining traveling waves F by combining with the initial conditions (3) and (2) of motion 1 Sum traveling wave G 1 Is represented by the formula (6):
in formula (6):
F 1 (x-v r t) represents the velocityDegree v r Is denoted as F 1 ；
G 1 (x+v l t) is velocity v l Left traveling wave of (1), denoted as G 1 (ii) a Xi is an integral variable; d is an integral constant;
combining the mixed boundary condition constraint equation system characterized by the formula (4) and the continuity condition characterized by the formula (7), respectively, the traveling wave G characterized by the formula (8) is obtained 2 And a traveling wave F characterized by the formula (9) 2 ：
G 2 (l 0 )＝G 1 (l 0 ) (7)
Wherein:
G 2 (x+v l t) is velocity v l Left traveling wave of (1), denoted as G 2 ；
F 2 (x-v r t) is a velocity v r Is denoted as F 2 ；
The left traveling wave and the right traveling wave of the first stage are obtained from the formula (6), the formula (8) and the formula (9) and expressed as the formula (10) and the formula (11):
the vibration displacement response of the rope equipment in the first stage is obtained by superposing the left traveling wave and the right traveling wave according to the equations (10) and (11) and is shown as the equation (12):
u(x,t)＝F(x-v r t)+G(x+v l t),0＜t＜t a (12)
and 5: acquiring the rope equipment in the second stage [ t ] by combining the motion equation, the traveling wave superposition formula (2) of the motion equation solution and the mixed boundary constraint equation a ,t b ]Transverse oscillatory displacement of [ c ], second stage [ t ] a ,t b ]Is related to the direction of movement of the rope arrangement, thus distinguishing v>0 and v<Two cases are discussed:
in the second stage [ t ] a ,t b ]In respect of v>0, as shown in FIG. 2b, the left traveling wave G is divided into traveling waves G 1 Traveling wave G 2 Sum traveling wave G 3 The right traveling wave F is a traveling wave F 2 (ii) a Travelling wave F 2 Being incident waves, G 3 Is F 2 At right border x = l 0 A reflected wave of (d);
obtaining G from the mixed boundary constraint equation and equation (2) 3 And F 2 Is characterized by the relationship of equation (13):
the continuity conditions are characterized by formula (14) and formula (15):
substituting formula (14) and formula (15) into formula (13) to obtain G characterized by formula (16) 3 ：
In a second stage [ t ] a ,t b ]In respect of v<0, as shown in FIG. 2c, the left traveling wave G is the traveling wave G 2 The right traveling wave F is divided into traveling waves F 1 Traveling wave F 2 And a traveling wave F 3 (ii) a Travelling wave F 3 Is a travelling wave G 2 Reflected waves at the left border x = 0; obtaining F using the hybrid boundary constraint equation and equation (2) 3 And G 2 Is as in formula (17):
in formula (17), G 2 (l 0 ) And F 1 (l 0 ) Satisfies continuity conditions, respectively, characterized by formula (18) and formula (19):
G 2 (l 0 )＝G 2 (l 0 +v l t b ) (18)
F 1 (l 0 )＝F 1 (l 0 -v r t b ) (19)
let x = l 0 ,t＝t b G is obtained by substituting formulae (6) and (8) with the combination of formulae (18) and (19) 2 (l 0 )、F 1 (l 0 ) Characterized by formula (20) and formula (21):
substituting formula (20) and formula (21) into formula (17) to obtain formula (22):
the second stage [ t ] is obtained from the formula (6), the formula (8), the formula (9), the formula (16) and the formula (22) a ,t b ]Left traveling wave sum ofThe right traveling wave is expressed as formula (23) and formula (24):
the second stage [ t ] is obtained by superposing the left traveling wave and the right traveling wave according to the equations (23) and (24) a ,t b ]The vibration displacement response of the rope apparatus of (4) is characterized by equation (25):
u(x,t)＝F(x-v r t)+G(x+v l t),t a ＜t＜t b (25)
step 6: according to a first phase [0, t a ]And a second stage [ t ] a ,t b ]Obtaining the moving rope device in the third stage t b ,T]Transverse vibrational displacement of [ c ], a third stage [ t ] b ,T]The state of motion of the travelling wave of (a) is independent of the direction of movement of the rope arrangement:
in a third stage [ t ] b ,T]In (D), as shown in FIG. 2d, the left traveling wave G is divided into traveling waves G 2 Sum traveling wave G 3 The right traveling wave F is divided into traveling waves F 2 And a traveling wave F 3 (ii) a Travelling wave G 3 Is a travelling wave F 2 At right border x = l 0 A reflected wave of (c); travelling wave F 3 Is a travelling wave G 2 Reflected waves at the left boundary x = 0;
the third stage [ t ] is obtained from formula (8), formula (9), formula (16) and formula (22) b ,T]The left traveling wave and the right traveling wave in (1) are represented by the following expressions (26) and (27):
the left traveling wave and the right traveling wave are superposed by the equations (26) and (27) to obtain a third stage [ t ] b ,T]The vibration displacement response of the rope arrangement of (4), as characterized by equation (28):
u(x,t)＝F(x-v r t)+G(x+v l t),t b ＜t＜T (28)
equations (12), (25) and (28) are piecewise analytical expressions of the lateral vibration displacement of the rope device in one vibration period T.
The method skillfully solves the problem that the transverse vibration solving process of the axial rope system is complex under the mixed boundary condition, the obtained vibration response of the rope equipment is more in line with the actual situation, and the limitations of a numerical calculation method and a Dalnbell principle method are avoided.
Claims (1)
1. A method for acquiring transverse vibration of an axial moving rope device under a mixed boundary condition, wherein the mixed boundary condition is that one end boundary is a typical boundary and the other end boundary is an atypical boundary in two end boundaries of the axial moving rope device; the method is characterized in that: the method for acquiring the transverse vibration of the axial moving rope device under the mixed boundary condition comprises the following steps:
obtaining a motion equation of axial movement rope equipment according to a Hamilton principle, and expressing the solution of the motion equation of the axial movement rope equipment as the superposition of two traveling waves, wherein the superposition of the two traveling waves refers to the superposition of a left traveling wave and a right traveling wave; and obtaining a mixed boundary constraint equation set;
deriving an initial representation of the two travelling waves from initial conditions of motion of the axially moving rope device;
determining the vibration period T of the rope equipment according to the motion rule of the traveling wave in the rope equipment and the reflection superposition rule of the traveling wave on the boundaries of the two ends of the rope equipment;
dividing a vibration period T into three stages, and combining a traveling wave superposition formula of the motion equation, the motion initial condition and the motion equation solution of the axial movement rope equipment with a mixed boundary constraint equation respectively to obtain a reflected wave response equation of each stage in the vibration period T; according to three different stages in a vibration cycle, respectively superposing incident waves and reflected waves on the boundary of the axial movement rope device to obtain transverse vibration displacement response;
the method for acquiring the transverse vibration of the axial moving rope device under the mixed boundary condition comprises the following steps:
step 1: obtaining an axial movement rope device motion equation according to the Hamilton principle as shown in the formula (1):
u tt +2vu xt +(v 2 -c 2 )u xx ＝0 (1)
x represents the axial coordinate of the rope device, t represents time, v represents the axial moving speed of the rope device;
u represents the rope lateral vibration displacement, u is a function of x and t, u = u (x, t);
u tt is the second partial derivative of u over t; u. of xx Is the second partial derivative of u over x; u. of xt Is the first partial derivative of u with respect to x and t, respectively;
c represents the velocity of the traveling wave;
the solution u (x, t) of equation (1) is expressed in the form of superposition of two travelling waves, as in equation (2):
u(x,t)＝F(x-v r t)+G(x+v l t) (2)
for the two end boundaries of the axially moving rope device, the left end boundary is defined as x =0 and the right end boundary is defined as x = l 0 ，l 0 Is the length of the rope;
v r is the velocity of the traveling wave traveling to the right in the rope relative to a fixed coordinate system;
v l is the velocity of the left traveling wave in the rope relative to a fixed coordinate system;
F(x-v r t) represents a velocity v r The right traveling wave of (a) is marked as F;
G(x+v l t) represents a velocity v l The left traveling wave of (a) is marked as G;
step 2: determining a motion initiation condition and a mixed boundary condition constraint equation set of the rope device:
when t =0 is set, the initial condition of the movement of the rope apparatus is as follows (3):
in formula (3):
u (x, 0) represents the lateral vibration displacement of the rope at time t =0, u t (x, 0) represents the first partial derivative of u (x, 0) with respect to t;
the function phi (x) is the initial transverse displacement at different positions on the rope device in a fixed coordinate system;
the function ψ (x) is the initial velocity at different positions on the rope arrangement in a fixed coordinate system;
the typical boundary of the mixed boundary condition constraint equation system at x =0 is a fixed boundary at x = l 0 The atypical boundary at (a) is a damped boundary, characterized by equation (4):
in formula (4):
u (0, t) represents the vibrational displacement of the rope arrangement at x = 0; η represents the damping coefficient;
u t (l 0 t) is expressed at x = l 0 The first partial derivative of position u with respect to t; ρ represents the linear density of the rope;
u x (l 0 t) is expressed at x = l 0 The first partial derivative of position u with respect to x; p represents the tension of the rope arrangement;
and step 3: the vibration period T is determined by equation (5):
dividing the vibration period T into three stages according to the motion rule of the traveling wave in the rope equipment and the reflection superposition rule of the traveling wave on the boundaries of two ends of the rope equipment, wherein the three stages are respectively a first stage [0, T ] a ]Second stage [ t ] a ,t b ]And thirdStage [ t ] a ,T]Wherein 0 is<t a <t b <T；
And 4, step 4: obtaining the first stage [0, t ] of the rope device by combining the motion initial condition and the mixed boundary constraint equation a ]The lateral vibration displacement of (2):
first stage [0, t a ]In, the left traveling wave G is divided into traveling waves G 1 Sum traveling wave G 2 The right traveling wave F is divided into traveling waves F 1 And a traveling wave F 2 (ii) a Wherein the traveling wave G 2 Is a travelling wave F 1 At right border x = l 0 Reflected wave of (F), travelling wave 2 Is a travelling wave G 1 Reflected waves at the left border x = 0;
respectively obtaining traveling waves F by combining initial motion conditions (3) and formula (2) 1 Sum traveling wave G 1 Is represented by the formula (6):
in formula (6):
F 1 (x-v r t) represents a velocity v r Is denoted as F 1 ；
G 1 (x+v l t) is velocity v l Left traveling wave of (1), denoted as G 1 (ii) a Xi is an integral variable; d is an integral constant;
combining the mixed boundary condition constraint equation system characterized by the formula (4) and the continuity condition characterized by the formula (7), respectively, the traveling wave G characterized by the formula (8) is obtained 2 And a traveling wave F characterized by the formula (9) 2
G 2 (l 0 )＝G 1 (l 0 ) (7)
Wherein:
G 2 (x+v l t) is velocity v l Left traveling wave of (2), denoted as G 2 ；
F 2 (x-v r t) is velocity v r Is denoted as F 2 ；
The left traveling wave and the right traveling wave of the first stage are obtained from the formula (6), the formula (8) and the formula (9) and expressed as the formula (10) and the formula (11):
the vibration displacement response of the rope equipment in the first stage is obtained by superposing the left traveling wave and the right traveling wave according to the equations (10) and (11) and is shown as the equation (12):
u(x,t)＝F(x-v r t)+G(x+v l t),0＜t＜t a (12)
and 5: acquiring the rope equipment in the second stage [ t ] by combining the motion equation, the traveling wave superposition formula (2) of the motion equation solution and the mixed boundary constraint equation a ,t b ]The lateral vibration displacement of (2):
in the second stage [ t ] a ,t b ]In respect of v>0, the left moving wave G is divided into traveling waves G 1 Traveling wave G 2 Sum traveling wave G 3 The right traveling wave F is a traveling wave F 2 (ii) a Travelling wave F 2 Is an incident wave, G 3 Is F 2 At right border x = l 0 A reflected wave of (c);
from the hybrid boundary constraint equation and equation (2) to obtain G 3 And F 2 Is characterized by the relation of (1) as in formula (13)：
The continuity conditions are characterized by formula (14) and formula (15):
substituting formula (14) and formula (15) into formula (13) to obtain G characterized by formula (16) 3 ：
In a second stage [ t ] a ,t b ]In respect of v<0, the left moving wave G is the traveling wave G 2 The right traveling wave F is divided into traveling waves F 1 Traveling wave F 2 And a traveling wave F 3 (ii) a Travelling wave F 3 Is a travelling wave G 2 Reflected waves at the left boundary x = 0; obtaining F using the hybrid boundary constraint equation and equation (2) 3 And G 2 Is as in formula (17):
in the formula (17), G 2 (l 0 ) And F 1 (l 0 ) Satisfies continuity conditions, respectively, characterized by formula (18) and formula (19):
G 2 (l 0 )＝G 2 (l 0 +v l t b ) (18)
F 1 (l 0 )＝F 1 (l 0 -v r t b ) (19)
let x = l 0 ,t＝t b G is obtained by substituting formulae (6) and (8) with the combination of formulae (18) and (19) 2 (l 0 )、F 1 (l 0 ) Characterized by formula (20) and formula (21):
substituting the formula (20) and the formula (21) into the formula (17) to obtain the formula (22):
the second stage [ t ] is obtained from the formula (6), the formula (8), the formula (9), the formula (16) and the formula (22) a ,t b ]The left moving wave and the right moving wave in (2) are expressed as formula (23) and formula (24):
the second stage [ t ] is obtained by superposing the left traveling wave and the right traveling wave according to the equations (23) and (24) a ,t b ]The vibration displacement response of the rope arrangement of (1) is characterized by equation (25):
u(x,t)＝F(x-v r t)+G(x+v l t),t a ＜t＜t b (25)
step 6: according to a first phase [0, t a ]And a second stage [ t ] a ,t b ]Obtaining a traveling wave expression ofMoving the rope arrangement in a third phase t b ,T]The lateral vibration displacement of (2):
in a third stage [ t ] b ,T]The middle and left moving wave G is divided into traveling waves G 2 Sum traveling wave G 3 The right traveling wave F is divided into traveling waves F 2 And a traveling wave F 3 (ii) a Travelling wave G 3 Is a travelling wave F 2 At right border x = l 0 A reflected wave of (d); travelling wave F 3 Is a travelling wave G 2 Reflected waves at the left boundary x = 0;
the third stage [ t ] is obtained from formula (8), formula (9), formula (16) and formula (22) b ,T]The left traveling wave and the right traveling wave in (1) are represented by the following expressions (26) and (27):
the left traveling wave and the right traveling wave are superposed by the equations (26) and (27) to obtain a third stage [ t ] b ,T]The vibrational displacement response of the rope arrangement of (a), as characterized by equation (28):
u(x,t)＝F(x-v r t)+G(x+v l t),t b ＜t＜T (28)
the expression (12), the expression (25) and the expression (28) are piecewise analytical expressions of the transverse vibration displacement of the rope device in one vibration period T.
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