CN112347576A - Vibration energy calculation method of axial movement rope equipment under mixed boundary condition - Google Patents
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Abstract
The invention discloses a vibration energy calculation method of axial moving rope equipment under a mixed boundary condition, which is characterized in that a motion equation of the axial moving rope equipment is obtained according to the Hamilton principle, and the displacement response of the motion equation is expressed as the superposition of two traveling waves; deriving expressions of two initial traveling waves according to the motion initial conditions of the axial movement rope equipment, obtaining a constraint equation of a mixed boundary, and then combining the constraint equation of the mixed boundary with the motion equation to obtain a reflected wave response equation of each stage; and finally representing 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 calculating the energy of the vibration displacement response obtained under the mixed boundary condition and various speed working conditions of the movable rope equipment, and can analyze the 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 movement 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 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 especially lateral vibration has a great impact 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 darbeyer 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-length chord. In practical engineering application of axial moving rope equipment, traveling waves in different directions can be reflected for multiple times at the boundary of the limited-length moving rope equipment under the condition of a complex mixed boundary 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 superposition of the multiple reflections of the traveling waves in the fixed-length moving rope equipment under the constraint condition of the mixed boundary cannot be solved by using a method of the Dalnbell principle.
Disclosure of Invention
The invention aims to avoid the defects of the prior art and provides a vibration energy calculation method of axial movement 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 movement rope equipment and the problem of instability of the vibration displacement response due to the increase of the movement speed; therefore, the method is suitable for calculating the energy of the vibration displacement response obtained under the mixed boundary condition and various speed working conditions of the movable rope device, and can analyze the energy change of the rope device under various working conditions.
The invention adopts the following technical scheme for solving the technical problems:
the invention relates to a vibration energy calculation method of axial moving rope equipment under a mixed boundary condition, wherein the mixed boundary condition is that one end boundary of two end boundaries of the axial moving rope equipment is an atypical boundary, the other end boundary is a typical boundary, the atypical boundary is used as a coordinate origin, the axial moving direction of the axial rope equipment is used as an x direction, and the transverse vibration direction is used as a u direction, so that a fixed coordinate system is established; the method is characterized in that: the vibration energy calculation method comprises the following steps:
step 1, obtaining a motion equation of the axial moving rope equipment by using a formula (1) according to a Hamilton principle, and expressing a solution u (x, t) of the motion equation of the axial moving rope equipment as the 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;
utt+2vuxt-(c2-v2)uxx＝0 (1)
in the formula (1), uttIs the second partial derivative of the rope lateral vibration displacement u with respect to time t; v represents the axial movement speed of the axially moving rope arrangement; u. ofxtThe first-order partial derivatives of the transverse vibration displacement u of the rope on the axial coordinate x and the time t of the axial moving rope device in a fixed coordinate system are respectively obtained; c represents the velocity of the traveling wave; u. ofxxIs the second partial derivative of the rope transverse vibration displacement u to the axial coordinate x of the axially moving rope arrangement;
u(x,t)＝F(x-vrt)+G(x+vlt) (2)
in the formula (2), vrIs the velocity of the right traveling wave relative to a fixed coordinate system; v. oflIs the velocity of the left traveling wave relative to a fixed coordinate system; f (x-v)rt) represents a velocity vrRight traveling wave of (2); g (x + v)lt) represents a velocity vlThe left traveling wave of (a);
step 2, let the atypical boundary at x ═ 0 be the spring-damping boundary, at x ═ l0The typical boundary at (a) is a fixed boundary, so that a mixed boundary constraint equation set is obtained by using the equation (3);
in equation (3), u (0, t) represents the vibrational displacement of the axially moving rope device at x ═ 0; u. oft(l0And t) is represented by0The first partial derivative of the transverse vibration displacement u of the cable to the time t; u. ofx(l0And t) is represented by0The first partial derivative of the transverse vibration displacement u of the cable to the axial coordinate x; η represents the atypical boundary damping coefficient of the axially moving rope device; 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 formula (4) and the formula (5):
G(l0+vlt)＝-F(l0-vrt) (4)
in the formula (5), F 'and G' respectively represent the derivatives of the two traveling waves with respect to time t;
two intermediate variables α, β are obtained using equation (6):
obtaining a simplified traveling wave relational expression by using an equation (7):
in the formula (7), s represents the displacement of the traveling right wave, and s ═ vrt；
Obtaining a general traveling wave relational expression by using the formula (8):
in the formula (8), f (x) represents a traveling wave relation expression of the right traveling wave with respect to the axial coordinate x; e.g. of the type-αsRepresents an integration factor;
and 4, giving a motion initial condition formula of the axial movement rope device by using the formula (9):
in equation (9), the function phi (x) is the initial lateral displacement at different positions on the axially moving rope device in a fixed coordinate system; the function psi (x) is the initial velocity of different positions on the axially moving rope device in a fixed coordinate system;
and according to the motion initial condition, obtaining initial expressions of two traveling waves by using an expression (10):
in the formula (10), ξ is an integral variable; g (x) represents the traveling wave relation expression of the left traveling wave relative to the axial coordinate x; c is an integration constant;
and step 5, determining the vibration period T of the axial movement rope equipment by using the formula (11) according to the motion rule of the two traveling waves in the axial movement rope equipment and the reflection superposition rule of the two traveling waves on the boundaries of the two ends of the axial movement rope equipment0；
Step 6, combining the motion equation, the motion initial condition and the solution u (x, T) of the motion equation of the axial moving rope equipment with the mixed boundary constraint equation set respectively, and carrying out vibration period T0The traveling waves of different stages are superposed, so that a transverse vibration displacement response type is obtained;
from the axial coordinate x at travelling-wave displacement1To the axial coordinate x2The left traveling wave G is divided into a first left traveling wave G1And a second left traveling wave G2The right traveling wave F is divided into a first right traveling wave F1And a second right traveling wave F2(ii) a Wherein the second left traveling wave G2Is the first right traveling wave F1At the right border x ═ l0Reflected wave of (d), second right travelingWave F2Is the first left traveling wave G1A reflected wave at the left boundary x ═ 0;
obtaining a first right traveling wave F by equation (12)1And a first left traveling wave G1Expression (c):
in the formula (12), F1(x-vrt) represents a velocity vrA first right traveling wave of (a); g1(x+vlt) is velocity vlA first left-shifted wave of (a); c is an integral constant;
the continuity condition of the two traveling waves at the atypical boundary where x ═ 0 is constructed using equation (13):
from equations (3) and (13), the second leftward traveling wave G is obtained by equations (14) and (15), respectively2And a second right traveling wave F2：
In formulae (14) and (15), G2(x+vlt) is velocity vlA second left traveling wave of (a); f2(x-vrt) is velocity vrA second right traveling wave of (a);
step 7, according to the first stage [0, ta]The traveling wave expression in (1) is obtained in the first stage [0, ta]The median traveling wave energy expression, whereby the total vibration energy of the axially moving rope arrangement is calculated:
step 7.1 of obtaining a second right traveling wave F2 by using the formula (16)Vibration energy
Step 7.2, obtaining the vibration energy of the first right traveling wave F1 by using the formula (17)
Step 7.3, utilizing the first left traveling wave G of the formula (18)1Vibration energy of
Step 7.4, obtaining a second left traveling wave G by using the formula (19)2Vibration energy of
And 7.5, obtaining total vibration energy E (t) by using the formula (20):
in the formula (20), Ek(t) is the potential energy of the spring, and
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.
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 energy expression of each traveling wave is represented through each traveling wave expression in three stages of one vibration period, so that the total vibration energy of the rope equipment can be calculated. The method is suitable for calculating the energy of the vibration displacement response obtained under the mixed boundary condition and various speed working conditions of the movable rope equipment, and the method for analyzing the energy change of the rope equipment under various working conditions can meet the requirement of testing the feasibility and the effectiveness of various numerical calculation methods for transverse vibration of the axial movable rope equipment.
Drawings
FIG. 1 is a spring-damper-fixed hybrid boundary model of the present invention;
FIG. 1a shows the first stage [0, t ] of the present inventiona]A traveling wave reflection superposition diagram;
FIG. 1b shows a second stage [ t ] of the present inventiona，tb]A traveling wave reflection superposition diagram;
FIG. 1c shows the first stage [ t ] of the present inventionb，T0]And (5) a traveling wave reflection superposition schematic diagram.
Detailed Description
In this embodiment, a vibration energy calculation method for an axially moving rope device under a mixed boundary condition is provided. As shown in fig. 1, the mixed boundary condition means that a fixed coordinate system is established with one of boundaries at both ends of the axial moving rope device as an atypical boundary and the other boundary as a typical boundary, with the atypical boundary as a coordinate origin, with the axial moving direction of the axial rope device as an x direction and with the lateral vibration direction as a u direction; the vibration energy calculation method comprises the following steps:
step 1, obtaining a motion equation of axial movement rope equipment by using a formula (1) according to a Hamilton principle, and expressing a solution u (x, t) of the motion equation of the axial movement rope equipment as the 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;
utt+2vuxt-(c2-v2)uxx＝0 (1)
in the formula (1), uttIs the second partial derivative of the rope lateral vibration displacement u with respect to time t; v represents the axial movement speed of the axially moving rope arrangement; u. ofxtThe first-order partial derivatives of the transverse vibration displacement u of the rope to the axial coordinate x and the time t of the axial moving rope device in a fixed coordinate system are respectively obtained; c represents the velocity of the traveling wave; u. ofxxIs the second partial derivative of the rope transverse vibration displacement u to the axial coordinate x of the axially moving rope arrangement;
u(x,t)＝F(x-vrt)+G(x+vlt) (2)
in the formula (2), vrIs the velocity of the right traveling wave relative to a fixed coordinate system; v. oflIs the velocity of the left traveling wave relative to a fixed coordinate system; f (x-v)rt) represents a velocity vrRight traveling wave of (2); g (x + v)lt) represents a velocity vlThe left traveling wave of (a);
step 2, let the atypical boundary at x ═ 0 be the spring-damping boundary, at x ═ l0The typical boundary of (A) is a fixed boundary, and thus is obtained by equation (3)Mixing a boundary constraint equation set;
in equation (3), u (0, t) represents the vibrational displacement of the axially moving rope device at x ═ 0; u. oft(l0And t) is represented by0The first partial derivative of the transverse vibration displacement u of the cable to the time t; u. ofx(l0And t) is represented by0The first partial derivative of the transverse vibration displacement u of the cable to the axial coordinate x; η represents the atypical boundary damping coefficient of the axially moving rope device; 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 formula (4) and the formula (5):
G(l0+vlt)＝-F(l0-vrt) (4)
in the formula (5), F 'and G' respectively represent the derivatives of the two traveling waves with respect to time t;
two intermediate variables α, β are obtained using equation (6):
obtaining a simplified traveling wave relational expression by using an equation (7):
in the formula (7), s represents the displacement of the traveling right wave, and s ═ vrt；
Using an integration factor e-αsTo give formula (8):
obtaining a general traveling wave relational expression by using the formula (9):
in the formula (9), f (x) represents a traveling wave relation expression of the right traveling wave with respect to the axial coordinate x; e.g. of the type-αsRepresents an integration factor;
and 4, setting a motion initial condition formula of the axial movement rope device by using the formula (10):
in the formula (10), the function phi (x) is the initial transverse displacement of different positions on the axially moving rope device in the fixed coordinate system; the function psi (x) is the initial velocity of different positions on the axially moving rope device in a fixed coordinate system;
according to the motion initial condition, an initial expression of two traveling waves is obtained by using an equation (11):
in the formula (11), ξ is an integral variable; g (x) represents the traveling wave relation 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 movement rope equipment by using the formula (12) according to the motion rule of the two traveling waves in the axial movement rope equipment and the reflection superposition rule of the two traveling waves on the boundaries of the two ends of the axial movement rope equipment0；
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, Ta]Second stage [ t ]a,tb]And a third stage [ t ]b,T0]Wherein 0 is<ta<tb<T0；
Step 6, combining the motion equation, the motion initial condition and the solution u (x, T) of the motion equation of the axial moving rope device with the mixed boundary constraint equation set respectively, and carrying out vibration period T0The traveling waves of different stages are superposed, so that a transverse vibration displacement response type is obtained;
step 6.1, as shown in FIG. 1a, first stage [0, ta]In, the left traveling wave G is divided into a first left traveling wave G1And a second left traveling wave G2The right traveling wave F is divided into a first right traveling wave F1And a second right traveling wave F2(ii) a Wherein the second left traveling wave G2Is the first right traveling wave F1At the right border x ═ l0Reflected wave of (F), second right-hand traveling wave2Is the first left traveling wave G1A reflected wave at the left boundary x ═ 0;
obtaining a first right traveling wave F by equation (13)1And a first left traveling wave G1Expression (c):
in the formula (13), F1(x-vrt) represents a velocity vrA first right traveling wave of (a); g1(x+vlt) is velocity vlA first left-shifted wave of (a); c is an integral constant;
the continuity condition of the two traveling waves on the atypical boundary at x ═ 0 is constructed using equation (14):
from equations (3) and (14), the second leftward traveling wave G is obtained by using equations (15) and (16), respectively2And a second right traveling wave F2：
In formulae (15) and (16), G2(x+vlt) is velocity vlA second left traveling wave of (a); f2(x-vrt) is velocity vrA second right traveling wave of (a);
step 6.2, as shown in FIG. 1b, in the second stage [ ta,tb]In, the left traveling wave G is divided into a first left traveling wave G1The second left traveling wave G2And a third left traveling wave G3The right traveling wave F is a second right traveling wave F2(ii) a Second right traveling wave F2For incident waves, third left-shift G3Is the second right shift F2At the right border x ═ l0A reflected wave of (c);
obtaining a third left shift G by equation (17)3And a second right shift F2The relational expression of (1):
in the formula (17), G3(x+vlt) is velocity vlA third left traveling wave of (a);
obtaining a third left shift G by equation (18)3And a first right shift F1The relational expression of (1):
step 6.3, as shown in FIG. 1c, in the third stage [ tb,T0]In, the left moving wave G is divided into a second left moving wave G2And a third left traveling wave G3The right traveling wave F is divided into a second right traveling wave F2And a third right traveling wave F3(ii) a Travelling wave G3Is the second right traveling wave F2At the right border x ═ l0A reflected wave of (c); third right traveling wave F3Is the second left traveling wave G2Reflected waves at the left border x ═ 0;
the third right-hand traveling wave F is obtained by the formula (19)3Expression:
obtaining a second left traveling wave G by using the formula (20)2On the left boundary expression:
obtaining traveling wave F by equation (21)2And F3The continuity conditions of (a) are:
F3(0,tb)＝F2(0,tb) (21)
the third right traveling wave F is obtained by the formula (22)3Expression:
in the formula (22), F3(x-vrt) represents a velocity vrA third right traveling wave of (a);
step 7, according to the first stage [0, ta]The traveling wave expression in (1) is obtained in the first stage [0, ta]The median traveling wave energy expression, whereby the total vibration energy of the axially moving rope arrangement is calculated:
step 7.1 potential energy of spring when spring-damper fastening system is on atypical boundary
The total system energy e (t) of the rope installation is obtained by using equation (23):
respectively deriving x and t by a traveling wave superposition formula of a formula (2) to obtain a formula (24):
in formula (24): u. oftIs the first order partial derivative of u over t; u. ofxIs the first order partial derivative of u over x;
from equations (23) and (24), an energy characterization of the rope can be obtained as shown in equation (25):
derivation from equation (16) yields equation (26):
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, utilizing the first left traveling wave G of the formula (29)1Vibration energy of
Step 7.5, obtaining a second left traveling wave G by using the formula (30)2Vibration energy of
Step 7.6, obtaining a second left traveling wave G by using the formula (31)2Vibration energy of
Step 7.7, obtaining a second left traveling wave G by using the formula (32)2Vibration energy of
And 7.8, obtaining the total vibration energy E (t) of the three stages by using the formula (33):
Claims (1)
1. a vibration energy calculation method of axial movement rope equipment under a mixed boundary condition is characterized in that one end boundary of two end boundaries of the axial movement rope equipment is an atypical boundary, the other end boundary is a typical boundary, the atypical boundary is used as a coordinate origin, the axial movement direction of the axial rope equipment is used as an x direction, and the transverse vibration direction is used as a u direction, and a fixed coordinate system is established; the method is characterized in that: the vibration energy calculation method comprises the following steps:
step 1, obtaining a motion equation of the axial moving rope equipment by using a formula (1) according to a Hamilton principle, and expressing a solution u (x, t) of the motion equation of the axial moving rope equipment as the 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;
utt+2vuxt-(c2-v2)uxx＝0 (1)
in the formula (1), uttIs the second partial derivative of the rope lateral vibration displacement u with respect to time t; v represents the axial movement speed of the axially moving rope arrangement; u. ofxtThe first-order partial derivatives of the transverse vibration displacement u of the rope on the axial coordinate x and the time t of the axial moving rope device in a fixed coordinate system are respectively obtained; c represents the velocity of the traveling wave; u. ofxxIs the second partial derivative of the rope transverse vibration displacement u to the axial coordinate x of the axially moving rope arrangement;
u(x,t)＝F(x-vrt)+G(x+vlt) (2)
in the formula (2), vrIs the velocity of the right traveling wave relative to a fixed coordinate system; v. oflIs the velocity of the left traveling wave relative to a fixed coordinate system; f (x-v)rt) represents a velocity vrRight traveling wave of (2); g (x + v)lt) represents a velocity vlThe left traveling wave of (a);
step 2, orderingThe atypical boundary at x-0 is the spring-damper boundary, at x-l0The typical boundary at (a) is a fixed boundary, so that a mixed boundary constraint equation set is obtained by using the equation (3);
in equation (3), u (0, t) represents the vibrational displacement of the axially moving rope device at x ═ 0; u. oft(l0And t) is represented by0The first partial derivative of the transverse vibration displacement u of the cable to the time t; u. ofx(l0And t) is represented by0The first partial derivative of the transverse vibration displacement u of the cable to the axial coordinate x; η represents the atypical boundary damping coefficient of the axially moving rope device; 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 formula (4) and the formula (5):
G(l0+vlt)＝-F(l0-vrt) (4)
in the formula (5), F 'and G' respectively represent the derivatives of the two traveling waves with respect to time t;
two intermediate variables α, β are obtained using equation (6):
obtaining a simplified traveling wave relational expression by using an equation (7):
in the formula (7), the reaction mixture is,s represents the displacement of the right traveling wave, and s ═ vrt；
Obtaining a general traveling wave relational expression by using the formula (8):
in the formula (8), f (x) represents a traveling wave relation expression of the right traveling wave with respect to the axial coordinate x; e.g. of the type-αsRepresents an integration factor;
and 4, giving a motion initial condition formula of the axial movement rope device by using the formula (9):
in equation (9), the function phi (x) is the initial lateral displacement at different positions on the axially moving rope device in a fixed coordinate system; the function psi (x) is the initial velocity of different positions on the axially moving rope device in a fixed coordinate system;
and according to the motion initial condition, obtaining initial expressions of two traveling waves by using an expression (10):
in the formula (10), ξ is an integral variable; g (x) represents the traveling wave relation expression of the left traveling wave relative to the axial coordinate x; c is an integration constant;
and step 5, determining the vibration period T of the axial movement rope equipment by using the formula (11) according to the motion rule of the two traveling waves in the axial movement rope equipment and the reflection superposition rule of the two traveling waves on the boundaries of the two ends of the axial movement rope equipment0；
Step 6, combining the motion equation, the motion initial condition and the solution u (x, T) of the motion equation of the axial moving rope equipment with the mixed boundary constraint equation set respectively, and carrying out vibration period T0The traveling waves of different stages are superposed, so that a transverse vibration displacement response type is obtained;
from the axial coordinate x at travelling-wave displacement1To the axial coordinate x2The left traveling wave G is divided into a first left traveling wave G1And a second left traveling wave G2The right traveling wave F is divided into a first right traveling wave F1And a second right traveling wave F2(ii) a Wherein the second left traveling wave G2Is the first right traveling wave F1At the right border x ═ l0Reflected wave of (F), second right-hand traveling wave2Is the first left traveling wave G1A reflected wave at the left boundary x ═ 0;
obtaining a first right traveling wave F by equation (12)1And a first left traveling wave G1Expression (c):
in the formula (12), F1(x-vrt) represents a velocity vrA first right traveling wave of (a); g1(x+vlt) is velocity vlA first left-shifted wave of (a); c is an integral constant;
the continuity condition of the two traveling waves at the atypical boundary where x ═ 0 is constructed using equation (13):
from equations (3) and (13), the second leftward traveling wave G is obtained by equations (14) and (15), respectively2And a second right traveling wave F2：
In formulae (14) and (15), G2(x+vlt) is velocity vlA second left traveling wave of (a); f2(x-vrt) is velocity vrA second right traveling wave of (a);
step 7, according to the first stage [0, ta]The traveling wave expression in (1) is obtained in the first stage [0, ta]The median traveling wave energy expression, whereby the total vibration energy of the axially moving rope arrangement is calculated:
step 7.1, obtaining the vibration energy of the second right traveling wave F2 by using the formula (16)
Step 7.2, obtaining the vibration energy of the first right traveling wave F1 by using the formula (17)
Step 7.3, utilizing the first left traveling wave G of the formula (18)1Vibration energy of
Step 7.4, obtaining a second left traveling wave G by using the formula (19)2Vibration energy of
And 7.5, obtaining total vibration energy E (t) by using the formula (20):
in the formula (20), Ek(t) is the potential energy of the spring, and
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN113642127A (en) * | 2021-08-13 | 2021-11-12 | 合肥工业大学 | Multi-cycle calculation method for vibration and energy of axial moving rope equipment |
Citations (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2004257439A (en) * | 2003-02-25 | 2004-09-16 | National Institute Of Advanced Industrial & Technology | Vibration suppressing device and vibration suppressing method |
| CN103991767A (en) * | 2013-02-14 | 2014-08-20 | 三菱电机株式会社 | Elevator apparatus and rope sway suppressing method therefor |
| CN106227934A (en) * | 2016-07-21 | 2016-12-14 | 合肥工业大学 | The method accurately obtaining elongated degree rope equipment oscillation crosswise |
| CN106250678A (en) * | 2016-07-21 | 2016-12-21 | 合肥工业大学 | Accurately obtain the robust method moving axially rope equipment oscillation crosswise |
| CN107808038A (en) * | 2017-10-12 | 2018-03-16 | 宁波大学 | A kind of method for solving of Arbitrary Boundaries constraints drag-line oscillation crosswise frequency |
| CN109614745A (en) * | 2018-12-28 | 2019-04-12 | 合肥工业大学 | Obtain the method that rope equipment oscillation crosswise is moved axially under mixed boundary condition |
| JP2020132427A (en) * | 2019-02-25 | 2020-08-31 | フジテック株式会社 | Rope vibration amount detection device |
| CN111859640A (en) * | 2020-07-06 | 2020-10-30 | 上海交通大学 | String vibration displacement prediction method and system based on wavelet picard iteration method |
-
2020
- 2020-11-03 CN CN202011212518.3A patent/CN112347576B/en active Active
Patent Citations (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2004257439A (en) * | 2003-02-25 | 2004-09-16 | National Institute Of Advanced Industrial & Technology | Vibration suppressing device and vibration suppressing method |
| CN103991767A (en) * | 2013-02-14 | 2014-08-20 | 三菱电机株式会社 | Elevator apparatus and rope sway suppressing method therefor |
| CN106227934A (en) * | 2016-07-21 | 2016-12-14 | 合肥工业大学 | The method accurately obtaining elongated degree rope equipment oscillation crosswise |
| CN106250678A (en) * | 2016-07-21 | 2016-12-21 | 合肥工业大学 | Accurately obtain the robust method moving axially rope equipment oscillation crosswise |
| CN107808038A (en) * | 2017-10-12 | 2018-03-16 | 宁波大学 | A kind of method for solving of Arbitrary Boundaries constraints drag-line oscillation crosswise frequency |
| CN109614745A (en) * | 2018-12-28 | 2019-04-12 | 合肥工业大学 | Obtain the method that rope equipment oscillation crosswise is moved axially under mixed boundary condition |
| JP2020132427A (en) * | 2019-02-25 | 2020-08-31 | フジテック株式会社 | Rope vibration amount detection device |
| CN111859640A (en) * | 2020-07-06 | 2020-10-30 | 上海交通大学 | String vibration displacement prediction method and system based on wavelet picard iteration method |
Non-Patent Citations (2)
| Title |
|---|
| CHEN ENWEI: "A superposition method of reflected wave for moving string vibration with onclassical boundary", TAYLOR & FRANCIS, vol. 42, no. 4 * |
| 陈恩伟;张凯;王军;魏浩征;陆益民;: "固定-阻尼边界定长轴向移动绳振动的行波边界反射叠加法", 振动工程学报, no. 05 * |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113642127A (en) * | 2021-08-13 | 2021-11-12 | 合肥工业大学 | Multi-cycle calculation method for vibration and energy of axial moving rope equipment |
| CN113642127B (en) * | 2021-08-13 | 2024-03-01 | 合肥工业大学 | Multi-period calculation method for vibration and energy of axially moving rope equipment |
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