CN106227934B - Accurately obtain the method for elongated degree rope equipment oscillation crosswise - Google Patents
Accurately obtain the method for elongated degree rope equipment oscillation crosswise Download PDFInfo
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Abstract
The invention discloses the methods that one kind accurately obtains elongated degree rope equipment oscillation crosswise, it moves axially the rope equipment that equipment is axial movement, the movement operating condition of rope equipment changes linearly for rope lengths, it is characterized in that the vibration period will be divided into three phases according to traveling wave locomotion rule in rope equipment；Obtain the boundary incidence wave and boundary echo of the rope equipment of three phases respectively in conjunction with traveling wave movement law in rope equipment and boundary condition；Boundary incidence wave and boundary echo are overlapped, obtain the rope equipment respectively in the lateral displacement of three phases.The Various Complex boundary condition for the mobile rope equipment that the present invention is changed linearly suitable for rope lengths and a variety of speed operating conditions, the vibration displacement response of acquisition is accurate, is able to satisfy inspection axial movement cord traverse and vibrates the feasibility of a variety of numerical computation methods and the needs of validity.
Description
Technical field
The invention belongs to machinery system dynamics modeling and vibration control fields, accurately obtain more particularly to one kind
The method for taking elongated degree rope equipment oscillation crosswise.
Background technique
Move axially rope equipment have operation efficiently, adaptive strong, bearing capacity is big, structure is simple, flexible controllable etc. excellent
Point, on numerous engineering fields have highly important application value, as tethered satellite hawser, dynamic conveyor belt, elevator cable,
Passenger and freight cableway etc..Noise and vibration along with the operating of these equipment, especially oscillation crosswise to the functions of these equipment and
Safety causes very big influence.The Transverse Vibration of A Roller of axial movement rope equipment is one and has been studied many years and has
The project of challenge, is still widely noticed so far.Traditional investigative technique is the partial differential driving side established based on Hamiton's principle
Journey and the finite elements kinetics equation established based on Lagrange's equation, are then utilized respectively numerical computation method, such as gal
Distant gold method, runge kutta method, Newmark method and time-varying state space equation method etc. solve above equation, are moved axially
The oscillation crosswise of rope model responds.Above method, when moving axially, rope device rate is higher, is close to or up to critical speed
When, vibration equipment amplitude can be made to increase extremely, error is caused to increase.DAlembert principle points out that the uniform string of endless is lateral
Vibration can be expressed as the superposition of two traveling waves in opposite direction, to obtain mobile rope equipment transverse-vibration using wave superposition theory
It is dynamic to have established theoretical basis, its advantage is that the vibratory response not unstability due to increase of movement speed.But the principle lays particular emphasis on solution
Vibration and energy variation characteristic of the traveling wave in the different boundary individual reflection of semi-infinite long string.It is set in actually axial movement rope
In standby engineer application, the traveling wave of different directions is having the mobile rope boundary of limit for length that multiple reflections can occur, and folded with incidence wave
Add the oscillation crosswise for constituting mobile rope, it is clear that not can solve limited long mobile rope using the method for dAlembert principle and set
Standby middle traveling wave multiple reflections are superimposed the accurate acquisition problem for the oscillation crosswise to be formed.
Summary of the invention
The present invention is to provide one kind to avoid above-mentioned existing deficiencies in the technology and accurately obtain elongated degree rope
The method of equipment oscillation crosswise can be suitable for avoid numerical computation method and the limitation of dAlembert principle method
The elongated a variety of typical boundary conditions of degree rope equipment.
The present invention adopts the following technical scheme that in order to solve the technical problem
The method that the present invention accurately obtains elongated degree rope equipment oscillation crosswise, the axial movement equipment are axial movement
Rope equipment, the movement operating condition of the rope equipment changes linearly for rope lengths, its main feature is that the method is by as follows
Process carries out:
The motion model for determining rope equipment obtains the equation of motion of the rope equipment；
Determine the movement primary condition and boundary condition of rope equipment；
Determine the vibration period T of rope equipment；
The vibration period T is divided for three phases according to traveling wave locomotion rule in the rope equipment: being [t respectivelyn-1,
tan], [tan,tbn] and [tbn,tn], wherein tn-1<tan<tbn<tn；
Obtain the boundary of the rope equipment of three phases respectively in conjunction with traveling wave movement law in rope equipment and boundary condition
Incidence wave and boundary echo；The boundary incidence wave and boundary echo are overlapped, obtain the rope equipment respectively
In the lateral displacement of three phases.
The present invention accurately obtain it is elongated degree rope equipment oscillation crosswise method the characteristics of be to carry out as follows:
Step 1: giving rope equipment according to the parameter including the line density of rope equipment, movement speed and tension
Motion model, for the motion model according to Hamiton's principle establish formula (1) shown in rope equipment the equation of motion:
ωtt+2vωxt+(v2-c2)ωxx=0 (1)
In formula (1):
X is the axial coordinate of rope equipment；T is the time；V is the axial movement speed of rope equipment；Row wave speed c is row
Spread speed of the wave in rope equipment, c=(P/ ρ)0.5, P is the tension of rope equipment；ρ is the line density of rope equipment；
ω is the transverse vibrational displacement of rope equipment, and ω is the function of x and t, ω=ω (x, t)；
ωttIt is second-order partial differential coefficient of the ω to t；ωxxIt is second-order partial differential coefficient of the ω to x；
ωxtIt is ω respectively to x and to the first-order partial derivative of t；
By the solution of formula (1), i.e. transverse vibrational displacement ω (x, t) is considered as the superposition for moving to right traveling wave and moving to left traveling wave, such as formula
(2):
ω (x, t)=F (x-vrt)+G(x+vlt) (2)
In formula (2):
vlFor the speed for moving to left traveling wave in the rope equipment relative to fixed coordinate system, vl=c-v；
vrFor the speed for moving to right traveling wave in the rope equipment relative to fixed coordinate system, vr=c+v；
F(x-vrIt t) be speed is vrMove to right traveling wave, be denoted as F；
G(x+vlIt t) be speed is vlMove to left traveling wave, be denoted as G；
F and G is any twice continuously differentiable function；
Step 2: determine the movement primary condition and boundary condition of rope equipment:
Set the movement primary condition such as formula (3) when t=0:
In formula (3): ωtIt is ω to the first-order partial derivative of t；FunctionFor different positions in rope equipment in fixed coordinate system
The initial lateral displacement set；Function ψ (x) is the initial velocity of different location in rope equipment in fixed coordinate system；L (t) is rope
Length of the rope equipment in moment t；
It is the fixed form in both ends shown in formula (4) that boundary condition, which is arranged:
Step 3: it is changed linearly for the movement operating condition of rope equipment for rope lengths, has formula (5) as follows:
L (t)=l0+vt (5)
L in formula (5)0For the initial length of rope；The duration T in n-th of periodnAre as follows:
In formula (5), ln-1Rope lengths when for (n-1)th end cycle；
The duration for obtaining each vibration period: T is calculated using formula (5) and formula (6)1, T2…Tn, n is natural number；
Step 4: the vibration period it will be divided into three phases according to the mobile changing rule of traveling wave:
Define tnThe time of original state is circulated back to by n times for system, then is had:
Tn=tn-tn-1 (8)
Work as v > 0, rope moves from left to right, i.e. rope elongation, then in cycle TnInterior three phases are [t respectivelyn-1,
tan], [tan,tbn] and [tbn,tn], wherein tn-1<tan<tbn<tn, it is as follows:
When v < 0, rope moves from right to left, i.e., rope lengths shorten, then the t in three phasesanAnd tbnValue is mutually handed over
It changes；
Step 5: obtaining rope equipment [t in the first stage in conjunction with movement primary condition and boundary conditionn-1,tan] transverse direction
Vibration displacement:
In first stage, traveling wave G points are moved to left as traveling wave G1With traveling wave G2, traveling wave F points are moved to right as traveling wave F1With traveling wave F2；Its
In, traveling wave G2It is traveling wave F1The back wave at the place boundary x=l (t) on the right, traveling wave F2It is traveling wave G1In the anti-of the place of left margin x=0
Ejected wave；Traveling wave F is respectively obtained in conjunction with movement primary condition and formula (2)1With traveling wave G1Expression formula such as formula (10) and formula (11):
Wherein, ξ is integration variable；K is integral constant；
Traveling wave F is obtained in conjunction with boundary condition and formula (2)2, traveling wave G2With traveling wave F1, traveling wave G1Relationship such as formula (12):
Formula (13) and formula (14) are obtained by formula (10), formula (11) and formula (12):
Then, first stage [tn-1,tan] move to right traveling wave F and move to left traveling wave G respectively as formula (15), formula (16) characterize:
The transverse vibrational displacement ω (x, t) of first stage mobile rope is obtained using formula (17)
ω (x, t)=F (x-vrt)+G(x+vlt),0≤x≤l(t)、tn-1≤t≤tan (17)
Step 6: obtaining mobile rope in second stage [t in conjunction with primary condition and boundary conditionan,tbn] oscillation crosswise
Displacement:
In second stage, for v > 0, traveling wave G points are moved to left as traveling wave G1, traveling wave G2With traveling wave G3, moving to right traveling wave F is
Traveling wave F2；Traveling wave F2For incidence wave, G3It is F2The back wave at the place boundary x=l (t) on the right；G is obtained by boundary condition and formula (2)3With
F2Relationship such as formula (18):
G3(x)=- F2(2ln-1-x) (18)
Convolution (13) and formula (18) obtain traveling wave G3(x+vlT) such as formula (19):
For v < 0, moving to left traveling wave G is traveling wave G2, traveling wave F points are moved to right as traveling wave F1, traveling wave F2With traveling wave F3；Traveling wave F3It is
Traveling wave G2Back wave at left margin x=0；F is obtained using boundary condition and formula (2)3And G2Relationship such as formula (20):
Convolution (14) and formula (20) obtain traveling wave F3(x-vrT) such as formula (21):
Then, second stage [tan,tbn] move to right traveling wave F and move to left traveling wave G respectively such as formula (22) and formula (23):
The transverse vibrational displacement ω (x, t) of the mobile rope of second stage is obtained using formula (24):
ω (x, t)=F (x-vrt)+G(x+vlt),0≤x≤l(t)、tan≤t≤tbn (24)
Step 7: obtaining mobile rope in phase III [t in conjunction with primary condition and boundary conditionbn,tn] oscillation crosswise position
It moves:
In the phase III, traveling wave G points are moved to left as traveling wave G2With traveling wave G3, traveling wave F points are moved to right as traveling wave F2With traveling wave F3,
Traveling wave G3It is traveling wave F2The back wave at the place boundary x=l (t) on the right, traveling wave F3It is traveling wave G2Back wave at left margin x=0；
It moves to right traveling wave F and moves to left traveling wave G respectively such as formula (25) and formula (26):
The transverse vibrational displacement ω (x, t) of phase III mobile rope is obtained using formula (27) are as follows:
ω (x, t)=F (x-vrt)+G(x+vlt),0≤x≤l(t)、tbn≤t≤tn (27)。
Compared with the prior art, the invention has the advantages that:
1, the method for the present invention process is simple, will be divided into three ranks the vibration period according to traveling wave locomotion rule in rope equipment
Section, the traveling wave of each stage different directions can reflect on the boundary at both ends；Then in conjunction with the mobile rule of traveling wave in rope equipment
Rule and boundary condition obtain the boundary incidence wave and boundary echo of three phases rope equipment respectively；Finally by boundary incidence wave
It is overlapped with boundary echo, obtains the rope equipment respectively in the lateral displacement of three phases, resulting rope equipment
Vibratory response is more in line with actual conditions.
2, the present invention is according to stringent mathematical expression, and considers the multiple reflections of the traveling wave of different directions, it is ensured that is obtained
The accuracy of the lateral displacement obtained.
3, the present invention can be adjusted boundary condition according to different boundaries, be suitble to a variety of typical boundary conditions.
Detailed description of the invention
(a) is initial time cord traverse vibration displacement in Fig. 1, (b) is initial left lateral wave, (c) is initial right lateral wave；Figure
(b's) and (c) is superposed to (a) in 1；
(a) is the first stage [0, t in Fig. 2a] cord traverse vibration displacement；It (b) is left lateral wave G1It is sent out at left margin x=0
The state of raw reflection, and obtain right lateral wave F2, (c) it is right lateral wave F1The state that the place boundary x=l (t) is reflected on the right, and
To left lateral wave G2；(b's) and (c) is superposed to (a) in Fig. 2；
When (a) is the axial movement speed v > 0 of rope equipment in Fig. 3, second stage [ta,tb] cord traverse vibration displacement,
It (b) is left lateral wave G1Occur to reflect at left margin x=0 and obtains right lateral wave F2, right lateral wave F2The place boundary x=l (t) is sent out on the right
It is raw to reflect and obtain left lateral wave G3State, be (c) the right lateral wave F in the first stage1The place boundary x=l (t) all occurs on the right
Left lateral wave G is obtained after transmitting2State；(b's) and (c) is superposed to (a) in Fig. 3；
When (a) is the axial movement speed v < 0 of rope equipment in Fig. 4, second stage [ta,tb] cord traverse vibration displacement；
It (b) is the left lateral wave G in the first stage1Right lateral wave F is obtained after all reflecting at left margin x=02State, (c) be
Right lateral wave F1The place boundary x=l (t) occurs to reflect and obtains left lateral wave G on the right2, left lateral wave G2It is reflected at left margin x=0
And obtain right lateral wave F3State；(b's) and (c) is superposed to (a) in Fig. 4；
(a) is phase III [t in Fig. 5b, T] and cord traverse vibration displacement；It (b) is the left lateral wave G in the first stage1On a left side
Right lateral wave F is obtained after all reflecting at the x=0 of boundary2, right lateral wave F2The place boundary x=l (t) occurs to reflect and obtains a left side on the right
Traveling wave G3State, (c) be right lateral wave F1The place boundary x=l (t) obtains left lateral wave G after all reflecting on the right2, left lateral wave G2
Occur to reflect at left margin x=0 and obtains right lateral wave F3State；(b's) and (c) is superposed to (a) in Fig. 5；
Specific embodiment
The present embodiment accurately obtains in the method for elongated degree rope equipment oscillation crosswise, and axial movement equipment is axial movement
Rope equipment, the characteristics of movement operating condition of rope equipment changes linearly for rope lengths, the method for the present invention is by following mistake
Cheng Jinhang:
It determines the motion model of rope equipment, obtains the equation of motion of rope equipment；
Determine the movement primary condition and boundary condition of rope equipment；
Determine the vibration period T of rope equipment；
The vibration period T is divided for three phases according to traveling wave locomotion rule in the rope equipment: being [t respectivelyn-1,
tan], [tan,tbn] and [tbn,tn], wherein tn-1<tan<tbn<tn；
Obtain the boundary of the rope equipment of three phases respectively in conjunction with traveling wave movement law in rope equipment and boundary condition
Incidence wave and boundary echo；The boundary incidence wave and boundary echo are overlapped, obtain the rope equipment respectively
In the lateral displacement of three phases.
The method that elongated degree rope equipment oscillation crosswise is accurately obtained in the present embodiment is to carry out as follows:
Step 1: giving rope equipment according to the parameter including the line density of rope equipment, movement speed and tension
Motion model, for motion model according to Hamiton's principle establish formula (1) shown in rope equipment the equation of motion:
ωtt+2vωxt+(v2-c2)ωxx=0 (1)
In formula (1):
X is the axial coordinate of rope equipment；T is the time；V is the axial movement speed of rope equipment；Row wave speed c is row
Spread speed of the wave in rope equipment, c=(P/ ρ)0.5, P is the tension of rope equipment；ρ is the line density of rope equipment；
ω is the transverse vibrational displacement of rope equipment, and ω is the function of x and t, ω=ω (x, t)；
ωttIt is second-order partial differential coefficient of the ω to t；ωxxIt is second-order partial differential coefficient of the ω to x；
ωxtIt is ω respectively to x and to the first-order partial derivative of t；
As shown in figure 1 shown in (a), (b) and (c), by the solution of formula (1), i.e., transverse vibrational displacement ω (x, t), which is considered as, moves to right traveling wave
With the superposition for moving to left traveling wave, such as formula (2):
ω (x, t)=F (x-vrt)+G(x+vlt) (2)
In formula (2):
vlFor the speed for moving to left traveling wave in the rope equipment relative to fixed coordinate system, vl=c-v；
vrFor the speed for moving to right traveling wave in the rope equipment relative to fixed coordinate system, vr=c+v；
F(x-vrIt t) be speed is vrMove to right traveling wave, be denoted as F；
G(x+vlIt t) be speed is vlMove to left traveling wave, be denoted as G；
F and G is any twice continuously differentiable function；
Step 2: determine the movement primary condition and boundary condition of rope equipment:
Set the movement primary condition such as formula (3) when t=0:
In formula (3): ωtIt is ω to the first-order partial derivative of t；FunctionFor different positions in rope equipment in fixed coordinate system
The initial lateral displacement set；Function ψ (x) is the initial velocity of different location in rope equipment in fixed coordinate system；L (t) is rope
Length of the rope equipment in moment t；
It is the fixed form in both ends shown in formula (4) that boundary condition, which is arranged:
Step 3: it is changed linearly for the movement operating condition of rope equipment for rope lengths, has formula (5) as follows:
L (t)=l0+vt (5)
L in formula (5)0For the initial length of rope；The duration T in n-th of periodnAre as follows:
In formula (5), ln-1Rope lengths when for (n-1)th end cycle；
The duration for obtaining each vibration period: T is calculated using formula (5) and formula (6)1, T2…Tn, n is natural number；
Step 4: the vibration period it will be divided into three phases according to the mobile changing rule of traveling wave:
Define tnThe time of original state is circulated back to by n times for system, then is had:
Tn=tn-tn-1 (8)
Work as v > 0, rope moves from left to right, i.e. rope elongation, then in cycle TnInterior three phases are [t respectivelyn-1,
tan], [tan,tbn] and [tbn,tn], wherein tn-1<tan<tbn<tn, it is as follows:
When v < 0, rope moves from right to left, i.e., rope lengths shorten, then the t in three phasesanAnd tbnValue is mutually handed over
It changes, as shown in formula (9-1):
Step 5: obtaining rope equipment [t in the first stage in conjunction with movement primary condition and boundary conditionn-1,tan] transverse direction
Vibration displacement:
In first stage, in Fig. 2 shown in (a), (b) and (c), traveling wave G points are moved to left as traveling wave G1With traveling wave G2, move to right row
Wave F points are traveling wave F1With traveling wave F2；Wherein, traveling wave G2It is traveling wave F1The back wave at the place boundary x=l (t) on the right, in Fig. 2 (c)
It is shown；Traveling wave F2It is traveling wave G1Back wave at the place of left margin x=0, in Fig. 2 shown in (b)；In conjunction with movement primary condition and
Formula (2) respectively obtains traveling wave F1With traveling wave G1Expression formula such as formula (10) and formula (11):
Wherein, ξ is integration variable；K is integral constant；
Traveling wave F is obtained in conjunction with boundary condition and formula (2)2, traveling wave G2With traveling wave F1, traveling wave G1Relationship such as formula (12):
Formula (13) and formula (14) are obtained by formula (10), formula (11) and formula (12):
Then, first stage [tn-1,tan] move to right traveling wave F and move to left traveling wave G respectively as formula (15), formula (16) characterize:
The transverse vibrational displacement ω (x, t) of first stage mobile rope is obtained using formula (17)
ω (x, t)=F (x-vrt)+G(x+vlt),0≤x≤l(t)、tn-1≤t≤tan (17)
Step 6: obtaining mobile rope in second stage [t in conjunction with primary condition and boundary conditionan,tbn] oscillation crosswise
Displacement, the traveling wave locomotion state in this stage is related with rope equipment moving direction, therefore v>0 and v<0 two kinds of situation is divided to carry out
It discusses:
In second stage, for v > 0, in Fig. 3 shown in (a), (b) and (c), traveling wave G points are moved to left as traveling wave G1, traveling wave
G2With traveling wave G3, moving to right traveling wave F is traveling wave F2；Traveling wave F2For incidence wave, G3It is F2The back wave at the place boundary x=l (t) on the right,
In Fig. 3 shown in (b)；G is obtained by boundary condition and formula (2)3And F2Relationship such as formula (18):
G3(x)=- F2(2ln-1-x) (18)
Convolution (13) and formula (18) obtain traveling wave G3(x+vlT) such as formula (19):
For v < 0, in Fig. 4 shown in (a), (b) and (c), moving to left traveling wave G is traveling wave G2, traveling wave F points are moved to right as traveling wave
F1, traveling wave F2With traveling wave F3；Traveling wave F3It is traveling wave G2Back wave at left margin x=0, in Fig. 4 shown in (c)；Utilize boundary
Condition and formula (2) obtain F3And G2Relationship such as formula (20):
Convolution (14) and formula (20) obtain traveling wave F3(x-vrT) such as formula (21):
Then, second stage [tan,tbn] move to right traveling wave F and move to left traveling wave G respectively such as formula (22) and formula (23):
The transverse vibrational displacement ω (x, t) of the mobile rope of second stage is obtained using formula (24):
ω (x, t)=F (x-vrt)+G(x+vlt),0≤x≤l(t)、tan≤t≤tbn (24)
Step 7: obtaining mobile rope in phase III [t in conjunction with primary condition and boundary conditionbn,tn] oscillation crosswise position
It moves, the traveling wave locomotion state in this stage is unrelated with rope equipment moving direction, but for different moving directions, second
When step transition is to three phases, the change procedure of traveling wave locomotion state is different, as v > 0, (b), (c) in Fig. 3
(b), (c) are compared one by one in Fig. 5；As v < 0, (b), (c) in Fig. 4 in (b), (c) and Fig. 5 are compared one by one:
In the phase III, in Fig. 5 shown in (a), (b) and (c), traveling wave G points are moved to left as traveling wave G2With traveling wave G3, move to right
Traveling wave F points are traveling wave F2With traveling wave F3, traveling wave G3It is traveling wave F2The back wave at the place boundary x=l (t) on the right, (c) institute in Fig. 5
Show；Traveling wave F3It is traveling wave G2Back wave at left margin x=0, in Fig. 5 shown in (b)；It moves to right traveling wave F and moves to left traveling wave G and divide
Not such as formula (25) and formula (26):
The transverse vibrational displacement ω (x, t) of phase III mobile rope is obtained using formula (27) are as follows:
ω (x, t)=F (x-vrt)+G(x+vlt),0≤x≤l(t)、tbn≤t≤tn (27)。
The method of the present invention process is simple, and the rope vibration equipment response of acquisition is more in line with actual conditions, avoids numerical value
The limitation of calculation method and dAlembert principle method is suitable for the elongated a variety of typical boundary conditions of degree rope equipment.
Claims (1)
1. the method that one kind accurately obtains elongated degree rope equipment oscillation crosswise, axial movement equipment is that the rope of axial movement is set
Standby, the movement operating condition of the rope equipment changes linearly for rope lengths, it is characterized in that the method carries out according to the following procedure:
The motion model for determining rope equipment obtains the equation of motion of the rope equipment；
Determine the movement primary condition and boundary condition of rope equipment；
Determine the vibration period T of rope equipment；
The vibration period T is divided for three phases according to traveling wave locomotion rule in the rope equipment: being [t respectivelyn-1,tan],
[tan,tbn] and [tbn,tn], wherein tn-1<tan<tbn<tn；
The boundary for obtaining the rope equipment of three phases respectively in conjunction with traveling wave movement law in rope equipment and boundary condition is incident
Wave and boundary echo；The boundary incidence wave and boundary echo are overlapped, obtain the rope equipment respectively three
The lateral displacement in a stage；
The method for accurately obtaining elongated degree rope equipment oscillation crosswise carries out as follows:
Step 1: the movement of rope equipment is given according to the parameter including the line density of rope equipment, movement speed and tension
Model, for the motion model according to Hamiton's principle establish formula (1) shown in rope equipment the equation of motion:
ωtt+2vωxt+(v2-c2)ωxx=0 (1)
In formula (1):
X is the axial coordinate of rope equipment；T is the time；V is the axial movement speed of rope equipment；Row wave speed c is that traveling wave exists
Spread speed in rope equipment, c=(P/ ρ)0.5, P is the tension of rope equipment；ρ is the line density of rope equipment；
ω is the transverse vibrational displacement of rope equipment, and ω is the function of x and t, ω=ω (x, t)；
ωttIt is second-order partial differential coefficient of the ω to t；ωxxIt is second-order partial differential coefficient of the ω to x；
ωxtIt is ω respectively to x and to the first-order partial derivative of t；
By the solution of formula (1), i.e. transverse vibrational displacement ω (x, t) is considered as the superposition for moving to right traveling wave and moving to left traveling wave, such as formula (2):
ω (x, t)=F (x-vrt)+G(x+vlt) (2)
In formula (2):
vlFor the speed for moving to left traveling wave in the rope equipment relative to fixed coordinate system, vl=c-v；
vrFor the speed for moving to right traveling wave in the rope equipment relative to fixed coordinate system, vr=c+v；
F(x-vrIt t) be speed is vrMove to right traveling wave, be denoted as F；
G(x+vlIt t) be speed is vlMove to left traveling wave, be denoted as G；
F and G is any twice continuously differentiable function；
Step 2: determine the movement primary condition and boundary condition of rope equipment:
Set the movement primary condition such as formula (3) when t=0:
In formula (3): ωtIt is ω to the first-order partial derivative of t；FunctionFor different location in rope equipment in fixed coordinate system
Initial lateral displacement；Function ψ (x) is the initial velocity of different location in rope equipment in fixed coordinate system；L (t) sets for rope
The standby length in moment t；
It is the fixed form in both ends shown in formula (4) that boundary condition, which is arranged:
Step 3: it is changed linearly for the movement operating condition of rope equipment for rope lengths, has formula (5) as follows:
L (t)=l0+vt (5)
L in formula (5)0For the initial length of rope；The duration T in n-th of periodnAre as follows:
In formula (5), ln-1Rope lengths when for (n-1)th end cycle；
The duration for obtaining each vibration period: T is calculated using formula (5) and formula (6)1, T2...Tn, n is natural number；
Step 4: the vibration period it will be divided into three phases according to the mobile changing rule of traveling wave:
Define tnThe time of original state is circulated back to by n times for system, then is had:
Tn=tn-tn-1 (8)
Work as v > 0, rope moves from left to right, i.e. rope elongation, then in cycle TnInterior three phases are [t respectivelyn-1,tan],
[tan,tbn] and [tbn,tn], wherein tn-1<tan<tbn<tn, it is as follows:
When v < 0, rope moves from right to left, i.e., rope lengths shorten, then the t in three phasesanAnd tbnValue is exchanged with each other；
Step 5: obtaining rope equipment [t in the first stage in conjunction with movement primary condition and boundary conditionn-1,tan] oscillation crosswise
Displacement:
In first stage, traveling wave G points are moved to left as traveling wave G1With traveling wave G2, traveling wave F points are moved to right as traveling wave F1With traveling wave F2；Wherein, row
Wave G2It is traveling wave F1The back wave at the place boundary x=l (t) on the right, traveling wave F2It is traveling wave G1Back wave at the place of left margin x=0；
Traveling wave F is respectively obtained in conjunction with movement primary condition and formula (2)1With traveling wave G1Expression formula such as formula (10) and formula (11):
Wherein, ξ is integration variable；K is integral constant；
Traveling wave F is obtained in conjunction with boundary condition and formula (2)2, traveling wave G2With traveling wave F1, traveling wave G1Relationship such as formula (12):
Formula (13) and formula (14) are obtained by formula (10), formula (11) and formula (12):
Then, first stage [tn-1,tan] move to right traveling wave F and move to left traveling wave G respectively as formula (15), formula (16) characterize:
The transverse vibrational displacement ω (x, t) of first stage mobile rope is obtained using formula (17)
ω (x, t)=F (x-vrt)+G(x+vlT), 0≤x≤l (t), tn-1≤t≤tan (17)
Step 6: obtaining mobile rope in second stage [t in conjunction with primary condition and boundary conditionan,tbn] transverse vibrational displacement:
In second stage, for v > 0, traveling wave G points are moved to left as traveling wave G1, traveling wave G2With traveling wave G3, moving to right traveling wave F is traveling wave
F2；Traveling wave F2For incidence wave, G3It is F2The back wave at the place boundary x=l (t) on the right；G is obtained by boundary condition and formula (2)3And F2's
Relationship such as formula (18):
G3(x)=- F2(2ln-1-x) (18)
Convolution (13) and formula (18) obtain traveling wave G3(x+vlT) such as formula (19):
For v < 0, moving to left traveling wave G is traveling wave G2, traveling wave F points are moved to right as traveling wave F1, traveling wave F2With traveling wave F3；Traveling wave F3It is traveling wave
G2Back wave at left margin x=0；F is obtained using boundary condition and formula (2)3And G2Relationship such as formula (20):
Convolution (14) and formula (20) obtain traveling wave F3(x-vrT) such as formula (21):
Then, second stage [tan,tbn] move to right traveling wave F and move to left traveling wave G respectively such as formula (22) and formula (23):
The transverse vibrational displacement ω (x, t) of the mobile rope of second stage is obtained using formula (24):
ω (x, t)=F (x-vrt)+G(x+vlT), 0≤x≤l (t), tan≤t≤tbn (24)
Step 7: obtaining mobile rope in phase III [t in conjunction with primary condition and boundary conditionbn,tn] transverse vibrational displacement:
In the phase III, traveling wave G points are moved to left as traveling wave G2With traveling wave G3, traveling wave F points are moved to right as traveling wave F2With traveling wave F3, traveling wave G3It is capable
Wave F2The back wave at the place boundary x=l (t) on the right, traveling wave F3It is traveling wave G2Back wave at left margin x=0；Move to right traveling wave F and
Traveling wave G is moved to left respectively such as formula (25) and formula (26):
The transverse vibrational displacement ω (x, t) of phase III mobile rope is obtained using formula (27) are as follows:
ω (x, t)=F (x-vrt)+G(x+vlT), 0≤x≤l (t), tbn≤t≤tn (27)。
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| US8162110B2 (en) * | 2008-06-19 | 2012-04-24 | Thyssenkrupp Elevator Capital Corporation | Rope tension equalizer and load monitor |
| CN104280112A (en) * | 2013-01-25 | 2015-01-14 | 合肥工业大学 | Measuring method for axial movement rope transverse vibration measurement system |
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| CN104280112A (en) * | 2013-01-25 | 2015-01-14 | 合肥工业大学 | Measuring method for axial movement rope transverse vibration measurement system |
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