CN106227934A - The method accurately obtaining elongated degree rope equipment oscillation crosswise - Google Patents

Abstract

The method that the invention discloses a kind of accurate acquisition elongated degree rope equipment oscillation crosswise, it moves axially equipment is the rope equipment moved axially, the motion operating mode of rope equipment is that rope lengths linearly changes, and it is characterized in that the vibration period to be divided into three phases according to row wave motion rule in rope equipment；Border incidence wave and the boundary echo of the rope equipment of three phases is obtained respectively in conjunction with row ripple movement law and boundary condition in rope equipment；Border incidence wave and boundary echo are overlapped, obtain the described rope equipment lateral displacement at three phases respectively.The present invention is applicable to the Various Complex boundary condition of the mobile rope equipment that rope lengths linearly changes and multiple speed operating mode, the vibration displacement response obtained accurately, can meet inspection and move axially feasibility and the needs of effectiveness of the cord traverse multiple numerical computation method of vibration.

Description

The method accurately obtaining elongated degree rope equipment oscillation crosswise

Technical field

The invention belongs to machinery system dynamics modeling and vibration control field, accurately obtain more particularly to one The method taking elongated degree rope equipment oscillation crosswise.

Background technology

Move axially rope equipment have operation efficiently, self adaptation is strong, bearing capacity is big, simple in construction, the most controlled etc. excellent Point, has highly important using value on numerous engineering fields, 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 function of these equipment and Safety causes the biggest impact.Moving axially the Transverse Vibration of A Roller of rope equipment is studied a lot of years has Challenging problem, the most concerned.Traditional investigative technique is the partial differential driving side set up based on Hamiton's principle Journey and the finite elements kinetics equation set up based on Lagrange's equation, be then utilized respectively numerical computation method, such as gal The Liao Dynasty's gold method, runge kutta method, Newmark method and time-varying state space equation method etc. solve above equation, it is thus achieved that move axially The oscillation crosswise response of rope model.Above method, when moving axially, rope device rate is higher, close to or up critical velocity Time, the abnormal increase of vibration equipment amplitude can be made, cause error to increase.DAlembert principle points out that the uniform string of a musical instrument of endless is horizontal Vibration can be expressed as the superposition of two row ripples in opposite direction, for utilizing ripple superposition theory to obtain mobile rope equipment transverse-vibration Moving and established theoretical basis, its advantage is the vibratory response not unstability because of the increase of translational speed.But this principle lays particular emphasis on solution Row ripple is in the vibration of the different boundary individual reflection of the semi-infinite long string of a musical instrument and energy variation characteristic.Set at the actual rope that moves axially In standby engineer applied, the row ripple of different directions can occur multiple reflections there being limit for length to move rope boundary, and folds with incidence wave Add the oscillation crosswise constituting mobile rope, it is clear that utilizing the method for dAlembert principle to fail to solution has limit for length to move rope to set The problem that accurately obtains of the oscillation crosswise that standby middle row ripple multiple reflections superposition is formed.

Summary of the invention

The present invention is for avoiding the weak point existing for above-mentioned prior art, it is provided that a kind of accurately elongated degree rope of acquisition The method of equipment oscillation crosswise, to avoid numerical computation method and the limitation of dAlembert principle method so that it is can be adapted to The elongated multiple typical boundary condition of degree rope equipment.

The present invention solves that technical problem adopts the following technical scheme that

The present invention accurately obtains the method for elongated degree rope equipment oscillation crosswise, described in move axially equipment for moving axially Rope equipment, the motion operating mode of described rope equipment is that rope lengths linearly changes, and is characterized in that described method is by as follows Process is carried out:

Determine the motion model of rope equipment, it is thus achieved that the equation of motion of described rope equipment；

Determine motion initial condition and the boundary condition of rope equipment；

Determine the vibration period T of rope equipment；

According to row wave motion rule in described rope equipment, described vibration period T is divided into three phases: be [t respectively_n-1, t_an], [t_an,t_bn] and [t_bn,t_n], wherein t_n-1<t_an<t_bn<t_n；

The border of the rope equipment of three phases is obtained respectively in conjunction with row ripple movement law and boundary condition in rope equipment Incidence wave and boundary echo；Described border incidence wave and boundary echo are overlapped, obtain described rope equipment respectively Lateral displacement at three phases.

The feature of the method that the present invention accurately obtains elongated degree rope equipment oscillation crosswise is to carry out as follows:

Step 1: give rope equipment according to the parameter including line density, translational speed and the tension force of rope equipment Motion model, sets up the equation of motion of the rope equipment shown in formula (1) for described motion model according to Hamiton's principle:

ω_tt+2vω_xt+(v²-c²)ω_xx=0 (1)

In formula (1):

X is the axial coordinate of rope equipment；T is the time；V be rope equipment move axially speed；Row wave speed c is row Ripple spread speed in rope equipment, c=(P/ ρ)^0.5, P is the tension force 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, and ω=ω (x, t)；

ω_ttIt it is the ω second-order partial differential coefficient to t；ω_xxIt it is the ω second-order partial differential coefficient to x；

ω_xtIt is that ω is respectively to x and the first-order partial derivative to t；

By the solution of formula (1), i.e. (x t) is considered as right travelling wave and the superposition of left travelling wave, such as formula to transverse vibrational displacement ω (2):

ω (x, t)=F (x-v_rt)+G(x+v_lt) (2)

In formula (2):

v_lFor the speed relative to travelling wave left in the rope equipment of fixed coordinate system, v_l=c v；

v_rFor the speed relative to travelling wave right in the rope equipment of fixed coordinate system, v_r=c+v；

F(x-v_rT) be speed be v_rRight travelling wave, be designated as F；

G(x+v_lT) be speed be v_lLeft travelling wave, be designated as G；

F and G is any twice continuously differentiable function；

Step 2: determine motion initial condition and the boundary condition of rope equipment:

Motion initial condition such as formula (3) during setting t=0:

In formula (3): ω_tFor the ω first-order partial derivative to t；FunctionFor not coordination on rope equipment in fixed coordinate system The initial lateral displacement put；Function ψ (x) is the initial velocity of diverse location on rope equipment in fixed coordinate system；L (t) is rope Rope equipment is in the length of moment t；

Arranging boundary condition is the form that the two ends shown in formula (4) are fixed:

$\left\{\begin{array}{c}\mathrm{\&omega;}(0,t)=0\\ \mathrm{\&omega;}\left(l\right(t),t)=0\end{array}\right.---\left(4\right)$

Step 3: the motion operating mode for rope equipment is that rope lengths linearly changes, and has formula (5) as follows:

L (t)=l₀+vt (5)

L in formula (5)₀Initial length for rope；The duration T in the n-th cycle_nFor:

$T_n=\frac{2l_{n-1}}{c-v},n=1,2,3,\mathrm{...}---\left(6\right)$

In formula (5), l_n-1It it is rope lengths during (n-1)th end cycle；

$l_{n-1}=\left\{\begin{array}{cc}{l}_0+v{\&Sigma;}_{i=1}^{n-1}{T}_i,& n=2,3,\mathrm{...}\\ l_0,& n=1\end{array}\right.---\left(7\right)$

Formula (5) and formula (6) is utilized to calculate the duration obtaining each vibration period: T₁, T₂…T_n, n is natural number；

Step 4: move Changing Pattern according to row ripple and will be divided into three phases the vibration period:

Definition t_nFor system through being circulated back to the time of original state for n time, then have:

T_n=t_n-t_n-1 (8)

Work as v > 0, rope moves from left to right, i.e. rope elongation, then in cycle T_nInterior three phases is [t respectively_n-1, t_an], [t_an,t_bn] and [t_bn,t_n], wherein t_n-1<t_an<t_bn<t_n, as follows:

$\left\{\begin{array}{c}{t}_{an}=\frac{l_{n-1}}{c}+t_{n-1}\\ t_{bn}=\frac{l_{n-1}}{c-v}+t_{n-1}\\ t_n=\frac{2l_{n-1}}{c-v}+t_{n-1}\end{array}\right.,v>0,t_0=0,n=1,2,3,\mathrm{...}---\left(9\right)$

As v < 0, rope moves from right to left, i.e. rope lengths shortens, then t in three phases_anAnd t_bnValue is mutually handed over Change；

Step 5: combine motion initial condition and boundary condition obtains rope equipment at first stage [t_n-1,t_an] horizontal Vibration displacement:

In first stage, left travelling wave G is divided into row ripple G₁With row ripple G₂, right travelling wave F is divided into row ripple F₁With row ripple F₂；Its In, row ripple G₂It is row ripple F₁The echo at place of boundary x=l (t) on the right, row ripple F₂It is row ripple G₁Anti-at the place of left margin x=0 Ejected wave；Row ripple F is respectively obtained in conjunction with motion initial condition and formula (2)₁With row ripple G₁Expression formula such as formula (10) and formula (11):

Wherein, ξ is integration variable；K is integral constant；

Row ripple F is obtained in conjunction with boundary condition and formula (2)₂, row ripple G₂With row ripple F₁, row ripple G₁Relation such as formula (12):

$\left\{\begin{array}{c}{F}_2\left(x\right)=-G_1(-x\frac{v_l}{v_r})\\ G_2\left(x\right)=-F_1(2l_{n-1}-x)\end{array}\right.---\left(12\right)$

Formula (13) and formula (14) is obtained by formula (10), formula (11) and formula (12):

Then, first stage [t_n-1,t_an] right travelling wave F and left travelling wave G characterizes such as formula (15), formula (16) respectively:

$F(x-v_{r}t)=\left\{\begin{array}{cc}{F}_1(x-v_{r}t),& v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)\\ F_2(x-v_{r}t),& 0\&le;x\&le;v_l(t-t_{n-1})\end{array}\right.---\left(15\right)$

$G(x+v_{l}t)=\{\begin{array}{cc}{G}_1(x+v_{l}t),& 0\&le;x\&le;l_{n-1}-v_l(t-t_{n-1})\\ G_2(x+v_{l}t),& l_{n-1}-v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)\end{array}---\left(16\right)$

Utilize formula (17) obtain the first stage move rope transverse vibrational displacement ω (x, t)

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_n-1≤t≤t_an (17)

Step 6: combine initial condition and boundary condition obtains mobile rope at second stage [t_an,t_bn] oscillation crosswise Displacement:

In second stage, for v > 0, left travelling wave G is divided into row ripple G₁, row ripple G₂With row ripple G₃, right travelling wave F is Row ripple F₂；Row ripple F₂For incidence wave, G₃It is F₂The echo at place of boundary x=l (t) on the right；G is obtained by boundary condition and formula (2)₃With F₂Relation such as formula (18):

G₃(x)=-F₂(2l_n-1-x) (18)

Convolution (13) and formula (18), it is thus achieved that row ripple G₃(x+v_lT) such as formula (19):

Row ripple G is for v < 0, left travelling wave G₂, right travelling wave F is divided into row ripple F₁, row ripple F₂With row ripple F₃；Row ripple F₃It is Row ripple G₂Echo at left margin x=0；Boundary condition and formula (2) is utilized to obtain F₃And G₂Relation such as formula (20):

$F_3\left(x\right)=-G_2(-\frac{v_l}{v_r}x)---\left(20\right)$

Convolution (14) and formula (20), it is thus achieved that row ripple F₃(x-v_rT) such as formula (21):

Then, second stage [t_an,t_bn] right travelling wave F and left travelling wave G is respectively such as formula (22) and formula (23):

$F(x-v_{r}t)=\left\{\begin{array}{cc}{F}_2(x-v_{r}t),& 0\&le;x\&le;l\left(t\right),v>0\\ F_3(x-v_{r}t),& 0\&le;x\&le;v_r(t-t_{an}),v<0\\ F_2(x-v_{r}t),& v_r(t-t_{an})\&le;x\&le;v_r(t-t_{n-1}),v<0\\ F_1(x-v_{r}t),& v_r(t-t_{n-1})\&le;x\&le;l\left(t\right),v<0\end{array}\right.---\left(22\right)$

$G(x+v_{l}t)=\left\{\begin{array}{cc}{G}_1(x+v_{l}t),& 0\&le;x\&le;l_{n-1}-v_l(t-t_{n-1}),v>0\\ G_2(x+v_{l}t),& l_{n-1}-v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)-c(t-t_{an}),v>0\\ G_3(x+v_{l}t),& l\left(t\right)-c(t-t_{an})\&le;x\&le;l\left(t\right),v>0\\ G_2(x+v_{l}t),& 0\&le;x\&le;l\left(t\right),v<0\end{array}\right.---\left(23\right)$

Utilize formula (24) obtain second stage move rope transverse vibrational displacement ω (x, t):

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_an≤t≤t_bn (24)

Step 7: combine initial condition and boundary condition obtains mobile rope at phase III [t_bn,t_n] oscillation crosswise position Move:

In the phase III, left travelling wave G is divided into row ripple G₂With row ripple G₃, right travelling wave F is divided into row ripple F₂With row ripple F₃, Row ripple G₃It is row ripple F₂The echo at place of boundary x=l (t) on the right, row ripple F₃It is row ripple G₂Echo at left margin x=0；

Right travelling wave F and left travelling wave G is respectively such as formula (25) and formula (26):

$F(x-v_{r}t)=\{\begin{array}{cc}{F}_2(x-v_{r}t),& l\left(t\right)-v_r(t-t_{bn})\&le;x\&le;l\left(t\right)\\ F_3(x-v_{r}t),& 0\&le;x\&le;l\left(t\right)-v_r(t-t_{bn})\end{array}---\left(25\right)$

$G(x+v_{l}t)=\{\begin{array}{cc}{G}_2(x+v_{l}t),& 0\&le;x\&le;l\left(t\right)-c(t-t_{an})\\ G_3(x+v_{l}t),& l\left(t\right)-c(t-t_{an})\&le;x\&le;l\left(t\right)\end{array}---\left(26\right)$

Utilize formula (27) obtain the phase III move rope transverse vibrational displacement ω (x, t) be:

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_bn≤t≤t_n (27)。

Compared with the prior art, the present invention has the beneficial effect that:

1, the inventive method process is simple, will be divided into three rank the vibration period according to row wave motion rule in rope equipment Section, the row ripple of each stage different directions can reflect on the border at two ends；Rule are moved then in conjunction with row ripple in rope equipment Rule and boundary condition obtain border incidence wave and the boundary echo of three phases rope equipment respectively；Finally by border incidence wave It is overlapped with boundary echo, obtains the described rope equipment lateral displacement at three phases, the rope equipment of gained respectively Vibratory response more conforms to practical situation.

2, the present invention is according to strict mathematical expression, and considers the multiple reflections of the row ripple of different directions, it is ensured that obtained The accuracy of the lateral displacement obtained.

3, boundary condition can be adjusted by the present invention according to different borders, is suitable for multiple typical boundary condition.

Accompanying drawing explanation

In Fig. 1, (a) is initial time cord traverse vibration displacement, and (b) is initial left lateral ripple, and (c) is initial right lateral ripple；Figure In 1, (b's) and (c) is superposed to (a)；

In Fig. 2, (a) is the first stage [0, t_a] cord traverse vibration displacement；B () is left lateral ripple G₁Send out at left margin x=0 The state of raw reflection, and obtain right lateral ripple F₂, (c) is right lateral ripple F₁There is the state of reflection in place of boundary x=l (t) on the right, and obtains To left lateral ripple G₂；In Fig. 2, (b's) and (c) is superposed to (a)；

In Fig. 3 (a) be rope equipment move axially speed v > 0 time, second stage [t_a,t_b] cord traverse vibration displacement, B () is left lateral ripple G₁At left margin x=0, there is reflection and obtain right lateral ripple F₂, right lateral ripple F₂Place of boundary x=l (t) is sent out on the right Raw reflection also obtains left lateral ripple G₃State, (c) is the right lateral ripple F in the first stage₁Place of boundary x=l (t) all occurs on the right Left lateral ripple G is obtained after transmitting₂State；In Fig. 3, (b's) and (c) is superposed to (a)；

In Fig. 4 (a) be rope equipment move axially speed v < when 0, second stage [t_a,t_b] cord traverse vibration displacement； B () is the left lateral ripple G in the first stage₁Right lateral ripple F is obtained after all reflecting at left margin x=0₂State, (c) is Right lateral ripple F₁There is reflection and obtain left lateral ripple G in place of boundary x=l (t) on the right₂, left lateral ripple G₂Reflect at left margin x=0 And obtain right lateral ripple F₃State；In Fig. 4, (b's) and (c) is superposed to (a)；

In Fig. 5, (a) is phase III [t_b, T] and cord traverse vibration displacement；B () is the left lateral ripple G in the first stage₁On a left side Right lateral ripple F is obtained after all reflecting at the x=0 of border₂, right lateral ripple F₂There is reflection and obtain a left side in place of boundary x=l (t) on the right Row ripple G₃State, (c) is right lateral ripple F₁Place of boundary x=l (t) obtains left lateral ripple G after all reflecting on the right₂, left lateral ripple G₂ At left margin x=0, there is reflection and obtain right lateral ripple F₃State；In Fig. 5, (b's) and (c) is superposed to (a)；

Detailed description of the invention

The present embodiment accurately obtains in the method for elongated degree rope equipment oscillation crosswise, and the equipment that moves axially is for moving axially Rope equipment, the motion operating mode of rope equipment is that rope lengths linearly changes, and the feature of the inventive method is by following mistake Cheng Jinhang:

Determine the motion model of rope equipment, it is thus achieved that the equation of motion of rope equipment；

Determine motion initial condition and the boundary condition of rope equipment；

Determine the vibration period T of rope equipment；

According to row wave motion rule in described rope equipment, described vibration period T is divided into three phases: be [t respectively_n-1, t_an], [t_an,t_bn] and [t_bn,t_n], wherein t_n-1<t_an<t_bn<t_n；

The border of the rope equipment of three phases is obtained respectively in conjunction with row ripple movement law and boundary condition in rope equipment Incidence wave and boundary echo；Described border incidence wave and boundary echo are overlapped, obtain described rope equipment respectively Lateral displacement at three phases.

The method accurately obtaining elongated degree rope equipment oscillation crosswise in the present embodiment is to carry out as follows:

Step 1: give rope equipment according to the parameter including line density, translational speed and the tension force of rope equipment Motion model, sets up the equation of motion of the rope equipment shown in formula (1) for motion model according to Hamiton's principle:

ω_tt+2vω_xt+(v²-c²)ω_xx=0 (1)

In formula (1):

X is the axial coordinate of rope equipment；T is the time；V be rope equipment move axially speed；Row wave speed c is row Ripple spread speed in rope equipment, c=(P/ ρ)^0.5, P is the tension force 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, and ω=ω (x, t)；

ω_ttIt it is the ω second-order partial differential coefficient to t；ω_xxIt it is the ω second-order partial differential coefficient to x；

ω_xtIt is that ω is respectively to x and the first-order partial derivative to t；

As shown in (a), (b) and (c) in Fig. 1, by the solution of formula (1), i.e. (x t) is considered as right travelling wave to transverse vibrational displacement ω With the superposition of left travelling wave, such as formula (2):

ω (x, t)=F (x-v_rt)+G(x+v_lt) (2)

In formula (2):

v_lFor the speed relative to travelling wave left in the rope equipment of fixed coordinate system, v_l=c v；

v_rFor the speed relative to travelling wave right in the rope equipment of fixed coordinate system, v_r=c+v；

F(x-v_rT) be speed be v_rRight travelling wave, be designated as F；

G(x+v_lT) be speed be v_lLeft travelling wave, be designated as G；

F and G is any twice continuously differentiable function；

Step 2: determine motion initial condition and the boundary condition of rope equipment:

Motion initial condition such as formula (3) during setting t=0:

In formula (3): ω_tFor the ω first-order partial derivative to t；FunctionFor not coordination on rope equipment in fixed coordinate system The initial lateral displacement put；Function ψ (x) is the initial velocity of diverse location on rope equipment in fixed coordinate system；L (t) is rope Rope equipment is in the length of moment t；

Arranging boundary condition is the form that the two ends shown in formula (4) are fixed:

$\left\{\begin{array}{c}\mathrm{\&omega;}(0,t)=0\\ \mathrm{\&omega;}\left(l\right(t),t)=0\end{array}\right.---\left(4\right)$

Step 3: the motion operating mode for rope equipment is that rope lengths linearly changes, and has formula (5) as follows:

L (t)=l₀+vt (5)

L in formula (5)₀Initial length for rope；The duration T in the n-th cycle_nFor:

$T_n=\frac{2l_{n-1}}{c-v},n=1,2,3,\mathrm{...}---\left(6\right)$

In formula (5), l_n-1It it is rope lengths during (n-1)th end cycle；

$l_{n-1}=\left\{\begin{array}{cc}{l}_0+v{\&Sigma;}_{i=1}^{n-1}{T}_i,& n=2,3,\mathrm{...}\\ l_0,& n=1\end{array}\right.---\left(7\right)$

Formula (5) and formula (6) is utilized to calculate the duration obtaining each vibration period: T₁, T₂…T_n, n is natural number；

Step 4: move Changing Pattern according to row ripple and will be divided into three phases the vibration period:

Definition t_nFor system through being circulated back to the time of original state for n time, then have:

T_n=t_n-t_n-1 (8)

Work as v > 0, rope moves from left to right, i.e. rope elongation, then in cycle T_nInterior three phases is [t respectively_n-1, t_an], [t_an,t_bn] and [t_bn,t_n], wherein t_n-1<t_an<t_bn<t_n, as follows:

$\left\{\begin{array}{c}{t}_{an}=\frac{l_{n-1}}{c}+t_{n-1}\\ t_{bn}=\frac{l_{n-1}}{c-v}+t_{n-1}\\ t_n=\frac{2l_{n-1}}{c-v}+t_{n-1}\end{array}\right.,v>0,t_0=0,n=1,2,3,\mathrm{...}---\left(9\right)$

As v < 0, rope moves from right to left, i.e. rope lengths shortens, then t in three phases_anAnd t_bnValue is mutually handed over Change, as shown in formula (9-1):

$\left\{\begin{array}{c}{t}_{an}=\frac{l_{n-1}}{c-v}+t_{n-1}\\ t_{bn}=\frac{l_{n-1}}{c}+t_{n-1}\\ t_n=\frac{2l_{n-1}}{c-v}+t_{n-1}\end{array}\right.,v<0,t_0=0,n=1,2,3,\mathrm{...}---(9-1)$

Step 5: combine motion initial condition and boundary condition obtains rope equipment at first stage [t_n-1,t_an] horizontal Vibration displacement:

In first stage, as shown in (a), (b) and (c) in Fig. 2, left travelling wave G is divided into row ripple G₁With row ripple G₂, move to right row Ripple F is divided into row ripple F₁With row ripple F₂；Wherein, row ripple G₂It is row ripple F₁The echo at place of boundary x=l (t) on the right, as in Fig. 2 (c) Shown in；Row ripple F₂It is row ripple G₁At the echo at the place of left margin x=0, as shown in (b) in Fig. 2；In conjunction with motion initial condition and Formula (2) respectively obtains row ripple F₁With row ripple G₁Expression formula such as formula (10) and formula (11):

Wherein, ξ is integration variable；K is integral constant；

Row ripple F is obtained in conjunction with boundary condition and formula (2)₂, row ripple G₂With row ripple F₁, row ripple G₁Relation such as formula (12):

$\left\{\begin{array}{c}{F}_2\left(x\right)=-G_1(-x\frac{v_l}{v_r})\\ G_2\left(x\right)=-F_1(2l_{n-1}-x)\end{array}\right.---\left(12\right)$

Formula (13) and formula (14) is obtained by formula (10), formula (11) and formula (12):

Then, first stage [t_n-1,t_an] right travelling wave F and left travelling wave G characterizes such as formula (15), formula (16) respectively:

$F(x-v_{r}t)=\left\{\begin{array}{cc}{F}_1(x-v_{r}t),& v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)\\ F_2(x-v_{r}t),& 0\&le;x\&le;v_l(t-t_{n-1})\end{array}\right.---\left(15\right)$

$G(x+v_{l}t)=\{\begin{array}{cc}{G}_1(x+v_{l}t),& 0\&le;x\&le;l_{n-1}-v_l(t-t_{n-1})\\ G_2(x+v_{l}t),& l_{n-1}-v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)\end{array}---\left(16\right)$

Utilize formula (17) obtain the first stage move rope transverse vibrational displacement ω (x, t)

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_n-1≤t≤t_an (17)

Step 6: combine initial condition and boundary condition obtains mobile rope at second stage [t_an,t_bn] oscillation crosswise Displacement, the row wave motion state in this stage is relevant with rope equipment moving direction, therefore divides v>0 and v<0 two kind of situation to carry out Discuss:

In second stage, for v > 0, as shown in (a), (b) and (c) in Fig. 3, left travelling wave G is divided into row ripple G₁, row ripple G₂With row ripple G₃, right travelling wave F is row ripple F₂；Row ripple F₂For incidence wave, G₃It is F₂The echo at place of boundary x=l (t) on the right, As shown in (b) in Fig. 3；G is obtained by boundary condition and formula (2)₃And F₂Relation such as formula (18):

G₃(x)=-F₂(2l_n-1-x) (18)

Convolution (13) and formula (18), it is thus achieved that row ripple G₃(x+v_lT) such as formula (19):

For v < 0, as shown in (a), (b) and (c) in Fig. 4, left travelling wave G is row ripple G₂, right travelling wave F is divided into row ripple F₁, row ripple F₂With row ripple F₃；Row ripple F₃It is row ripple G₂Echo at left margin x=0, as shown in (c) in Fig. 4；Utilize border Condition and formula (2) obtain F₃And G₂Relation such as formula (20):

$F_3\left(x\right)=-G_2(-\frac{v_l}{v_r}x)---\left(20\right)$

Convolution (14) and formula (20), it is thus achieved that row ripple F₃(x-v_rT) such as formula (21):

Then, second stage [t_an,t_bn] right travelling wave F and left travelling wave G is respectively such as formula (22) and formula (23):

$F(x-v_{r}t)=\left\{\begin{array}{cc}{F}_2(x-v_{r}t),& 0\&le;x\&le;l\left(t\right),v>0\\ F_3(x-v_{r}t),& 0\&le;x\&le;v_r(t-t_{an}),v<0\\ F_2(x-v_{r}t),& v_r(t-t_{an})\&le;x\&le;v_r(t-t_{n-1}),v<0\\ F_1(x-v_{r}t),& v_r(t-t_{n-1})\&le;x\&le;l\left(t\right),v<0\end{array}\right.---\left(22\right)$

$G(x+v_{l}t)=\left\{\begin{array}{cc}{G}_1(x+v_{l}t),& 0\&le;x\&le;l_{n-1}-v_l(t-t_{n-1}),v>0\\ G_2(x+v_{l}t),& l_{n-1}-v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)-c(t-t_{an}),v>0\\ G_3(x+v_{l}t),& l\left(t\right)-c(t-t_{an})\&le;x\&le;l\left(t\right),v>0\\ G_2(x+v_{l}t),& 0\&le;x\&le;l\left(t\right)v<0,\end{array}\right.---\left(23\right)$

Utilize formula (24) obtain second stage move rope transverse vibrational displacement ω (x, t):

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_an≤t≤t_bn (24)

Step 7: combine initial condition and boundary condition obtains mobile rope at phase III [t_bn,t_n] oscillation crosswise position Moving, the row wave motion 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 row wave motion state is different, as v > 0 time, such as (b), (c) in Fig. 3 Contrast one by one with (b), (c) in Fig. 5；When v < when 0, as (b) in Fig. 4, (c) contrast one by one with (b), (c) in Fig. 5:

In the phase III, as shown in (a), (b) and (c) in Fig. 5, left travelling wave G is divided into row ripple G₂With row ripple G₃, move to right Row ripple F is divided into row ripple F₂With row ripple F₃, row ripple G₃It is row ripple F₂The echo at place of boundary x=l (t) on the right, such as (c) institute in Fig. 5 Show；Row ripple F₃It is row ripple G₂Echo at left margin x=0, as shown in (b) in Fig. 5；Right travelling wave F and left travelling wave G divides Not such as formula (25) and formula (26):

$F(x-v_{r}t)=\{\begin{array}{cc}{F}_2(x-v_{r}t),& l\left(t\right)-v_r(t-t_{bn})\&le;x\&le;l\left(t\right)\\ F_3(x-v_{r}t),& 0\&le;x\&le;l\left(t\right)-v_r(t-t_{bn})\end{array}---\left(25\right)$

$G(x+v_{l}t)=\{\begin{array}{cc}{G}_2(x+v_{l}t),& 0\&le;x\&le;l\left(t\right)-c(t-t_{an})\\ G_3(x+v_{l}t),& l\left(t\right)-c(t-t_{an})\&le;x\&le;l\left(t\right)\end{array}---\left(26\right)$

Utilize formula (27) obtain the phase III move rope transverse vibrational displacement ω (x, t) be:

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_bn≤t≤t_n (27)。

The inventive method process is simple, it is thus achieved that the response of rope vibration equipment more conform to practical situation, it is to avoid numerical value Computational methods and the limitation of dAlembert principle method, be suitable to elongated degree rope equipment multiple typical boundary condition.

Claims (2)

1. a method for accurate acquisition elongated degree rope equipment oscillation crosswise, described in move axially equipment be the rope moved axially Rope equipment, the motion operating mode of described rope equipment is that rope lengths linearly changes, and it is characterized in that described method according to the following procedure Carry out:

Determine the motion model of rope equipment, it is thus achieved that the equation of motion of described rope equipment；

Determine motion initial condition and the boundary condition of rope equipment；

Determine the vibration period T of rope equipment；

According to row wave motion rule in described rope equipment, described vibration period T is divided into three phases: be [t respectively_n-1,t_an], [t_an,t_bn] and [t_bn,t_n], wherein t_n-1<t_an<t_bn<t_n；

The border of the rope equipment obtaining three phases in conjunction with row ripple movement law and boundary condition in rope equipment respectively is incident Ripple and boundary echo；Described border incidence wave and boundary echo are overlapped, obtain described rope equipment respectively three The lateral displacement in individual stage.

The method of accurate acquisition the most according to claim 1 elongated degree rope equipment oscillation crosswise, is characterized in that by as follows Step is carried out:

Step 1: give the motion of rope equipment according to the parameter including line density, translational speed and the tension force of rope equipment Model, sets up the equation of motion of the rope equipment shown in formula (1) for described motion model according to Hamiton's principle:

ω_tt+2vω_xt+(v²-c²)ω_xx=0 (1)

In formula (1):

X is the axial coordinate of rope equipment；T is the time；V be rope equipment move axially speed；Row wave speed c is that row ripple exists Spread speed in rope equipment, c=(P/ ρ)^0.5, P is the tension force 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, and ω=ω (x, t)；

ω_ttIt it is the ω second-order partial differential coefficient to t；ω_xxIt it is the ω second-order partial differential coefficient to x；

ω_xtIt is that ω is respectively to x and the first-order partial derivative to t；

By the solution of formula (1), i.e. transverse vibrational displacement ω (x, t) is considered as right travelling wave and the superposition of left travelling wave, such as formula (2):

ω (x, t)=F (x-v_rt)+G(x+v_lt) (2)

In formula (2):

v_lFor the speed relative to travelling wave left in the rope equipment of fixed coordinate system, v_l=c-v；

v_rFor the speed relative to travelling wave right in the rope equipment of fixed coordinate system, v_r=c+v；

F(x-v_rT) be speed be v_rRight travelling wave, be designated as F；

G(x+v_lT) be speed be v_lLeft travelling wave, be designated as G；

F and G is any twice continuously differentiable function；

Step 2: determine motion initial condition and the boundary condition of rope equipment:

Motion initial condition such as formula (3) during setting t=0:

In formula (3): ω_tFor the ω first-order partial derivative to t；FunctionFor diverse location on rope equipment in fixed coordinate system Initial lateral displacement；Function ψ (x) is the initial velocity of diverse location on rope equipment in fixed coordinate system；L (t) is that rope sets The standby length at moment t；

Arranging boundary condition is the form that the two ends shown in formula (4) are fixed:

$\left\{\begin{array}{c}\mathrm{\&omega;}(0,t)=0\\ \mathrm{\&omega;}\left(l\right(t),t)=0\end{array}\right.---\left(4\right)$

Step 3: the motion operating mode for rope equipment is that rope lengths linearly changes, and has formula (5) as follows:

L (t)=l₀+vt (5)

L in formula (5)₀Initial length for rope；The duration T in the n-th cycle_nFor:

$T_n=\frac{2l_{n-1}}{c-v},n=1,2,3,\mathrm{...}---\left(6\right)$

In formula (5), l_n-1It it is rope lengths during (n-1)th end cycle；

$l_{n-1}=\left\{\begin{array}{cc}{l}_0+{\mathrm{v\&Sigma;}}_{i=1}^{n-1}{T}_i,& n=2,3,\mathrm{...}\\ l_0,& n=1\end{array}\right.---\left(7\right)$

Formula (5) and formula (6) is utilized to calculate the duration obtaining each vibration period: T₁, T₂…T_n, n is natural number；

Step 4: move Changing Pattern according to row ripple and will be divided into three phases the vibration period:

Definition t_nFor system through being circulated back to the time of original state for n time, then have:

T_n=t_n-t_n-1 (8)

Work as v > 0, rope moves from left to right, i.e. rope elongation, then in cycle T_nInterior three phases is [t respectively_n-1,t_an], [t_an,t_bn] and [t_bn,t_n], wherein t_n-1<t_an<t_bn<t_n, as follows:

$\left\{\begin{array}{c}{t}_{an}=\frac{l_{n-1}}{c}+t_{n-1}\\ t_{bn}=\frac{l_{n-1}}{c-v}+t_{n-1}\\ t_n=\frac{2l_{n-1}}{c-v}+t_{n-1}\end{array}\right.,v>0,t_0=0,n=1,2,3,\mathrm{...}---\left(9\right)$

As v < 0, rope moves from right to left, i.e. rope lengths shortens, then t in three phases_anAnd t_bnValue is exchanged with each other；

Step 5: combine motion initial condition and boundary condition obtains rope equipment at first stage [t_n-1,t_an] oscillation crosswise Displacement:

In first stage, left travelling wave G is divided into row ripple G₁With row ripple G₂, right travelling wave F is divided into row ripple F₁With row ripple F₂；Wherein, OK Ripple G₂It is row ripple F₁The echo at place of boundary x=l (t) on the right, row ripple F₂It is row ripple G₁Echo at the place of left margin x=0； Row ripple F is respectively obtained in conjunction with motion initial condition and formula (2)₁With row ripple G₁Expression formula such as formula (10) and formula (11):

Wherein, ξ is integration variable；K is integral constant；

Row ripple F is obtained in conjunction with boundary condition and formula (2)₂, row ripple G₂With row ripple F₁, row ripple G₁Relation such as formula (12):

$\left\{\begin{array}{c}{F}_2\left(x\right)=-G_1(-x\frac{v_l}{v_r})\\ G_2\left(x\right)=-F_1(2l_{n-1}-x)\end{array}\right.---\left(12\right)$

Formula (13) and formula (14) is obtained by formula (10), formula (11) and formula (12):

Then, first stage [t_n-1,t_an] right travelling wave F and left travelling wave G characterizes such as formula (15), formula (16) respectively:

$F(x-v_{r}t)=\left\{\begin{array}{cc}{F}_1(x-v_{r}t),& v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)\\ F_2(x-v_{r}t),& 0\&le;x\&le;v_l(t-t_{n-1})\end{array}\right.---\left(15\right)$

$G(x+v_{l}t)=\left\{\begin{array}{cc}{G}_1(x+v_{l}t),& 0\&le;x\&le;l_{n-1}-v_l(t-t_{n-1})\\ G_2(x+v_{l}t),& l_{n-1}-v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)\end{array}\right.---\left(16\right)$

Utilize formula (17) obtain the first stage move rope transverse vibrational displacement ω (x, t)

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_n-1≤t≤t_an (17)

Step 6: combine initial condition and boundary condition obtains mobile rope at second stage [t_an,t_bn] transverse vibrational displacement:

In second stage, for v > 0, left travelling wave G is divided into row ripple G₁, row ripple G₂With row ripple G₃, right travelling wave F is row ripple F₂；Row ripple F₂For incidence wave, G₃It is F₂The echo at place of boundary x=l (t) on the right；G is obtained by boundary condition and formula (2)₃And F₂'s Relation such as formula (18):

G₃(x)=-F₂(2l_n-1-x) (18)

Convolution (13) and formula (18), it is thus achieved that row ripple G₃(x+v_lT) such as formula (19):

Row ripple G is for v < 0, left travelling wave G₂, right travelling wave F is divided into row ripple F₁, row ripple F₂With row ripple F₃；Row ripple F₃It it is row ripple G₂Echo at left margin x=0；Boundary condition and formula (2) is utilized to obtain F₃And G₂Relation such as formula (20):

$F_3\left(x\right)=-G_2(-\frac{v_l}{v_r}x)---\left(20\right)$

Convolution (14) and formula (20), it is thus achieved that row ripple F₃(x-v_rT) such as formula (21):

Then, second stage [t_an,t_bn] right travelling wave F and left travelling wave G is respectively such as formula (22) and formula (23):

$F(x-v_{r}t)=\left\{\begin{array}{cc}{F}_2(x-v_{r}t),& 0\&le;x\&le;l\left(t\right),v>0\\ F_3(x-v_{r}t),& 0\&le;x\&le;v_r(t-t_{an}),v<0\\ F_2(x-v_{r}t),& v_r(t-t_{an})\&le;x\&le;v_r(t-t_{n-1}),v<0\\ F_1(x-v_{r}t),& v_r(t-t_{n-1})\&le;x\&le;l\left(t\right),v<0\end{array}\right.---\left(22\right)$

$G(x+v_{l}t)=\left\{\begin{array}{cc}{G}_1(x+v_{l}t),& 0\&le;x\&le;l_{n-1}-v_l(t-t_{n-1}),v>0\\ G_2(x+v_{l}t),& l_{n-1}-v_l(t-t_{n-1})\&le;x\&le;l\left(t\right)-c(t-t_{an}),v>0\\ G_3(x+v_{l}t),& l\left(t\right)-c(t-t_{an})\&le;x\&le;l\left(t\right),v>0\\ G_2(x+v_{l}t),& 0\&le;x\&le;l\left(t\right),v<0\end{array}\right.---\left(23\right)$

Utilize formula (24) obtain second stage move rope transverse vibrational displacement ω (x, t):

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_an≤t≤t_bn (24)

Step 7: combine initial condition and boundary condition obtains mobile rope at phase III [t_bn,t_n] transverse vibrational displacement:

In the phase III, left travelling wave G is divided into row ripple G₂With row ripple G₃, right travelling wave F is divided into row ripple F₂With row ripple F₃, row ripple G₃ It is row ripple F₂The echo at place of boundary x=l (t) on the right, row ripple F₃It is row ripple G₂Echo at left margin x=0；

Right travelling wave F and left travelling wave G is respectively such as formula (25) and formula (26):

$F(x-v_{r}t)=\{\begin{array}{cc}{F}_2(x-v_{r}t),& l\left(t\right)-v_r(t-t_{bn})\&le;x\&le;l\left(t\right)\\ F_3(x-v_{r}t),& 0\&le;x\&le;l\left(t\right)-v_r(t-t_{bn})\end{array}---\left(25\right)$

$G(x+v_{l}t)=\{\begin{array}{cc}{G}_2(x+v_{l}t),& 0\&le;x\&le;l\left(t\right)-c(t-t_{an})\\ G_3(x+v_{l}t),& l\left(t\right)-c(t-t_{an})\&le;x\&le;l\left(t\right)\end{array}---\left(26\right)$

Utilize formula (27) obtain the phase III move rope transverse vibrational displacement ω (x, t) be:

ω (x, t)=F (x-v_rt)+G(x+v_lt),0≤x≤l(t)、t_bn≤t≤t_n (27)。