CN106323528A - High-precision vibration measurement method for cable tension - Google Patents

Abstract

The invention discloses a high-precision vibration measurement method for cable tension, and the method comprises the following steps: 1, building a stay cable vibration equation combining a sag effect, a rigidness effect and an inclination angle through employing a stay cable vibration model; 2, carrying out the discretization of the vibration equation through employing a finite difference method, obtaining a matrix expression, calculating the matrix eigenvalues, and obtaining the vibration frequency under given cable tension; 3, carrying out the iterative solving based on the matrix eigenvalues, and obtaining the system parameters under the given vibration frequency: cable tension, anti-bending rigidness and axial rigidness. The vibration equation gives consideration to the impact from the rigidness, the sag, the inclination angle and the boundary rotation rigidness, and the multi-order frequency is used for calculation. The method can carry out the recognition of a plurality of system parameters, and there is no need to carry out the rough assumption of the anti-bending rigidness and boundary conditions of the stay cable, thereby improving the testing precision.

Description

A kind of high accuracy dynamic measurement of pile of rope tensility

Technical field

The present invention relates to a kind of high-precision rope tensility dynamic measurement of pile, belong to technical field of bridge engineering.

Background technology

Suspension cable is one of main bearing carrier of cable-stayed bridge, and the size of its tension force is directly connected to cable-stayed bridge main-beam and tower The force-bearing situation of post, therefore, the whether accurate of cable tension test is directly connected to being smoothed out of CONSTRUCTION OF CABLE-STAYED BRIDGE control The safe operation after coming into operation is built up with cable-stayed bridge.

Cable tension test method has oil gauge method, pressure transducer method, frequency of vibration method and magnetic flux transducer method etc..Wherein Frequency of vibration method because it is simple, quickly and be suitable to cable-stayed bridge and build up the cable tension test after coming into operation and obtain extensively application. Its principle is the natural frequency of vibration of rope and rigidity is mainly determined by the tension force of rope, and has quantitative relation with rope tensility.By measuring The drag-line natural frequency of vibration, can calculate rope tensility.But it was verified that the Suo Li that existing vibratory drilling method obtains often deposits with actual Suo Li At certain error, main cause is that rope bending rigidity, cable sag effect, inclination angle and boundary condition etc. are considered deficiency.

Summary of the invention

It is an object of the invention to overcome deficiency of the prior art, it is provided that the high accuracy dynamic measurement of pile of a kind of rope tensility, solve In prior art of having determined, frequency of vibration method does not accounts for bending rigidity, cable sag effect and inclination angle factor affects its measurement result standard The really technical problem of property.

For solving above-mentioned technical problem, the invention provides the high accuracy dynamic measurement of pile of a kind of rope tensility, it is characterized in that, including Following steps:

Step one, by the model of vibration of suspension cable, sets up and combines cable sag effect, rigidity effect and the drag-line of inclination angle factor Vibration equation；

Step 2, uses finite difference calculus to carry out discrete to vibration equation, obtains its matrix expression；Calculate matrix character Value, obtains the frequency of vibration under given rope tensility；

Step 3, feature based value matrix is iterated solving, it is thus achieved that include that rope tensility, bending resistance are firm under given frequency of vibration Degree and the systematic parameter of axial rigidity.

Further, in step one, described combination cable sag effect, rigidity effect and the inhaul cable vibration side of inclination angle factor Cheng Wei:

$\frac{d^2}{{\mathrm{dx}}^2}\left(EI\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}\right)-H\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}-H^{\′}\frac{d\mathrm{\φ}}{dx}+\frac{{\∫}_0^L\frac{d^{2}y}{{\mathrm{dx}}^2}\mathrm{\φ}dx}{{\∫}_0^L\frac{{(ds/dx)}^3}{EA}dx}\frac{d^{2}y}{{\mathrm{dx}}^2}={\mathrm{m\ω}}^2\mathrm{\φ}$

Wherein, EI is drag-line bending stiffness, and φ represents that the vibration shape, H represent that the rope tensility under drag-line resting state is in the x-direction Component, y is the amount of deflection under drag-line resting state；L represents the rope projected length in x direction, and EA represents axial rigidity, and m is unit The quality of length rope；Ds=(dx²+dy²)^1/2Being the length of drag-line infinitesimal, ω is undamped frequency.

Further, in step 2, finite difference calculus vibration equation is carried out discrete after obtain its matrix expression and be:

(K-ω²M) w=0

K=K₁+K₂；w^T={ w₁,w₂,...,w_n}

Wherein, n represents the drag-line interstitial content of division, drag-line n decile, and the projected length in x direction of each sections is Being designated as the column vector that a, w are non-zeros, therefore, the ranks of frequency of vibration coefficient matrix are equal to zero, i.e. | K-ω²M |=0, to this formula Carry out eigenvalue and solve the frequency of vibration that can solve drag-line；Wherein K₁Representing linear stiffness matrix, its expression formula is as follows:

$K_1=\left[\begin{array}{cccccc}Q& U& W& & & \\ D& S& U& W& & \\ V& D& S& U& \_& \\ & V& \_& \_& \_& W\\ & & \_& \_& S& U\\ & & & V& D& T\end{array}\right]$

Wherein:

$S=\frac{1}{a^4}(-2{\mathrm{EI}}_{i+1}+10{\mathrm{EI}}_i-2{\mathrm{EI}}_{i-1})+\frac{2H_i}{a^2},D=\frac{1}{a^4}(2{\mathrm{EI}}_{i+1}-6{\mathrm{EI}}_i)-\frac{H_i}{a^2}+\frac{H_{i+1}-H_{i-1}}{4a^2},$

$U=\frac{1}{a^4}(-6{\mathrm{EI}}_i+2{\mathrm{EI}}_{i-1})-\frac{H_i}{a^2}-\frac{H_{i+1}-H_{i-1}}{4a^2},V=-\frac{1}{2a^4}(2{\mathrm{EI}}_{i+1}-6{\mathrm{EI}}_i-{\mathrm{EI}}_{i-1}),$

$W=\frac{1}{2a^4}({\mathrm{EI}}_{i+1}+2{\mathrm{EI}}_i-{\mathrm{EI}}_{i-1}),Q=S+\frac{K_{rot-1}a-{\mathrm{EI}}_0}{K_{rot-1}a+2{\mathrm{EI}}_0}V(i=1)$

$T=S+\frac{K_{rot-2}a-2{\mathrm{EI}}_{n+1}}{K_{rot-2}a+2{\mathrm{EI}}_{n+1}}W(i=n)$

Wherein, EI_iRepresent the bending stiffness at node i；EI₀And EI_n+1Represent the bending stiffness at two ends；K_rot-1 and K_rot-2Represent the rigidity of two ends rotation spring；K₂Representing nonlinear stiffness matrix, its expression formula is as follows:

K₂=rs^T

r^T={ r₁,r₂,...,r_n} s^T={ s₁,s₂,...,s_n}

Wherein

The mass matrix M of rope is a diagonal matrix:

M=diag{m₁,m₂,...,m_n}

Wherein m_iRepresent the i-th near nodal territorial unit linear mass.

Further, X represents the parameter vector that systematic parameter forms: X=[H EI EA m K_rot-1 K_rot-2]^T, wherein K₁ Bending stiffness and rope tensility derivative can be directly given its solution formula:

$\frac{\∂S}{\∂\left(EI\right)}=\frac{6}{a^4},\frac{\∂S}{\∂\left(H\right)}=\frac{2}{a^2},\frac{\∂D}{\∂\left(EI\right)}=-\frac{4}{a^4},\frac{\∂D}{\∂\left(H\right)}=-\frac{1}{a^2}$

$\frac{\∂U}{\∂\left(EI\right)}=-\frac{4}{a^4},\frac{\∂U}{\∂\left(H\right)}=-\frac{1}{a^2},\frac{\∂V}{\∂\left(EI\right)}=\frac{1}{a^4},\frac{\∂W}{\∂\left(EI\right)}=\frac{1}{a^4}$

$\frac{\∂Q}{\∂\left(EI\right)}=\frac{6}{a^4}+\frac{K_{rot-1}a-2{\mathrm{EI}}_0}{a^4(K_{rot-1}a+2{\mathrm{EI}}_0)}-\frac{4K_{rot-1}a}{{(K_{rot-1}a+2{\mathrm{EI}}_0)}^2}V(i=1)$

$\frac{\∂Q}{\∂\left(H\right)}=\frac{\∂S}{\∂\left(H\right)}\frac{2}{a^2},\frac{\∂Q}{\∂\left(K_{rot-1}\right)}=\frac{4{\mathrm{EI}}_{0}a}{{(K_{rot-1}a+2{\mathrm{EI}}_0)}^2}V(i=1)$

$\frac{\∂T}{\∂\left(EI\right)}=\frac{6}{a^4}+\frac{K_{rot-2}a-2{\mathrm{EI}}_{n+1}}{a^4(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}-\frac{4K_{rot-2}a}{{(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}^2}W(i=n)$

$\frac{\∂T}{\∂\left(H\right)}=\frac{\∂S}{\∂\left(H\right)}\frac{2}{a^2},\frac{\∂T}{\∂\left(K_{rot-2}\right)}=\frac{4{\mathrm{EI}}_{n+1}a}{{(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}^2}W(i=n)$

K₂Axial rigidity is solved as follows:

$\frac{d\left(r_i\right)}{d\left(EA\right)}=s_i{\left(\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{t}_i^3\right)}^{-1}$

After obtaining the drag-line frequency derivative to systematic parameter, can using the following method rope kinetic parameter be repaiied Just:

α Δ X=Δ λ

$\mathrm{\α}=\left[\begin{array}{cccc}\∂{\mathrm{\λ}}_1/\∂x_1& \∂{\mathrm{\λ}}_2/\∂x_1& ...& \∂{\mathrm{\λ}}_N/\∂x_1\\ \∂{\mathrm{\λ}}_2/\∂x_1& \∂{\mathrm{\λ}}_2/\∂x_2& ...& \∂{\mathrm{\λ}}_N/\∂x_2\\ ...& ...& ...& ...\\ \∂{\mathrm{\λ}}_N/\∂x_1& \∂{\mathrm{\λ}}_N/\∂x_2& ...& \∂{\mathrm{\λ}}_N/\∂x_m\end{array}\right]$

Δ X={ Δ x₁Δx₂...Δx_m}^T, Δ λ={ Δ λ₁Δλ₂...Δλ_N}^T

Wherein m represents the quantity of parameter to be identified；N represents the frequency number recorded.Δλ_k=λ_k,measured-λ_k,calculated (X^k),X^kRepresent the result that kth time is revised；

When number of parameters to be identified is consistent with the frequency number recorded (m=N), above formula can be with direct solution:

Δ X=α^-1Δλ

When number of parameters to be identified is inconsistent with the frequency number recorded, above formula can solve with method of least square:

Δ X=(α^Tα)^-1α^TΔλ

After obtaining parameter error, parameter vector is modified: X_k+1=X_k+ΔX；Constantly this makeover process of iteration, directly To the parameter vector obtaining convergence.

Compared with prior art, the present invention is reached to provide the benefit that: consider in the vibration equation of (1) present invention just Degree, sag, inclination angle and the impact of border rotational stiffness；(2) have employed many order frequencies to calculate, multiple systems can be joined Number is identified, it is not necessary to the bending rigidity of rope and boundary condition etc. are carried out rough hypothesis；(3) the inventive method identification rope is opened While power, it is also possible to correct other systematic parameter, this can improve the precision of test in actual applications further.

Accompanying drawing explanation

Fig. 1 is inclination inhaul cable vibration model in the embodiment of the present invention.

Detailed description of the invention

The invention will be further described below.Herein below is only used for clearly illustrating the technical side of the present invention Case, and can not limit the scope of the invention with this.

The high accuracy dynamic measurement of pile of a kind of rope tensility of the present invention, comprises the following steps:

Step one, by the model of vibration of suspension cable, sets up and combines cable sag effect, rigidity effect and the drag-line of inclination angle factor Vibration equation.

Inclined Cable Vibration model and coordinate system are as it is shown in figure 1, set up x coordinate along the chord length direction of rope, and vertical direction sets up y Coordinate, in figure, θ represents the inclination angle of drag-line；Variable y represents the Static Correction of rope, and it is to be caused by the deadweight of rope；U and η is respectively Represent the dynamic displacement owing to vibrating x and the y direction caused；Variable H represents the rope rope tensility when resting state, h represent due to The increment of the rope tensility that vibration causes；Variable K_{rot_1}、K_{rot_2}Representing the rope rotational stiffness at two ends, they are used for considering two ends It is not consolidation neither hinged, and there is the situation of elastic stiffness.

The equation of motion of known inclination drag-line is as follows:

$\frac{\∂}{\∂s}\[(T+\mathrm{\τ})\frac{\∂v}{\∂s}\]-\frac{{\∂}^2\left(EI\mathrm{\κ}\right)}{\∂s^2}+mgcos\mathrm{\θ}=m\frac{{\∂}^2\mathrm{\η}}{\∂t^2}---\left(1a\right)$

$\frac{\∂}{\∂s}\[(T+\mathrm{\τ})(\frac{\∂x}{\∂s}+\frac{\∂u}{\∂s})\]+mgsin\mathrm{\θ}=m\frac{{\∂}^{2}u}{\∂t^2}---\left(1b\right)$

Wherein EI is drag-line bending stiffness, and m is the quality of unit length rope；T=T (x) is the rope under drag-line resting state Tension force, y are the amount of deflection under drag-line resting state, and (x, (x t) produces τ=τ for motion t) to produce additional Suo Li, η=η for motion Additional amount of deflection；V (x)=y+ η is drag-line vertical deflection summation, and u=u (x) is drag-line displacement in the x-direction, and θ represents inclining of rope Rake angle；κ is the bending curvature of drag-line, ds=(dx²+dy²)^1/2It is the length of drag-line infinitesimal.Above equation is applicable to different The rope of tight degree, can be derived by Hamilton's principle.The suspension cable related in the present invention, its sag ratio is generally smaller than 1/10 (big sag rope described in engineering is also in this scope).In this case, the differential of arc in formula (1) can be reduced to right The differential of x coordinate, therefore, the Nonlinear Equations of Motion formula (1) tilting drag-line can be reduced to:

$\frac{\∂}{\∂x}\[(H+h)\frac{\∂v}{\∂x}\]-\frac{{\∂}^2\left({\mathrm{EIv}}^{\′\′}\right)}{\∂x^2}+mgcos\mathrm{\θ}=m\frac{{\∂}^2\mathrm{\η}}{\∂t^2}---\left(2a\right)$

$\frac{\∂}{\∂x}\[(H+h)(1+\frac{\∂u}{\∂x})\]+mgsin\mathrm{\θ}=m\frac{{\∂}^{2}u}{\∂t^2}---\left(2b\right)$

Wherein (x t) represents T (x) and τ=τ (x, t) component in the x-direction to H=H (x) and h=h respectively.Corresponding static(al) Equation is:

$\frac{\∂}{\∂x}\[H\left(x\right)\frac{\∂y}{\∂x}\]-\frac{\∂\left({\mathrm{EIy}}^{\′\′}\right)}{\∂x}+mgcos\mathrm{\θ}=0---\left(3a\right)$

$\frac{\∂H\left(x\right)}{\∂x}+mg\sin \mathrm{\θ}=0---\left(3b\right)$

The Nonlinear Equations of Motion formula (2) of drag-line is deducted corresponding equation of statics formula (3), the vibration of drag-line can be obtained Equation is:

$H\left(x\right)\frac{{\∂}^2\mathrm{\η}}{\∂x^2}+H^{\′}\left(x\right)\frac{\∂\mathrm{\η}}{\∂x}+h\frac{d^{2}y}{{\mathrm{dx}}^2}-\frac{{\∂}^2\left({\mathrm{EI\η}}^{\′\′}\right)}{\∂x^2}=m\frac{{\∂}^2\mathrm{\η}}{\∂t^2}---\left(4a\right)$

$\frac{\∂}{\∂x}\[H\frac{\∂u}{\∂x}+h(1+\frac{\∂u}{\∂x})\]=m\frac{{\∂}^{2}u}{\∂t^2}---\left(4b\right)$

Can be obtained by formula (3b): H (x)=H₀-mgxsinθ(H₀It is the H-number that goes out of zero position).From the point of view of in theory, Rope tensility increment h is had an impact by longitudinal vibration, but the extensional vibration of reality is the faintest, negligible, so by formula (4b) can obtain h is the amount unrelated with x, i.e. h (x, t)=h (t).For solving formula (4a), to the dynamic displacement η of drag-line, (x t) makees variable Separate:

η (x, t)=φ (x) q (t) (5)

Wherein φ=φ (x) represents the vibration shape, and q=q (t) represents with the vibration shape for decomposing the component of base.Shaking of real drag-line Move and can not directly do such separation, but the result being overlapped with some first order modes.For undamped drag-line, q is One amount decayed the most in time:

Q (t)=e^iωt (6)

Wherein ω is undamped frequency.By separating variables, formula (4a) can change and is expressed as:

$\frac{d^2}{{\mathrm{dx}}^2}\left(EI\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}\right)q-H\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}q-H^{\′}\frac{d\mathrm{\φ}}{dx}q-h\frac{d^{2}y}{{\mathrm{dx}}^2}={\mathrm{m\ω}}^2\mathrm{\φ}q---\left(7\right)$

According to tilting the physics of drag-line model and geometrical relationship, it is known that Irvine (1992) has made following derivation:

$\frac{\mathrm{\τ}}{EA}=\frac{h}{EA}\frac{ds}{dx}==\frac{{\mathrm{ds}}^{\′}-ds}{ds}---\left(8\right)$

Wherein EA represents axial rigidity, ds²=dx²+dy², ds '²=(dx+du)²+(dy+dη)².Therefore:

$\frac{h{(ds/dx)}^3}{EA}=\frac{\∂u}{\∂x}+\frac{dy}{dx}\frac{\∂\mathrm{\η}}{\∂x}---\left(9\right)$

Owing to h=h (t) is independently of the equivalent of x coordinate, are quadratured and can obtain in formula (9) two ends:

$h=\frac{{\∫}_0^L(\frac{\∂u}{\∂x}+\frac{dy}{dx}\frac{\∂\mathrm{\η}}{\∂x})dx}{{\∫}_0^L\frac{{(ds/dx)}^3}{EA}dx}---\left(10\right)$

Wherein L represents the rope projected length in x direction.(include knot hinged, firm and deposit when two ends do not have displacement of the lines when In situations such as rotation springs), can be as follows in the hope of the expression formula of h by integration by parts:

$h=\frac{-{\∫}_0^L\frac{d^{2}y}{{\mathrm{dx}}^2}\mathrm{\η}dx}{{\∫}_0^L\frac{{(ds/dx)}^3}{EA}dx}=\frac{-{\∫}_0^L\frac{d^{2}y}{{\mathrm{dx}}^2}\mathrm{\φ}dx}{{\∫}_0^L\frac{{(ds/dx)}^3}{EA}dx}q=h^{\′}q---\left(11\right)$

Wherein h ' is constant.Formula (11) is substituted into formula (7), eliminates q (t), can obtain combining cable sag effect, rigidity effect With the Mode Equation tilting drag-line of inclination angle factor it is:

$\frac{d^2}{{\mathrm{dx}}^2}\left(EI\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}\right)-H\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}-H^{\′}\frac{d\mathrm{\φ}}{dx}+\frac{{\∫}_0^L\frac{d^{2}y}{{\mathrm{dx}}^2}\mathrm{\φ}dx}{{\∫}_0^L\frac{{(ds/dx)}^3}{EA}dx}\frac{d^{2}y}{{\mathrm{dx}}^2}={\mathrm{m\ω}}^2\mathrm{\φ}---\left(12\right)$

Step 2, uses finite difference calculus to carry out discrete to vibration equation, obtains its matrix expression, calculate matrix character Value, obtains the frequency of vibration under given rope tensility.

Owing to rope is relatively big from focusing on chord length durection component, the rope tensility tilting drag-line is change in chord length durection component H, This will produce impact to the natural frequency of vibration and the vibration shape of rope.Equation (12) is the differential equation of variable coefficient, must carry out numerical solution, under Face uses finite difference calculus to carry out discrete to formula (12).By drag-line n decile, the projected length in x direction of each sections is note For a, and use following difference scheme to replace differential:

$\frac{d\mathrm{\φ}}{dx}=\frac{{\mathrm{\φ}}_{i+1}-{\mathrm{\φ}}_{i-1}}{2a},\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}=\frac{{\mathrm{\φ}}_{i+1}-2{\mathrm{\φ}}_i+{\mathrm{\φ}}_{i-1}}{a^2}---(13a,b)$

$\frac{d^3\mathrm{\φ}}{{\mathrm{dx}}^3}=\frac{{\mathrm{\φ}}_{i+2}-2{\mathrm{\φ}}_{i+1}+2{\mathrm{\φ}}_{i-1}-{\mathrm{\φ}}_{i-2}}{2a^3},\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^4}=\frac{{\mathrm{\φ}}_{i+2}-4{\mathrm{\φ}}_{i+1}+6{\mathrm{\φ}}_i-4{\mathrm{\φ}}_{i-1}+{\mathrm{\φ}}_{i-2}}{a^4}---(13c,d)$

Formula (13) is substituted into formula (12), can obtain the drag-line Mode Equation of matrix form:

(K-ω²M) w=0 (14a)

K=K₁+K₂；w^T={ w₁,w₂,...,w_n} (14b,c)

Wherein n represents the drag-line interstitial content (except two end nodes) of division；The transposition computing of subscript T' representing matrix.Formula (14) it is typical algebra characterisitic value problem, wherein ω²With eigenvalue and the characteristic vector that w is respectively system, here, its thing The quadratic sum vibration shape of reason meaning representation frequency.Obviously, w is the column vector (w=0 represents situation about not vibrating) of non-zero, therefore, In formula (14a), the determinant of coefficient matrix is equal to zero, it may be assumed that

|K-ω²M |=0 (15)

Formula (15) is carried out eigenvalue and solves the frequency of vibration that can solve system.Wherein K₁Represent linear stiffness matrix, Its expression formula is as follows:

$K_1=\left[\begin{array}{cccccc}Q& U& W& & & \\ D& S& U& W& & \\ V& D& S& U& \_& \\ & V& \_& \_& \_& W\\ & & \_& \_& S& U\\ & & & V& D& T\end{array}\right]---\left(16\right)$

Wherein:

$S=\frac{1}{a^4}(-2{\mathrm{EI}}_{i+1}+10{\mathrm{EI}}_i-2{\mathrm{EI}}_{i-1})+\frac{2H_i}{a^2},D=\frac{1}{a^4}(2{\mathrm{EI}}_{i+1}-6{\mathrm{EI}}_i)-\frac{H_i}{a^2}+\frac{H_{i+1}-H_{i-1}}{4a^2},$

$U=\frac{1}{a^4}(-6{\mathrm{EI}}_i,+2{\mathrm{EI}}_{i-1})-\frac{H_i}{a^2}-\frac{H_{i+1}-H_{i-1}}{4a^2},V=-\frac{1}{2a^4}({\mathrm{EI}}_{i+1}-2{\mathrm{EI}}_i-{\mathrm{EI}}_{i-1}),$

$W=\frac{1}{2a^4}({\mathrm{EI}}_{i+1}+2{\mathrm{EI}}_i-{\mathrm{EI}}_{i-1}),Q=S+\frac{K_{rot-1}a-2{\mathrm{EI}}_0}{K_{rot-1}a+2{\mathrm{EI}}_0}V(i=1)$

$T=S+\frac{K_{rot-2}a-2{\mathrm{EI}}_{n+1}}{K_{rot-2}a+2{\mathrm{EI}}_{n+1}}W(i=n)$

Wherein EI_iRepresent the bending stiffness at node i；EI₀And EI_n+1Represent the bending stiffness at two ends；K_rot-1 and K_rot-2Represent the rigidity of two ends rotation spring.K₂Representing nonlinear stiffness matrix, its expression formula is as follows:

K₂=rs^T (17a)

r^T={ r₁,r₂,...,r_n}s^T={ s₁,s₂,...,s_n} (17b,c)

Wherein

The mass matrix M of rope is a diagonal matrix:

M=diag{m₁,m₂,...,m_n(18) wherein m_iRepresent the i-th near nodal territorial unit linear mass.Formula (14) rigidity of rope, sag, inclination angle and the impact of two ends boundary condition are considered in.For jump stay, rigidity and quality Matrix is all symmetrical matrix.For the drag-line tilted, element D and U is generally inconsistent, but divides at equal length In the case of drag-line unit, element D and U numerically remains equal, and K is still symmetrical matrix.

Step 3, feature based value matrix is iterated solving, it is thus achieved that include that rope tensility, bending resistance are firm under given frequency of vibration Degree and the systematic parameter of axial rigidity.

From above formula, the frequency of vibration of suspension cable is corresponding with rope tensility.More than solve at given drag-line Rope tensility and other kinetic parameter in the case of, the problem accurately solving the natural frequency of vibration of drag-line.Set forth below how to lead to Overfrequency carrys out inverting and includes the systematic parameter of rope tensility, bending stiffness, axial rigidity, quality and rotational stiffness.To this end, we need Matrix exgenvalue to be calculated is for the derivative of systematic parameter.Matrix exgenvalue derivative has in the field such as Modifying model, fault diagnosis Important effect.Formula (14a) is expressed as following form:

[K (X)-λ (X) M (X)] w (X)=0 (19)

Wherein λ (X)=ω²(X) it is system features value；W (X) represents the vibration shape, for the column vector of non-zero；X represents that system is joined The parameter vector that array becomes:

X=[H EI EA m K_rot-1 K_rot-2]^T (20)

If λ₁It is that formula (19) is at X=X^*The eigenvalue at place, w₁(X^*) it is corresponding to λ₁Characteristic vector (non-zero).To formula (19) Two ends are to x_kDerivation:

$\[K\left(X^{*}\right)-{\mathrm{\λ}}_{1}M\left(X^{*}\right)\]\frac{\∂w_1\left(X^{*}\right)}{\∂x_k}=\[{\mathrm{\λ}}_1\frac{\∂M\left(X^{*}\right)}{\∂x_k}-\frac{\∂K\left(X^{*}\right)}{\∂x_k}\]w_1\left(X^{*}\right)+M\left(X^{*}\right)w_1\left(X^{*}\right)\frac{\∂{\mathrm{\λ}}_1\left(X^{*}\right)}{\∂x_k}---\left(21\right)$

Wherein x_kRepresent the kth component of X.Then at formula (21) two ends premultiplication matrixIn view of K (X) and M (X) it is all symmetrical matrix, formula (21) left side item premultiplicationRear vanishing.Diagonal matrix in view of M, therefore just like Lower relation:

$w_1^{T}M\left(X^{*}\right)w_1=\underset{i=1}{\overset{n}{\Σ}}{m}_i{w}_{1i}^2>0---\left(22\right)$

Then formula (21) can be reduced to:

$\frac{\∂{\mathrm{\λ}}_1\left(X^{*}\right)}{\∂x_k}={\[w_1^{T}M\left(X^{*}\right)w_1\]}^{-1}{w}_1^T\[\frac{\∂K\left(X^{*}\right)}{\∂x_k}-{\mathrm{\λ}}_1\frac{\∂M\left(X^{*}\right)}{\∂x_k}\]w_1---\left(23\right)$

In the dynamic trait problem of rope, multiplex eigenvalue is not had to occur, so here not for multiplex eigenvalue Situation is discussed.When applying equation (23), it would be desirable to calculate system stiffness and the derivative of rope tensility matrix, wherein K₁Lead Number can directly give its solution formula:

$\frac{\∂S}{\∂\left(EI\right)}=\frac{6}{a^4},\frac{\∂S}{\∂\left(H\right)}=\frac{2}{a^2},\frac{\∂D}{\∂\left(EI\right)}=-\frac{4}{a^4},\frac{\∂D}{\∂\left(H\right)}=-\frac{1}{a^2}---\left(24a\right)$

$\frac{\∂U}{\∂\left(EI\right)}=-\frac{4}{a^4},\frac{\∂U}{\∂\left(H\right)}=-\frac{1}{a^2},\frac{\∂V}{\∂\left(EI\right)}=\frac{1}{a^4},\frac{\∂W}{\∂\left(EI\right)}=\frac{1}{a^4}---\left(24b\right)$

$\frac{\∂Q}{\∂\left(EI\right)}=\frac{6}{a^4}+\frac{K_{rot-1}a-2{\mathrm{EI}}_0}{a^4(K_{rot-1}a+2{\mathrm{EI}}_0)}-\frac{4K_{rot-1}a}{{(K_{rot-1}a+2{\mathrm{EI}}_0)}^2}V(i=1)---\left(24c\right)$

$\frac{\∂Q}{\∂\left(H\right)}=\frac{\∂S}{\∂\left(H\right)}\frac{2}{a^2},\frac{\∂Q}{\∂\left(K_{rot-1}\right)}=\frac{4{\mathrm{EI}}_{0}a}{{(K_{rot-1}a+2{\mathrm{EI}}_0)}^2}V(i=1)---\left(24d\right)$

$\frac{\∂T}{\∂\left(EI\right)}=\frac{6}{a^4}+\frac{K_{rot-2}a-2{\mathrm{EI}}_{n+1}}{a^4(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}-\frac{4K_{rot-2}a}{{(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}^2}W(i=n)---\left(24e\right)$

$\frac{\∂T}{\∂\left(H\right)}=\frac{\∂S}{\∂\left(H\right)}\frac{2}{a^2},\frac{\∂T}{\∂\left(K_{rot-2}\right)}=\frac{4{\mathrm{EI}}_{n+1}a}{{(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}^2}W(i=n)---\left(24f\right)$

Stiffness matrix K₂It is that Static Correction is to systematic parameter in dead load (gravity) effect lower big displacement generation by rope Derivative solves as follows:

$\frac{dY}{{\mathrm{dx}}_k}=-{K_1}^{-1}\frac{d\left(K_1\right)}{\∂x_k}Y---\left(25\right)$

Wherein Y={y₁,y₂,…,y_n}^TRepresent Static Correction vector, y_iRepresent the Static Correction of i-th node.To formula (17a) two End derivation, can obtain:

$\frac{{\mathrm{dK}}_2(i,j)}{{\mathrm{dx}}_k}=\frac{d\left(r_i{s}_j\right)}{{\mathrm{dx}}_k}=\frac{d\left(r_i\right)}{{\mathrm{dx}}_k}{s}_j+r_i\frac{d\left(s_j\right)}{{\mathrm{dx}}_k}---\left(26\right)$

Wherein

$\frac{d\left(s_j\right)}{{\mathrm{dx}}_k}=\frac{1}{a^2}\[\frac{d\left(y_{j+1}\right)}{{\mathrm{dx}}_k}-2\frac{d\left(y\right)}{{\mathrm{dx}}_k}+\frac{d\left(y_{j-1}\right)}{{\mathrm{dx}}_k}\]---\left(27a\right)$

$\frac{d\left(r_i\right)}{{\mathrm{dx}}_k}=\frac{d\left(s_i\right)}{{\mathrm{dx}}_k}EA{\left(\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{t}_i^3\right)}^{-1}-{\left(\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{t}_i^3\right)}^{-2}\left(\underset{i=1}{\overset{n}{\Σ}}3t_i^2\frac{{\mathrm{dt}}_i}{{\mathrm{dx}}_k}\right){\mathrm{EAs}}_i---\left(27b\right)$

$\frac{{\mathrm{dt}}_i}{{\mathrm{dx}}_k}=\frac{1}{2a}{\[1+{\left(\frac{y_{i+1}-y_{i-1}}{2a}\right)}^2\]}^{-1/2}(\frac{{\mathrm{dy}}_{i+1}}{{\mathrm{dx}}_k}-\frac{{\mathrm{dy}}_{i-1}}{{\mathrm{dx}}_k})---\left(27c\right)$

Wherein i, j=1,2 ..., n；N is the dimension of stiffness matrix.Formula (25～27) provides one and solves stiffness matrix K₂To rope tensility, bending rigidity and the method for border rotational stiffness derivative.Additionally, K₂Axial rigidity is solved as follows:

$\frac{d\left(r_i\right)}{d\left(EA\right)}=s_i{\left(\underset{i=1}{\overset{n}{\Σ}}{t}_i^3\right)}^{-1}---\left(28\right)$

Either with or without the every systematic parameter listed, its derivative is zero.

After obtaining the drag-line frequency derivative to systematic parameter, can using the following method rope kinetic parameter be repaiied Just:

α Δ X=Δ λ (29a)

$\mathrm{\α}=\left[\begin{array}{cccc}\∂{\mathrm{\λ}}_1/\∂x_1& \∂{\mathrm{\λ}}_2/\∂x_1& ...& \∂{\mathrm{\λ}}_N/\∂x_1\\ \∂{\mathrm{\λ}}_2/\∂x_1& \∂{\mathrm{\λ}}_2/\∂x_2& ...& \∂{\mathrm{\λ}}_N/\∂x_2\\ ...& ...& ...& ...\\ \∂{\mathrm{\λ}}_N/\∂x_1& \∂{\mathrm{\λ}}_N/\∂x_2& ...& \∂{\mathrm{\λ}}_N/\∂x_m\end{array}\right]---\left(29b\right)$

Δ X={ Δ x₁ Δx₂...Δx_m}^T, Δ λ={ Δ λ₁ Δλ₂...Δλ_N}^T (29c,d)

Wherein m represents the quantity of parameter to be identified；N represents the frequency number recorded.Δλ_k=λ_k,measured-λ_k,calculated(X^k), X^kRepresent the result that kth time is revised.

When number of parameters to be identified is consistent with the frequency number recorded (m=N), formula (29a) can be with direct solution:

Δ X=α^-1Δλ (30)

When number of parameters to be identified is inconsistent with the frequency number recorded, formula (29a) can solve with method of least square:

Δ X=(α^Tα)^-1α^TΔλ (31)

After obtaining parameter error, parameter vector is modified: X_k+1=X_k+ΔX.Constantly repeat this makeover process, directly To the parameter vector obtaining convergence.

Embodiment

The application of the method is described below in conjunction with simply example, and the Practical Formula proposed with Zui et al. (1996) Method contrasts.The Practical Formula that Zui et al. (1996) proposes is as follows:

$\left\{\begin{array}{c}{S}_0=m{\left(f_{2}l\right)}^2\[1-1.40\frac{c}{f_2}-1.10{\left(\frac{c}{f_2}\right)}^2\],\mathrm{\ξ}\≥60\\ S_0=m{\left(f_{2}l\right)}^2\[1.03-6.33\frac{c}{f_2}-1.58{\left(\frac{c}{f_2}\right)}^2\],17\≤\mathrm{\ξ}\≤60\\ S_0=m{\left(f_{2}l\right)}^2\[1-1.40\frac{c}{f_2}-1.10{\left(\frac{c}{f_2}\right)}^2\],0\≤\mathrm{\ξ}\≤17\end{array}\right.---\left(32\right)$

Wherein:

Table 1 gives the parameter of 4 drag-lines, and in table 1, the 1st drag-line represents the situation of little sag and Low rigidity, and the 2nd is drawn Suo represents the situation of big sag and moderate stiffness, and the 3rd drag-line represents little sag and the situation of high rigidity, and the 4th Gen Suo represents big Sag and the situation of high rigidity.Table 2 gives these drag-lines frequency of vibration in the case of different inclination angle, in table 2 bracket Data are drawn by Practical Formula；Situation 1 refers to that drag-line is level；Situation 2 refers to that drag-line inclination angle is 30 °；Situation 3 refers to drag-line Inclination angle is 60 °；Boundary condition is clamped；Table medium frequency unit is Hz.

Table 3 gives the recognition result of Cable tension force.In table 3, boundary condition is clamped；The inclination angle of drag-line is 30°.Due to the effect of gravity, the rope tensility tilting drag-line is obeyed linear distribution, is given average rope in table 3 along string of a musical instrument direction Tension force.The effect at inclination angle makes drag-line lose the strict symmetrical or antisymmetric vibration shape, by the vibration shape close to symmetrical in table 3 Frequency as the 1st order frequency, the vibration shape close to antisymmetric frequency as the 2nd order frequency.Wherein No. 1, No. 3 and No. 4 drag-lines use Front two order frequencies be given in table 2 carry out Suo Li identification.And for No. 2 ropes, the derivative of frequency is attached at H=7.259e+005N Closely tending to level, this makes iterative also be difficult to provide high-precision result, gives the 3rd rank using No. 2 ropes in table 3 The solving result of frequency.Compare from table 3 understand, Practical Formula estimation result No. 2 ropes are had 7% error, and this Inventive method can obtain high-precision result.

Another problem that Practical Formula exists is, the method needs accurately to know the information such as the rigidity of drag-line, and requirement Boundary condition is only hinged or has just connect, but this is often difficult at Practical Project.Method in this paper can comprise draws The boundary condition of the sag of rope, rigidity, inclination angle and complexity, and can be with the many order frequencies recorded to rope tensility, rigidity and limit Boundary's rigidity is identified.Here, as a example by No. 4 ropes, the example of Multiparameter is given.If the tension force of drag-line, bending rigidity, axle It is respectively 10%, 20%, 10% and 20% to rigidity, border rotational stiffness with the relative error of data in table 2, applies the present invention The Multiparameter method proposed, table 4 gives the result of identification.Result shows, the method that the present invention proposes can be efficiently accurate Really identify the tension force of rope.The method that the present invention proposes is while identifying rope tensility, it is also possible to carry out other systematic parameter Correcting, this can improve the precision of test in actual applications further.

The parameter of 1, four typical drag-lines of table

Table 2, the calculating frequency of inclination drag-line

Table 3, the tension force identification of inclination drag-line

Table 4, the Multiparameter of inclination drag-line

The above is only the example explanation of the present invention, it is noted that come for those skilled in the art Saying, on the premise of without departing from the technology of the present invention principle, it is also possible to make some improvement and modification, these improve and modification also should It is considered as protection scope of the present invention.

Claims (4)

1. a high accuracy dynamic measurement of pile for rope tensility, is characterized in that, comprise the following steps:

Step one, by the model of vibration of suspension cable, sets up and combines cable sag effect, rigidity effect and the inhaul cable vibration of inclination angle factor Equation；

Step 2, uses finite difference calculus to carry out discrete to vibration equation, obtains its matrix expression；Calculate matrix exgenvalue, Obtain the frequency of vibration under given rope tensility；

Step 3, feature based value matrix is iterated solving, it is thus achieved that include under given frequency of vibration rope tensility, bending rigidity and The systematic parameter of axial rigidity.

The high accuracy dynamic measurement of pile of a kind of rope tensility the most according to claim 1, is characterized in that, in step one, and described knot The inhaul cable vibration equation closing cable sag effect, rigidity effect and inclination angle factor is:

$\frac{d^2}{{\mathrm{dx}}^2}\left(EI\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}\right)-H\frac{d^2\mathrm{\φ}}{{\mathrm{dx}}^2}-H^{\′}\frac{d\mathrm{\φ}}{dx}+\frac{{\∫}_0^L\frac{d^{2}y}{{\mathrm{dx}}^2}\mathrm{\φ}dx}{{\∫}_0^L\frac{{\left(ds/dx\right)}^3}{EA}dx}\frac{d^{2}y}{{\mathrm{dx}}^2}={\mathrm{m\ω}}^2\mathrm{\φ}$

Wherein, EI is drag-line bending stiffness, φ represent the vibration shape, H represent the rope tensility under drag-line resting state in the x-direction point Amount, y is the amount of deflection under drag-line resting state；L represents the rope projected length in x direction, and EA represents axial rigidity, and m is that unit is long The quality of degree rope；Ds=(dx²+dy²)^1/2Being the length of drag-line infinitesimal, ω is undamped frequency.

The high accuracy dynamic measurement of pile of a kind of rope tensility the most according to claim 2, is characterized in that, in step 2, and finite difference Point-score vibration equation is carried out discrete after obtain its matrix expression and be:

(K-ω²M) w=0

K=K₁+K₂；w^T={ w₁,w₂,...,w_n}

Wherein, n represents the drag-line interstitial content of division, drag-line n decile, and the projected length in x direction of each sections is for being designated as A, w are the column vectors of non-zero, and therefore, the ranks of frequency of vibration coefficient matrix are equal to zero, i.e. | K-ω²M |=0, this formula is carried out Eigenvalue solves the frequency of vibration that can solve drag-line；Wherein K₁Representing linear stiffness matrix, its expression formula is as follows:

$K_1=\left[\begin{array}{cccccc}Q& U& W& & & \\ D& S& U& W& & \\ V& D& S& U& \_& \\ & V& \_& \_& \_& W\\ & & \_& \_& S& U\\ & & & V& D& T\end{array}\right]$

Wherein:

$S=\frac{1}{a^4}\left(-2{\mathrm{EI}}_{i+1}+10{\mathrm{EI}}_i-2{\mathrm{EI}}_{i-1}\right)+\frac{2H_i}{a^2},D=\frac{1}{a^4}\left(2{\mathrm{EI}}_{i+1}-6{\mathrm{EI}}_i\right)-\frac{H_i}{a^2}+\frac{H_{i+1}-H_{i-1}}{4a^2},$

$U=\frac{1}{a^4}\left(-6{\mathrm{EI}}_i+2{\mathrm{EI}}_{i-1}\right)-\frac{H_i}{a^2}-\frac{H_{i+1}-H_{i-1}}{4a^2},V=-\frac{1}{2a^4}\left({\mathrm{EI}}_{i+1}-2{\mathrm{EI}}_i-{\mathrm{EI}}_{i-1}\right),$

$W=\frac{1}{2a^4}\left({\mathrm{EI}}_{i+1}+2{\mathrm{EI}}_i-{\mathrm{EI}}_{i-1}\right),Q=S+\frac{K_{rot-1}a-2{\mathrm{EI}}_0}{K_{rot-1}a+2{\mathrm{EI}}_0}V\left(i=1\right)$

$T=S+\frac{K_{rot-2}a-2{\mathrm{EI}}_{n+1}}{K_{rot-2}a+2{\mathrm{EI}}_{n+1}}W(i=n)$

Wherein, EI_iRepresent the bending stiffness at node i；EI₀And EI_n+1Represent the bending stiffness at two ends；K_rot-1and K_rot-2Table Show the rigidity of two ends rotation spring；K₂Representing nonlinear stiffness matrix, its expression formula is as follows:

K₂=rs^T

r^T={ r₁,r₂,...,r_n}s^T={ s₁,s₂,...,s_n}

Wherein

The mass matrix M of rope is a diagonal matrix:

M=diag{m₁,m₂,...,m_n}

Wherein m_iRepresent the i-th near nodal territorial unit linear mass.

The high accuracy dynamic measurement of pile of a kind of rope tensility the most according to claim 3, is characterized in that, K₁Bending stiffness and rope are opened Power derivative can directly give its solution formula:

$\frac{\∂S}{\∂\left(EI\right)}=\frac{6}{a^4},\frac{\∂S}{\∂\left(H\right)}=\frac{2}{a^2},\frac{\∂D}{\∂\left(EI\right)}=-\frac{4}{a^4},\frac{\∂D}{\∂\left(H\right)}=-\frac{1}{a^2}$

$\frac{\∂U}{\∂\left(EI\right)}=-\frac{4}{a^4},\frac{\∂U}{\∂\left(H\right)}=-\frac{1}{a^2},\frac{\∂V}{\∂\left(EI\right)}=\frac{1}{a^4},\frac{\∂W}{\∂\left(EI\right)}=\frac{1}{a^4}$

$\frac{\∂Q}{\∂\left(EI\right)}=\frac{6}{a^4}+\frac{K_{rot-1}a-2{\mathrm{EI}}_0}{a^4\left(K_{rot-1}a+2{\mathrm{EI}}_0\right)}-\frac{4K_{rot-1}a}{{\left(K_{rot-1}a+2{\mathrm{EI}}_0\right)}^2}V\left(i=1\right)$

$\frac{\∂Q}{\∂\left(H\right)}=\frac{\∂S}{\∂\left(H\right)}\frac{2}{a^2},\frac{\∂Q}{\∂\left(K_{rot-1}\right)}=\frac{4{\mathrm{EI}}_{0}a}{{\left(K_{rot-1}a+2{\mathrm{EI}}_0\right)}^2}V\left(i=1\right)$

$\frac{\∂T}{\∂\left(EI\right)}=\frac{6}{a^4}+\frac{K_{rot-2}a-2{\mathrm{EI}}_{n+1}}{a^4(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}-\frac{4K_{rot-2}a}{{(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}^2}W(i=n)$

$\frac{\∂T}{\∂\left(H\right)}=\frac{\∂S}{\∂\left(H\right)}\frac{2}{a^2},\frac{\∂T}{\∂\left(K_{rot-2}\right)}=\frac{4{\mathrm{EI}}_{n+1}a}{{(K_{rot-2}a+2{\mathrm{EI}}_{n+1})}^2}W(i=n)$

K₂Axial rigidity is solved as follows:

$\frac{d\left(r_i\right)}{d\left(EA\right)}=s_i{\left(\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{t}_i^3\right)}^{-1}$

After obtaining the drag-line frequency derivative to systematic parameter, can using the following method rope kinetic parameter be modified:

α Δ X=Δ λ

$\mathrm{\α}=\left[\begin{array}{cccc}\∂{\mathrm{\λ}}_1/\∂x_1& \∂{\mathrm{\λ}}_2/\∂x_1& ...& \∂{\mathrm{\λ}}_N/\∂x_1\\ \∂{\mathrm{\λ}}_2/\∂x_1& \∂{\mathrm{\λ}}_2/\∂x_2& ...& \∂{\mathrm{\λ}}_N/\∂x_2\\ ...& ...& ...& ...\\ \∂{\mathrm{\λ}}_N/\∂x_1& \∂{\mathrm{\λ}}_N/\∂x_2& ...& \∂{\mathrm{\λ}}_N/\∂x_m\end{array}\right]$

Δ X={ Δ x₁ Δx₂ ... Δx_m}^T, Δ λ={ Δ λ₁ Δλ₂ ... Δλ_N}^T

Wherein m represents the quantity of parameter to be identified；N represents the frequency number recorded.Δλ_k=λ_k,measured-λ_k,calculated(X^k),X^k Represent the result that kth time is revised；

When number of parameters to be identified is consistent with the frequency number recorded (m=N), above formula can be with direct solution:

Δ X=α^-1Δλ

When number of parameters to be identified is inconsistent with the frequency number recorded, above formula can solve with method of least square:

Δ X=(α^Tα)^-1α^TΔλ

After obtaining parameter error, parameter vector is modified: X_k+1=X_k+ΔX；Constantly this makeover process of iteration, until Parameter vector to convergence.