CN106595932A - Truncated total least squares-based bridge floor multiple axle moving load identifying method - Google Patents

Abstract

The invention discloses a truncated total least squares-based bridge floor multiple axle moving load identifying method comprising the following steps: 1) m displacement sensors are adhered to corresponding places x1, x2, ... xm on a bottom face of a bridge; measured displacement of a bridge floor multiple axle moving vehicle load fk(t) in time t at a position x is v(x,t); k=1, 2, 3..., is the number of vehicle axles; 2) a vibration differential equation is built; 3) an equation (1) is solved; 4) a system equation via which a multiple axle moving load is identified through displacement response is built when the bridge is under a load of a k axle vehicle; 5) truncated total least squares is used for obtaining an accurate value of the multiple axle moving load. The bridge displacement response measuring method disclosed in the invention is simple and high in accuracy, high feasibility can be realized, and the method can be widely used for identifying moving loads on all kinds of bridges.

Description

Recognition methodss based on the bridge floor multiaxis traveling load of truncated total least squares

Technical field

The invention belongs to bridge floor moving load identification technical field, more particularly to one kind is by bridge displacement identification bridge floor multiaxis The method of traveling load.

Background technology

Bridges in Our Country present situation is " rebuilding light supporting ", and from 1999 to 2013, domestic media disclosed report China because various The bridge that reason collapses up to more than 110 seats, wherein the bridge collapse still not caused including Wenchuan earthquake.Cause bridge damnification and break Bad reason can be summarized as external factor and internal factor, wherein in external factor due to automobile overload cause bridge fatigue damage and Durability reduces occupying leading position, and internal factor is then mainly bridge itself bearing capacity to be reduced and strength of materials degeneration.

With the explosive growth of highway in China traffic, the actual vehicle flowrate for bearing of many bridges increases compared with Earlier designs value A lot, the increase of speed and car weight can have a negative impact to bridge, the multiple-axle vehicle and large-scale multiple-axle vehicle especially overloads Appearance substantially exacerbate the risk of bridge collapse.

Highway in China transfinites to stand and make many work in control vehicle overload method, but measuring method is to adopt ground mostly at present Pound technology, i.e., the measurement weighed to realize gross combination weight by stopping.Under the trend of development rapid transit, how in vehicle row During sailing, accurate vehicle load has important practical meaning in engineering, especially the accurate measurement to each axle load of multiaxis lorry Safety and durability to protecting bridge all has very great help.

It is identified for conventional two-axle car more than existing moving load identification technology, it is impossible to which multiple-axle vehicle load is entered Row identification, therefore it is badly in need of a kind of method that can be identified to bridge floor multiaxis mobile vehicle load.

The content of the invention

It is an object of the invention to provide a kind of only need to measure the identification bridge floor multiaxis by bridge displacement is responded rapidly and efficiently Mobile vehicle load, accuracy of identification is high and does not affect bridge floor vehicle normal pass.

2., to reach above-mentioned purpose, the technical solution used in the present invention is：It is a kind of to be based on truncated total least squares

Bridge floor multiaxis traveling load recognition methodss, it is characterised in that：Comprise the following steps：

1), in bridge bottom surface correspondence position x₁,x₂,…x_mM displacement transducer is pasted at place respectively, measures the shifting of bridge floor multiaxis Dynamic vehicular load f_kT () displacement of t at the x position is v (x, t), k=1,2,3 ..., be the vehicle number of axle；

2), set up vehicle-bridge system oscillatory differential equation：Bridge length is taken for L, bending rigidity is EI, bridge unit length matter Measure as ρ, it is considered to which viscous damping simultaneously takes damped coefficient for C, ignores detrusion and the rotary inertia of bridge, bridge floor multiaxis locomotive Load fk (t) is moved right from beam left end supporting with speed c, then the oscillatory differential equation of vehicle-bridge system is：

Wherein δ (x-ct) is Dirac function；

The boundary condition of equation (1) is：

V (0, t)=0, v (L, t)=0,V (x, 0)=0,

3), equation (1) is solved；

4) bridge, is set up under k axle Vehicle Loads, multiaxis traveling load system equation is recognized by dynamic respond：

v_(m×1)=S_(m×k)·f_(k×1) (2)

v_(m×1)For traveling load f_kT () is in x₁,x₂,…x_mThe actual displacement at place, and m >=k；S_(m×k)For known system square Battle array；f_(k×1)For required k axle traveling loads；

The discrete form of formula (2) is expressed as：

Wherein

5) exact value of multiaxis traveling load, is tried to achieve using truncated total least squares；

Sytem matrix S and dynamic respond v in equation (2) is solved using truncated total least squares, is calculated first and is increased (S, singular value decomposition v) is for wide matrix：

Parameter in formula (4) is the parameter of singular value decomposition method for expressing；

Choose Truncation Parameters b to meet：

B≤min (n, rank (S, v)) (5)

Q=n-b+1 is taken, matrix in block form V is defined

Wherein V₁₁∈R^n×b, V₁₂∈R^n×q, v₂₂∈R^1×b, V₂₂∈R^1×q；

Here V₁₁, V₁₂, V₂₁, V₂₂All it is the matrix in block form inside matrix V, matrix V is divided into into four little matrixes exactly；R It is matrix total collection, V₁₁It is the matrix of a n rows b row, equally, V₂₂It is exactly the matrix of a 1 row q row, n, q, b are for table Show the line number and columns of matrix, in order to give one memory space of matrix in the calculation；

Then it is by the multiaxis traveling load that truncated total least squares are tried to achieve

Wherein V₂₂=(v_{N+1, b+1}..., v_{N+1, n+1})≠0。

Described step 3) in it is described to comprising the following steps that of solving to equation (1)：

Based on modal superposition principle, it is assumed that the n-th order Mode Shape function of bridge isThen equation (1) Solution be expressed as：

Matrix form is：

Here n be mode number, q_nT () (n=1,2 ... ∞) are n-th order modal displacements, equation (12) is substituted into equation (1), And x is integrated in [0, L], using boundary condition and Dirac function characteristic, vehicle-bridge system oscillatory differential equation q_n T () is expressed as：

HereFor q_nThe second dervative of (t),For q_nThe first derivative of (t), Respectively circular frequency, damping rate and bridge floor mobile vehicle load mod table Up to formula；

As vehicle has k axletree, and k-th axletree to the distance of first axletree isThen equation (14) It is written as：

Then correspond to the modal displacement at m measuring point to be expressed as by equation (13)：

X on bridge₁,x₂,…x_mThe speed at place is tried to achieve by the once differentiation of displacement：

Further, x on bridge₁,x₂,…x_mThe acceleration at place is tried to achieve by the second differential of displacement：

Similarly, x on beam₁,x₂,…x_mThe moment of flexure at place can utilize relational expressionTry to achieve：

If f₁,f₂,…,f_kFor each axle correspondence load of known k axles vehicle, ignore the impact of damping, then the solution of equation (1) can It is expressed as：

Wherein

The present invention can measure the method letter of bridge displacement response by measuring bridge displacement response identification multiaxis traveling load List and precision is higher, therefore there is good feasibility and accuracy of identification energy by bridge displacement response identification bridge floor traveling load Guarantee is accessed, dynamic respond need to only be obtained using method proposed by the present invention and be can recognize that bridge floor multiaxis traveling load, therefore Recognition methodss proposed by the present invention have good feasibility, can be widely applied to the moving load identification of all kinds bridge. In actual measurement process, vehicle traveling and surrounding enviroment unavoidably produce certain noise, and these noise signals are rung to displacement Ying Junyou certain interference, it is to ensure that recognition methodss are accurate effectively crucial to remove impact of the noise jamming to accuracy of identification.Cut Disconnected total least square eliminates noise polluted signal by blocking to measurement signal, effectively and recognition methodss is done Disturb, improve the accuracy of identification and noiseproof feature of recognition methodss.

Description of the drawings

Fig. 1 is method of the present invention flow chart.

Specific embodiment

As shown in figure 1, a kind of the invention discloses bridge floor multiaxis traveling load based on truncated total least squares Recognition methodss, comprise the following steps：

1), in bridge bottom surface correspondence position x₁,x₂,…x_mM displacement transducer is pasted at place respectively, measures the shifting of bridge floor multiaxis Dynamic vehicular load f_kT () displacement of t at the x position is the vehicle number of axle for v (x, t), k=1,2,3 ...；

2), set up vehicle-bridge system oscillatory differential equation：Bridge length is taken for L, bending rigidity is EI, bridge unit length matter Measure as ρ, it is considered to which viscous damping simultaneously takes damped coefficient for C, ignores detrusion and the rotary inertia of bridge, bridge floor multiaxis locomotive Load f_kT () is moved right from beam left end supporting with speed c, then the oscillatory differential equation of vehicle-bridge system is：

Wherein δ (x-ct) is Dirac function；

The boundary condition of equation (1) is：

V (0, t)=0, v (L, t)=0,V (x, 0)=0,

3), equation (1) is solved；

31), based on modal superposition principle, it is assumed that the n-th order Mode Shape function of beam isThen equation (1) solution is represented by：

Matrix form is：

Here n be mode number, q_nT () (n=1,2 ... ∞) are n-th order modal displacements, equation (12) is substituted into equation (1), And x is integrated in [0, L], using boundary condition and Dirac function characteristic, vehicle-bridge system oscillatory differential equation q_n T () is expressed as：

HereFor q_nThe second dervative of (t),For q_nThe first derivative of (t), Respectively circular frequency, damping rate and bridge floor mobile vehicle load mod table Up to formula.

As vehicle have k axletree, and k-th axletree to first axletree distance be x_k(x₁=0), then equation (14) It is written as：

Then correspond to the modal displacement at m measuring point to be expressed as by equation (13)：

X on bridge₁,x₂,…x_mThe speed at place is tried to achieve by the once differentiation of displacement：

Further, x on bridge₁,x₂,…x_mThe acceleration at place is tried to achieve by the second differential of displacement：

Similarly, x on beam₁,x₂,…x_mThe moment of flexure at place can utilize relational expressionTry to achieve：

If f₁,f₂,…,f_kFor each axle correspondence load of known k axles vehicle, ignore the impact of damping, then the solution of equation (1) can It is expressed as：

Wherein

4) bridge, is set up under k axle Vehicle Loads, multiaxis traveling load system equation is recognized by dynamic respond：

v_(m×1)=S_(m×k)·f_(k×1) (2)

v_(m×1)For traveling load f_kT () is in x₁,x₂,…x_mThe actual displacement at place (is exactly measured position in step (1) Move), and m >=k；S_(m×k)For known sytem matrix；f_(k×1)For required k axle traveling loads；

The discrete form of formula (2) is expressed as

Wherein

5) exact value of multiaxis traveling load, is tried to achieve using truncated total least squares；

Carrying out in solution procedure to equation (2), need that solving system matrix S's is inverse, to avoid sytem matrix morbid state from leading The accuracy of identification of cause is reduced, special to introduce the accuracy of identification that truncated total least squares improve multiple-axle vehicle time-histories load：

Sytem matrix S and dynamic respond v in equation (2) is solved using truncated total least squares, is calculated first and is increased (S, singular value decomposition v) is for wide matrix：

Parameter in formula (4) is the parameter of singular value decomposition method for expressing；

Choose Truncation Parameters b to meet：

B≤min (n, rank (S, v)) (5)

Q=n-b+1 is taken, matrix in block form V is defined

Wherein V₁₁∈R^n×b, V₁₂∈R^n×q, v₂₁∈R^1×b, V₂₂∈R^1×q；

Here V₁₁, V₁₂, V₂₁, V₂₂All it is the matrix in block form inside matrix V, matrix V is divided into into four little matrixes exactly；R It is matrix total collection, V₁₁It is the matrix of a n rows b row, equally, V₂₂It is exactly the matrix of a 1 row q row, n, q, b are for table Show the line number and columns of matrix, in order to give one memory space of matrix in the calculation；

Then it is by the multiaxis traveling load that truncated total least squares are tried to achieve

Wherein V₂₂=(v_{N+1, b+1}..., v_{N+1, n+1)}≠0。

Claims (2)

1. a kind of recognition methodss of the bridge floor multiaxis traveling load based on truncated total least squares, it is characterised in that：

Comprise the following steps：

1), in bridge bottom surface correspondence position x₁,x₂,…x_mM displacement transducer is pasted at place respectively, measures bridge floor multiaxis locomotive Load f_kT () displacement of t at the x position is v (x, t), k=1,2,3 ..., be the vehicle number of axle；

2), set up vehicle-bridge system oscillatory differential equation：Bridge length is taken for L, bending rigidity is EI, bridge linear mass is ρ, it is considered to which viscous damping simultaneously takes damped coefficient for C, ignores detrusion and the rotary inertia of bridge, bridge floor multiaxis mobile vehicle lotus Carry f_kT () is moved right from beam left end supporting with speed c, then the oscillatory differential equation of vehicle-bridge system is：

$\mathrm{\ρ}\·\frac{{\∂}^{2}v(x,t)}{\∂t^2}+C\·\frac{\∂v(x,t)}{\∂t}+EI\·\frac{{\∂}^{4}v(x,t)}{\∂x^4}=\mathrm{\δ}(x-ct)\·f\left(t\right)---\left(1\right)$

Wherein δ (x-ct) is Dirac function；

The boundary condition of equation (1) is：

V (0, t)=0, v (L, t)=0,V (x, 0)=0,

3), equation (1) is solved；

4) bridge, is set up under k axle Vehicle Loads, multiaxis traveling load system equation is recognized by dynamic respond：

v_(m×1)=S_(m×k)·f_(k×1) (2)

v_(m×1)For traveling load f_kT () is in x₁,x₂,…x_mThe actual displacement at place, and m >=k；S_(m×k)For known sytem matrix； f_(k×1)For required k axle traveling loads；

The discrete form of formula (2) is expressed as：

Wherein

5) exact value of multiaxis traveling load, is tried to achieve using truncated total least squares；

Sytem matrix S and dynamic respond v in equation (2) is solved using truncated total least squares, augmentation square is calculated first (S, singular value decomposition v) is battle array：

${\mathrm{U\σV}}^T={\Σ}_{i=1}^{n+1}{u}_i{\mathrm{\σ}}_i{v}_i,{\mathrm{\σ}}_1\≥{\mathrm{\σ}}_2\≥...\≥{\mathrm{\σ}}_{n+1}...\left(4\right)$

Parameter in formula (4) is the parameter of singular value decomposition method for expressing；

Choose Truncation Parameters b to meet：

B≤min (n, rank (S, v)) (5)

Q=n-b+1 is taken, matrix in block form V is defined

$V=\left[\begin{array}{cc}{V}_{11}& V_{12}\\ V_{21}& V_{22}\end{array}\right]---\left(6\right)$

Wherein v₁₁eR^n×b, C₁₂∈R^n×q, v₂₁∈R^1×b, V₂₂∈R^1×q；

Here V₁₁, V₁₂, V₂₁, V₂₂All it is the matrix in block form inside matrix V, matrix V is divided into into four little matrixes exactly；R is square Battle array total collection, V₁₁It is the matrix of a n rows b row, equally, V₂₂It is exactly the matrix of a 1 row q row, n, q, b are to represent square The line number and columns of battle array, in order to give one memory space of matrix in the calculation；

Then it is by the multiaxis traveling load that truncated total least squares are tried to achieve

$f_b=-V_{12}{V}_{22}^T\left|\right|V_{22}|{|}_2^{-2}---\left(7\right)$

Wherein V₂₂=(v_n+1,b+1,…,b_n+1,n+1)≠0。

2. the recognition methodss of the bridge floor multiaxis traveling load based on truncated total least squares as claimed in claim 1, its It is characterised by：Described step 3) in equation (1) is solved comprise the following steps that it is described：

Based on modal superposition principle, it is assumed that the n-th order Mode Shape function of bridge isThe then solution of equation (1) It is expressed as：

$v(x,t)=\underset{n=1}{\overset{\mathrm{\∞}}{\Σ}}{\mathrm{\φ}}_n\left(x\right)\·q_n\left(t\right)---\left(12\right)$

Matrix form is：

$v=\left\{\begin{array}{cccc}sin\frac{\mathrm{\π}x}{L}& sin\frac{2\mathrm{\π}x}{L}& ...& sin\frac{n\mathrm{\π}x}{L}\end{array}\right\}{\left\{\begin{array}{cccc}{q}_1& q_2& ...& q_n\end{array}\right\}}^T---\left(13\right)$

Here n be mode number, q_nT () (n=1,2 ... ∞) are n-th order modal displacements, by equation (12) substitution equation (1), and X is integrated in [0, L], using boundary condition and Dirac function characteristic, vehicle-bridge system oscillatory differential equation q_n(t) table It is shown as：

${\stackrel{\·\·}{q}}_n\left(t\right)+2{\mathrm{\ξ}}_n{\mathrm{\ω}}_n{\stackrel{\·}{q}}_n\left(t\right)+{\mathrm{\ω}}_n^2{q}_n\left(t\right)=\frac{2}{\mathrm{\ρ}L}{p}_n\left(t\right),(n=1,2,...,\mathrm{\∞})---\left(14\right)$

HereFor q_nThe second dervative of (t),For q_nThe first derivative of (t), Respectively circular frequency, damping rate and bridge floor mobile vehicle load mod table Up to formula；

As vehicle has k axletree, and k-th axletree to the distance of first axletree isThen equation (14) is written as：

Then correspond to the modal displacement at m measuring point to be expressed as by equation (13)：

X on bridge₁,x₂,…x_mThe speed at place is tried to achieve by the once differentiation of displacement：

Further, x on bridge₁,x₂,…x_mThe acceleration at place is tried to achieve by the second differential of displacement：

Similarly, x on beam₁,x₂,…x_mThe moment of flexure at place can utilize relational expressionTry to achieve：

If f₁,f₂,…,f_kFor each axle correspondence load of known k axles vehicle, ignore the impact of damping, then the solution of equation (1) can be represented For：

$v(x,t)=\frac{L^3}{48EI}\underset{i=1}{\overset{k}{\Σ}}{f}_i\underset{n=1}{\overset{\mathrm{\∞}}{\Σ}}\frac{1}{n^2(n^2-{\mathrm{\α}}^2)}\sin \frac{n\mathrm{\π}x}{L}(\sin \frac{n\mathrm{\π}(ct-{\stackrel{\‾}{x}}_i)}{L}-\frac{\mathrm{\α}}{n}{\mathrm{sin\ω}}_n(t-\frac{{\stackrel{\‾}{x}}_i}{c}))---\left(10\right)$

Wherein