CN103926836A - Shock absorption control method of aqueduct structure under action of random loads - Google Patents

Abstract

The invention discloses a shock absorption control method of an aqueduct structure under the action of random loads. The method includes the steps that r controllers are added onto an n-freedom-degree structure, and a semi-active control system motion equation of the structure under input of random loads is established; a virtual stimulation method is adopted for simulating the random loads to introduce a state vector, and a state equation of the structure is determined; a performance objective function of a system is defined, a minimum value of the performance objective function is used, and feedback gain can be obtained; semi-active control is performed on virtual response through a magnetorheological damper, and therefore random shock of the aqueduct structure can be effectively controlled. According to the method, the semi-active control method is adopted for calculating the aqueduct structure under the action of the random loads, and random features of actual stimulation are represented with the virtual stimulation method; the method is simple and convenient to operate and solves the problem that the semi-automatic control system is influenced because output variables are disturbed randomly.

Description

The shock-absorbing control method of a kind of analysis on aqueduct structure under random load effect

Technical field

The invention belongs to hydraulic engineering aqueduct active control technology field, relate in particular to the shock-absorbing control method of a kind of analysis on aqueduct structure under random load effect.

Background technology

At present, South to Northern Water Diversion Project of China is the large hydraulic engineering of building for alleviating the present situation of China North China and Water Resource in Northwest shortage, analysis on aqueduct structure is as the important component part in south water to north water delivery engineering, on Route, to apply one of more main crossing, and the water delivery line of work centerline passes through China's Areas of High Earthquake Intensity region, the seismic fortification intensity of the many sections of circuit reaches 8 degree, 49 of the total aqueducts of whole engineering water delivery general main canal, cumulative length reaches 5520m, and flow reaches 500m ³/ s, once there is earthquake, water delivery completely will be interrupted, and it closes on important trunk railways such as capital is wide, capital nine, may cause serious secondary disaster, so the quake-resistant safety of water delivery engineering be extremely important.

Because earthquake has the inconsistent feature of spatial and temporal distributions, randomness is very strong, controls the short-cut method that adopt at present for the damping of analysis on aqueduct structure more, chooses some definite earthquake motion input, and less to the discussion of randomness.

In addition, traditional random vibration law theory and computation process are all comparatively complicated, not high to the solution efficiency of non-linear dynamic problem, be difficult to extend to practical engineering application, make actual damping control effect and whether can obtain to a certain degree uncertain of existence, so adopt efficient means to have great theoretical and practical significance to the aqueduct vibration control under random load effect.It is very important that an effective Active Control Design is obtained good effect for half active control system, considers that output variable exists random perturbation, and controlled process is driven by process noise and control signal, can affect Semi-automatic control system.

Summary of the invention

The object of the embodiment of the present invention is to provide the shock-absorbing control method of a kind of analysis on aqueduct structure under random load effect, is intended to solve output variable and has random perturbation, affects the problem of Semi-automatic control system.

The embodiment of the present invention is achieved in that the shock-absorbing control method of a kind of analysis on aqueduct structure under random load effect, and the shock-absorbing control method of this analysis on aqueduct structure under random load effect comprises the following steps:

The first step is added r controller in the structure of n degree of freedom, and the structure half active control system equation of motion under the consistent input of earthquake motion is:

$M\stackrel{\·\·}{X}\left(t\right)+C\stackrel{\·}{X}\left(t\right)+\mathrm{KX}\left(t\right)=B_{s}U\left(t\right)+M{H}_0{\stackrel{\·\·}{X}}_g\left(t\right)$

X(t ₀)＝X ₀

$\stackrel{\·}{X}\left(t_0\right)={\stackrel{\·}{X}}_0$

In formula, M, C and K are respectively quality, damping and the stiffness matrix of structure n * n dimension; X (t), with respectively displacement, speed and the vector acceleration of structure n dimension; it is earthquake ground motion acceleration time-histories; U (t) is r * 1 dimension control vector; H ₀it is n * 1 dimension earthquake ground motion acceleration location matrix; B _sthat n * r ties up half ACTIVE CONTROL damper position matrix;

Second step, introduces state vector the equation of motion of structure can be expressed as state equation:

$\stackrel{\·}{Z}\left(t\right)=\mathrm{AZ}\left(t\right)+\mathrm{BU}\left(t\right)+{\stackrel{\·\·}{X}}_g\left(t\right)$

Z(t ₀)＝Z ₀

$Y=C\{\stackrel{\·}{Z}\left(t\right)-H{\stackrel{\·\·}{X}}_g\left(t\right)\}+v\left(t\right)$

Wherein, $\left[\begin{array}{cc}0& I\\ -M^{-1}K& -M^{-1}C\end{array}\right],B=\left[\begin{array}{c}0\\ M^{-1}{B}_s\end{array}\right],H=\left[\begin{array}{c}0\\ H_0\end{array}\right],$ C=[0I], Y is observation output, is that absolute acceleration is usingd as feedback herein, v (t) is for measuring noise;

The 3rd step, the performance objective function of define system is

$\underset{0}{\overset{\∞}{\∫}}{Z}^T\mathrm{QZ}+U^T\mathrm{RU}$

Wherein, Q and R are weight matrix, make performance objective function get minimum value, can be in the hope of feedback gain G=R ^-1b ^tp is r * 2n dimension feedback of status gain matrix; P is that 2n * 2n ties up matrix, can be by Riccati matrix algebra equation solution;

The 4th step, owing to being fed back to absolute acceleration, so need to introduce state estimation vector replace state vector Z, make for output feedback problem, keep optimum, state estimation vector by Kalman wave filter, produced:

$\stackrel{\stackrel{\·}{~}}{Z}\left(t\right)=A\tilde{Z}\left(t\right)+\mathrm{BU}\left(t\right)+L[Y\left(t\right)-\tilde{Y}\left(t\right)]$

Formula median filter gain L can be expressed as

L＝P ₁C ^TV ^-1

V=E[v (t) v ^t(t)] for measuring noise covariance matrix, E[] expression mathematical expectation, P ₁solution for following formula

AP ₁+P ₁A ^T+N-P ₁C ^TV ^-1CP ₁＝0

In formula, N=E[w (t) w ^t(t)] be input noise covariance matrix, Kalman wave filter makes the progressive covariance of state estimation error $\left(\underset{t\→\∞}{\lim }E\right\{[Z\left(t\right)-\tilde{Z}\left(t\right)][Z\left(t\right)-\tilde{Z}\left(t\right){]}^T\})$ Reach minimum.

Further, the shock-absorbing control method of this analysis on aqueduct structure under random load effect comprises the following steps by the analysis on aqueduct structure random earthquake response calculating of pseudo-excitation method:

Step 1, analysis on aqueduct structure is encouraged by earthquake single-point To Stationary Random Seismic shock excitation from spectrum for known, in order to calculate spectrum certainly or the cross-spectrum of 2 kinds of any response vectors { y (t) } and { z (t) }, can first constructing virtual harmonic excitation:

${\stackrel{\·\·}{x}}_g\left(t\right)=\sqrt{S_{{\stackrel{\·\·}{x}}_g}\left(\mathrm{\ω}\right)e^{\mathrm{i\ωt}}}$

Then calculate corresponding harmonic response y} and z}, the spectral power matrix that can obtain random response { y (t) } is:

[S _yy(ω)]＝{y} ^*{y} ^T

And the cross-power spectrum matrix of { y (t) } and { z (t) } is:

[S _yz(ω)]＝{y} ^*{z} ^T

Step 2, for the analysis on aqueduct structure of n degree of freedom at the equation of motion bearing under the excitation of single-point Stochastic earthquake is:

$\[M\]\left\{\stackrel{\·\·}{y}\right\}+\[C\]\left\{\stackrel{\·}{y}\right\}+\[K\]\left\{y\right\}=-\[M\]\left\{R\right\}{\stackrel{\·\·}{x}}_g\left(t\right)$

In formula: M, C, K is respectively analysis on aqueduct structure oeverall quality matrix, damping matrix and stiffness matrix, and y} is structure vector displacement, R} is influence matrix,

By formula ${\stackrel{\·\·}{x}}_g\left(t\right)=\sqrt{S_{{\stackrel{\·\·}{x}}_g}\left(\mathrm{\ω}\right)e^{\mathrm{i\ωt}}}$ Substitution formula $\[M\]\left\{\stackrel{\·\·}{y}\right\}+\[C\]\left\{\stackrel{\·}{y}\right\}+\[K\]\left\{y\right\}=-\[M\]\left\{R\right\}{\stackrel{\·\·}{x}}_g\left(t\right),$ Can obtain:

$\[M\]\left\{\stackrel{\·\·}{y}\right\}+\[C\]\left\{\stackrel{\·}{y}\right\}+\[K\]\left\{y\right\}=-\[M\]\left\{R\right\}\sqrt{S_{{\stackrel{\·\·}{x}}_g}\left(\mathrm{\ω}\right)e^{\mathrm{i\ωt}}}$

Steady state solution is:

{y(t)}＝{Y(ω)}e ^iωt

Displacement the spectral power matrix of y} is:

[S _yy(ω)]＝{y} ^*{y} ^T＝{Y(ω)} ^*{Y(ω)} ^T

All sides response that can obtain displacement according to the relation between variance and related function is:

${\mathrm{\σ}}_x^2={\∫}_{-\∞}^{+\∞}\left[S_{\mathrm{yy}}\left(\mathrm{\ω}\right)\right]\mathrm{d\ω}$

Step 3, seismic stimulation model:

Seismic acceleration power spectral density function adopts Clough-Penzien spectrum:

$S_f\left(\mathrm{\ω}\right)=\frac{{\mathrm{\ω}}_0^4+4{\mathrm{\ζ}}_0^2{\mathrm{\ω}}_0^2{\mathrm{\ω}}^2}{{({\mathrm{\ω}}^2-{\mathrm{\ω}}_0^2)}^2+4{\mathrm{\ζ}}_0^2{\mathrm{\ω}}_0^2{\mathrm{\ω}}^2}\×\frac{{\mathrm{\ω}}^4}{{({\mathrm{\ω}}^2-{\mathrm{\ω}}_f^2)}^4+4{\mathrm{\ζ}}_f^2{\mathrm{\ω}}_f^2{\mathrm{\ω}}^2}{S}_0$

Wherein: S ₀for the spectral intensity of white-noise excitation on basement rock, ζ ₀for the damping ratio of ground, ω ₀for the natural frequency of vibration of ground, ζ _fand ω _fbe respectively damping ratio and the natural frequency of vibration of Hi-pass filter, corresponding fortification intensity is the II class place of 7 degree.

The shock-absorbing control method of analysis on aqueduct structure provided by the invention under random load effect, adopt LQR/LQG control law to carry out ACTIVE CONTROL, LQG controller is comprised of optimum state feedback gain and Kalman wave filter two parts, wherein optimum state feedback gain adopts classical linear optimal active control algorithm LQR to try to achieve, method is simple, easy to operate, preferably resolve because output variable exists random perturbation, affect the problem of Semi-automatic control system.

Accompanying drawing explanation

Fig. 1 is the shock-absorbing control method process flow diagram of the analysis on aqueduct structure that provides of the embodiment of the present invention under random load effect.

Embodiment

In order to make object of the present invention, technical scheme and advantage clearer, below in conjunction with embodiment, the present invention is further elaborated.Should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not intended to limit the present invention.

Below in conjunction with drawings and the specific embodiments, application principle of the present invention is further described.

As shown in Figure 1, the shock-absorbing control method of the analysis on aqueduct structure of the embodiment of the present invention under random load effect comprises the following steps:

S101: add r controller in the structure of n degree of freedom, the structure half active control system equation of motion under the consistent input of earthquake motion;

S102: introduce state vector, determine the equation of motion of structure;

S103: the performance objective function of define system, make performance objective function get minimum value, can be in the hope of feedback gain;

S104: introduce state estimation vector and replace state vector, make to keep optimum for output feedback problem, state estimation vector is produced by Kalman wave filter, and Kalman wave filter makes the progressive covariance of state estimation error reach minimum.

Principle of work of the present invention and concrete grammar are:

Half Algorithm of Active Control of the present invention is on the design basis based on ACTIVE CONTROL, consider that output variable exists random perturbation, controlled process is driven by process noise and control signal, adopt LQR/LQG control law to carry out the design of ACTIVE CONTROL, LQG controller is comprised of optimum state feedback gain and Kalman wave filter two parts, wherein optimum state feedback gain adopts classical linear optimal active control algorithm LQR to try to achieve, and algorithm is as follows:

The first step first, is added r controller in the structure of n degree of freedom, and the structure half active control system equation of motion under the consistent input of earthquake motion is:

$M\stackrel{\·\·}{X}\left(t\right)+C\stackrel{\·}{X}\left(t\right)+\mathrm{KX}\left(t\right)=B_{s}U\left(t\right)+M{H}_0{\stackrel{\·\·}{X}}_g\left(t\right)$

X(t ₀)＝X ₀

$\stackrel{\·}{X}\left(t_0\right)={\stackrel{\·}{X}}_0$

In formula, M, C and K are respectively quality, damping and the stiffness matrix of structure n * n dimension; X (t), with respectively displacement, speed and the vector acceleration of structure n dimension; it is earthquake ground motion acceleration time-histories; U (t) is r * 1 dimension control vector; H ₀it is n * 1 dimension earthquake ground motion acceleration location matrix; B _sthat n * r ties up half ACTIVE CONTROL damper position matrix;

Second step, introduces state vector the equation of motion of structure can be expressed as state equation:

$\stackrel{\·}{Z}\left(t\right)=\mathrm{AZ}\left(t\right)+\mathrm{BU}\left(t\right)+{\stackrel{\·\·}{X}}_g\left(t\right)$

Z(t ₀)＝Z ₀

$Y=C\{\stackrel{\·}{Z}\left(t\right)-H{\stackrel{\·\·}{X}}_g\left(t\right)\}+v\left(t\right)$

Wherein, $\left[\begin{array}{cc}0& I\\ -M^{-1}K& -M^{-1}C\end{array}\right],B=\left[\begin{array}{c}0\\ M^{-1}{B}_s\end{array}\right],H=\left[\begin{array}{c}0\\ H_0\end{array}\right],$ C=[0I], Y is observation output, is that absolute acceleration is usingd as feedback herein, v (t) is for measuring noise;

The 3rd step, the performance objective function of define system is

$\underset{0}{\overset{\∞}{\∫}}{Z}^T\mathrm{QZ}+U^T\mathrm{RU}$

Wherein, Q and R are weight matrix, make performance objective function get minimum value, can be in the hope of feedback gain G=R ^-1b ^tp is r * 2n dimension feedback of status gain matrix; P is that 2n * 2n ties up matrix, can be by Riccati matrix algebra equation solution;

The 4th step, owing to being fed back to absolute acceleration, so need to introduce state estimation vector replace state vector Z, make for output feedback problem, keep optimum, state estimation vector by Kalman wave filter, produced:

$\stackrel{\stackrel{\·}{~}}{Z}\left(t\right)=A\tilde{Z}\left(t\right)+\mathrm{BU}\left(t\right)+L[Y\left(t\right)-\tilde{Y}\left(t\right)]$

Formula median filter gain L can be expressed as

L＝P ₁C ^TV ^-1

V=E[v (t) v ^t(t)] for measuring noise covariance matrix, E[] expression mathematical expectation, P ₁solution for following formula

AP ₁+P ₁A ^T+N-P ₁C ^TV ^-1CP ₁＝0

In formula, N=E[w (t) w ^t(t)] be input noise covariance matrix, Kalman wave filter makes the progressive covariance of state estimation error $\left(\underset{t\→\∞}{\lim }E\right\{[Z\left(t\right)-\tilde{Z}\left(t\right)][Z\left(t\right)-\tilde{Z}\left(t\right){]}^T\})$ Reach minimum.

The embodiment of the present invention comprises the following steps by the analysis on aqueduct structure random earthquake response calculating of pseudo-excitation method:

1. analysis on aqueduct structure is encouraged by earthquake single-point To Stationary Random Seismic shock excitation it is from spectrum for known, in order to calculate spectrum certainly or the cross-spectrum of 2 kinds of any response vectors { y (t) } and { z (t) }, can first constructing virtual harmonic excitation:

${\stackrel{\·\·}{x}}_g\left(t\right)=\sqrt{S_{{\stackrel{\·\·}{x}}_g}\left(\mathrm{\ω}\right)e^{\mathrm{i\ωt}}}\backslash *\mathrm{MERGEFORMAT}\left(1\right)$

Then calculate corresponding harmonic response y} and z}, the spectral power matrix that can obtain random response { y (t) } is:

[S _yy(ω)]＝{y} ^*{y} ^T\*MERGEFORMAT(2)

And the cross-power spectrum matrix of { y (t) } and { z (t) } is:

[S _yz(ω)]＝{y} ^*{z} ^T\*MERGEFORMAT(3)

2. the analysis on aqueduct structure for n degree of freedom at the equation of motion bearing under the excitation of single-point Stochastic earthquake is:

$\[M\]\left\{\stackrel{\·\·}{y}\right\}+\[C\]\left\{\stackrel{\·}{y}\right\}+\[K\]\left\{y\right\}=-\[M\]\left\{R\right\}{\stackrel{\·\·}{x}}_g\left(t\right)\backslash *\mathrm{MERGEFORMAT}\left(4\right)$

In formula: M, C, K is respectively analysis on aqueduct structure oeverall quality matrix, damping matrix and stiffness matrix, and { y} is structure vector displacement, and { R} is influence matrix.

By formula (1) substitution formula (4), can obtain:

$\[M\]\left\{\stackrel{\·\·}{y}\right\}+\[C\]\left\{\stackrel{\·}{y}\right\}+\[K\]\left\{y\right\}=-\[M\]\left\{R\right\}\sqrt{S_{{\stackrel{\·\·}{x}}_g}\left(\mathrm{\ω}\right)e^{\mathrm{i\ωt}}}\backslash *\mathrm{MERGEFORMAT}\left(5\right)$

Its steady state solution is:

{y(t)}＝{Y(ω)}e ^iωt\*MERGEFORMAT(6)

Displacement the spectral power matrix of y} is:

[S _yy(ω)]＝{y} ^*{y} ^T＝{Y(ω)} ^*{Y(ω)} ^T\*MERGEFORMAT(7)

All sides response that can obtain displacement according to the relation between variance and related function is:

${\mathrm{\σ}}_X^2={\∫}_{-\∞}^{+\∞}\left[s_{\mathrm{yy}}\left(\mathrm{\ω}\right)\right]\mathrm{d\ω}\backslash *\mathrm{MERGEFORMAT}\left(8\right)$

3. seismic stimulation model

Seismic acceleration power spectral density function adopts Clough-Penzien spectrum:

$\begin{array}{c}{S}_f\left(\mathrm{\ω}\right)=\frac{{\mathrm{\ω}}_0^4+4{\mathrm{\ζ}}_0^2{\mathrm{\ω}}_0^2{\mathrm{\ω}}^2}{{({\mathrm{\ω}}^2-{\mathrm{\ω}}_0^2)}^2+4{\mathrm{\ζ}}_0^2{\mathrm{\ω}}_0^2{\mathrm{\ω}}^2}\×\frac{{\mathrm{\ω}}^4}{{({\mathrm{\ω}}^2-{\mathrm{\ω}}_f^2)}^4+4{\mathrm{\ζ}}_f^2{\mathrm{\ω}}_f^2{\mathrm{\ω}}^2}{S}_0\\ \backslash *\mathrm{MERGEFORMAT}\left(9\right)\end{array}$

Wherein: S ₀for the spectral intensity of white-noise excitation on basement rock, ζ ₀for the damping ratio of ground, ω ₀for the natural frequency of vibration of ground, ζ _fand ω _fbe respectively damping ratio and the natural frequency of vibration of Hi-pass filter.Corresponding fortification intensity is the II class place of 7 degree, and parameter can be chosen according to table 1.

Table 1

The foregoing is only preferred embodiment of the present invention, not in order to limit the present invention, all any modifications of doing within the spirit and principles in the present invention, be equal to and replace and improvement etc., within all should being included in protection scope of the present invention.

Claims (2)

1. the shock-absorbing control method of analysis on aqueduct structure under random load effect, is characterized in that, the shock-absorbing control method of this analysis on aqueduct structure under random load effect comprises the following steps:

The first step is added r controller in the structure of n degree of freedom, and the structure half active control system equation of motion under the consistent input of earthquake motion is:

$M\stackrel{\·\·}{X}\left(t\right)+C\stackrel{\·}{X}\left(t\right)+\mathrm{KX}\left(t\right)=B_{s}U\left(t\right)+M{H}_0{\stackrel{\·\·}{X}}_g\left(t\right)$

X(t ₀)＝X ₀

$\stackrel{\·}{X}\left(t_0\right)={\stackrel{\·}{X}}_0$

In formula, M, C and K are respectively quality, damping and the stiffness matrix of structure n * n dimension; X (t), with respectively displacement, speed and the vector acceleration of structure n dimension; it is earthquake ground motion acceleration time-histories; U (t) is r * 1 dimension control vector; H ₀it is n * 1 dimension earthquake ground motion acceleration location matrix; B _sthat n * r ties up half ACTIVE CONTROL damper position matrix;

Second step, introduces state vector the equation of motion of structure can be expressed as state equation:

$\stackrel{\·}{Z}\left(t\right)=\mathrm{AZ}\left(t\right)+\mathrm{BU}\left(t\right)+{\stackrel{\·\·}{X}}_g\left(t\right)$

Z(t ₀)＝Z ₀

$Y=C\{\stackrel{\·}{Z}\left(t\right)-H{\stackrel{\·\·}{X}}_g\left(t\right)\}+v\left(t\right)$

Wherein, $\left[\begin{array}{cc}0& I\\ -M^{-1}K& -M^{-1}C\end{array}\right],B=\left[\begin{array}{c}0\\ M^{-1}{B}_s\end{array}\right],H=\left[\begin{array}{c}0\\ H_0\end{array}\right],$ C=[0I], Y is observation output, is that absolute acceleration is usingd as feedback herein, v (t) is for measuring noise;

The 3rd step, the performance objective function of define system is

$\underset{0}{\overset{\∞}{\∫}}{Z}^T\mathrm{QZ}+U^T\mathrm{RU}$

Wherein, Q and R are weight matrix, make performance objective function get minimum value, can be in the hope of feedback gain G=R ^-1b ^tp is r * 2n dimension feedback of status gain matrix; P is that 2n * 2n ties up matrix, can be by Riccati matrix algebra equation solution;

The 4th step, owing to being fed back to absolute acceleration, so need to introduce state estimation vector replace state vector Z, make for output feedback problem, keep optimum, state estimation vector by Kalman wave filter, produced:

$\stackrel{\stackrel{\·}{~}}{Z}\left(t\right)=A\tilde{Z}\left(t\right)+\mathrm{BU}\left(t\right)+L[Y\left(t\right)-\tilde{Y}\left(t\right)]$

Formula median filter gain L can be expressed as

L＝P ₁C ^TV ^-1

V=E[v (t) v ^t(t)] for measuring noise covariance matrix, E[] expression mathematical expectation, P ₁solution for following formula

AP ₁+P ₁A ^T+N-P ₁C ^TV ^-1CP ₁＝0

In formula, N=E[w (t) w ^t(t)] be input noise covariance matrix, Kalman wave filter makes the progressive covariance of state estimation error $\left(\underset{t\→\∞}{\lim }E\right\{[Z\left(t\right)-\tilde{Z}\left(t\right)][Z\left(t\right)-\tilde{Z}\left(t\right){]}^T\})$ Reach minimum.

2. the shock-absorbing control method of analysis on aqueduct structure as claimed in claim 1 under random load effect, it is characterized in that, the shock-absorbing control method of this analysis on aqueduct structure under random load effect comprises the following steps by the analysis on aqueduct structure random earthquake response calculating of pseudo-excitation method:

Step 1, analysis on aqueduct structure is encouraged by earthquake single-point To Stationary Random Seismic shock excitation from spectrum for known, in order to calculate spectrum certainly or the cross-spectrum of 2 kinds of any response vectors { y (t) } and { z (t) }, can first constructing virtual harmonic excitation:

${\stackrel{\·\·}{x}}_g\left(t\right)=\sqrt{S_{{\stackrel{\·\·}{x}}_g}\left(\mathrm{\ω}\right)e^{\mathrm{i\ωt}}}$

Then calculate corresponding harmonic response y} and z}, the spectral power matrix that can obtain random response { y (t) } is:

[S _yy(ω)]＝{y} ^*{y} ^T

And the cross-power spectrum matrix of { y (t) } and { z (t) } is:

[S _yz(ω)]＝{y} ^*{z} ^T

Step 2, for the analysis on aqueduct structure of n degree of freedom at the equation of motion bearing under the excitation of single-point Stochastic earthquake is:

$\[M\]\left\{\stackrel{\·\·}{y}\right\}+\[C\]\left\{\stackrel{\·}{y}\right\}+\[K\]\left\{y\right\}=-\[M\]\left\{R\right\}{\stackrel{\·\·}{x}}_g\left(t\right)$

In formula: M, C, K is respectively analysis on aqueduct structure oeverall quality matrix, damping matrix and stiffness matrix, and y} is structure vector displacement, R} is influence matrix,

By formula ${\stackrel{\·\·}{x}}_g\left(t\right)=\sqrt{S_{{\stackrel{\·\·}{x}}_g}\left(\mathrm{\ω}\right)e^{\mathrm{i\ωt}}}$ Substitution formula $\[M\]\left\{\stackrel{\·\·}{y}\right\}+\[C\]\left\{\stackrel{\·}{y}\right\}+\[K\]\left\{y\right\}=-\[M\]\left\{R\right\}{\stackrel{\·\·}{x}}_g\left(t\right),$ Can obtain:

$\[M\]\left\{\stackrel{\·\·}{y}\right\}+\[C\]\left\{\stackrel{\·}{y}\right\}+\[K\]\left\{y\right\}=-\[M\]\left\{R\right\}\sqrt{S_{{\stackrel{\·\·}{x}}_g}\left(\mathrm{\ω}\right)e^{\mathrm{i\ωt}}}$

Steady state solution is:

{y(t)}＝{Y(ω)}e ^iωt

Displacement the spectral power matrix of y} is:

[S _yy(ω)]＝{y} ^*{y} ^T＝{Y(ω)} ^*{Y(ω)} ^T

All sides response that can obtain displacement according to the relation between variance and related function is:

${\mathrm{\σ}}_x^2={\∫}_{-\∞}^{+\∞}\left[S_{\mathrm{yy}}\left(\mathrm{\ω}\right)\right]\mathrm{d\ω}$

Step 3, seismic stimulation model:

Seismic acceleration power spectral density function adopts Clough-Penzien spectrum:

$S_f\left(\mathrm{\ω}\right)=\frac{{\mathrm{\ω}}_0^4+4{\mathrm{\ζ}}_0^2{\mathrm{\ω}}_0^2{\mathrm{\ω}}^2}{{({\mathrm{\ω}}^2-{\mathrm{\ω}}_0^2)}^2+4{\mathrm{\ζ}}_0^2{\mathrm{\ω}}_0^2{\mathrm{\ω}}^2}\×\frac{{\mathrm{\ω}}^4}{{({\mathrm{\ω}}^2-{\mathrm{\ω}}_f^2)}^4+4{\mathrm{\ζ}}_f^2{\mathrm{\ω}}_f^2{\mathrm{\ω}}^2}{S}_0$

Wherein: S ₀for the spectral intensity of white-noise excitation on basement rock, ζ ₀for the damping ratio of ground, ω ₀for the natural frequency of vibration of ground, ζ _fand ω _fbe respectively damping ratio and the natural frequency of vibration of Hi-pass filter, corresponding fortification intensity is the II class place of 7 degree.