CN110083920B - Analysis method for random response of non-proportional damping structure under earthquake action - Google Patents
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Abstract
The invention discloses an analysis method of random response of a non-proportional damping structure under the action of earthquake, which comprises the following specific steps: 1. acquiring a unit impulse response of the structure with respect to an arbitrary excitation; 2. establishing a direct convolution formula for calculating a response power spectrum based on a random vibration theory; 3. carrying out POD characteristic orthogonal decomposition on the non-uniform modulation function excited by each point; 4. calculating an evolution frequency response matrix by using FFT (fast Fourier transform), and further determining an evolution power spectrum matrix and a variance matrix of the structural response; the invention explicitly expresses the response statistic as the convolution form about the impulse response, and utilizes the FFT technology to calculate the convolution, thereby greatly improving the response calculation efficiency; the method is suitable for linear structures, nonlinear structures and linear systems reduced by modal superposition.
Description
Technical Field
The invention belongs to the field of random vibration analysis, and particularly relates to a method for analyzing random response of a non-proportional damping structure under the action of earthquake.
Background
The non-proportional damping characteristic has a considerable influence on the structural dynamic response, and in some cases, the structural damping cannot be assumed to be proportional damping, such as a bottom concrete structure-upper steel structure hybrid structural system, a structure provided with a damper, a seismic isolation structure, and the like. In addition, in the nonlinear random vibration analysis based on equivalent linearization, an equivalent linear system is also a non-proportional damping system due to the introduction of an additional equivalent matrix.
Engineering structures are often subjected to non-stationary excitations such as earthquakes, and the analysis of such problems is usually aided by non-stationary random vibration theory. The non-stationary random response analysis of the non-proportional damping structure generally needs to be performed by means of complex modal analysis, however, the complex modal analysis of the large-scale multi-degree-of-freedom structure is relatively complex to calculate, and a modal superposition method has good approximation for displacement response, but has large errors for internal stress and stress. In order to obtain more accurate results, a large number of mode shapes are required to be calculated, and the calculation efficiency is reduced. Therefore, the current non-stationary response analysis of such structures is dominated by direct integration. The virtual excitation method proposed by great forest households converts the problem of non-stable random vibration into deterministic instantaneous power time course analysis of the structure, and the solving process becomes quite intuitive, so that the method is widely applied. However, as the structure tends to be large and complicated, the calculation of direct integration by using a Newmark-beta method cannot meet the requirement of efficiency, so that the problem of non-stationary random vibration is solved by combining a fine integration method proposed by Tokawa 21232and a virtual excitation method by forest and the like, and the calculation efficiency is greatly improved. Although this method improves the computational efficiency of a single time course analysis, the total number of time course analyses is not reduced. For large span flexible structures with dense frequency distributions, thousands of time-course analyses may be required, while the above approach still faces challenges.
Disclosure of Invention
In order to improve the efficiency of non-stationary response analysis of a non-proportional damping structure, the invention provides an analysis method of random response of the non-proportional damping structure under the action of earthquake, which comprises the following specific steps:
step 1: the unit impulse response of the structure with respect to any excitation is obtained.
Under the action of N-point seismic excitation X (t), the dynamic equation of a structural system with N degrees of freedom is as follows:
in the formula: y (t))，
And
respectively representing an Nx 1 order displacement vector, a velocity vector and an acceleration vector of the structure; m, C and K are respectively an NxN order mass matrix, a non-proportional damping matrix and a stiffness matrix; Γ is a load distribution matrix of Nxn order containing only elements 0 and 1, which expands N-dimensional load vectors to N-dimensional load vectors;
based on the equivalent initial condition method for solving the causal system unit impulse response function, the unit impulse response of the structure with respect to any excitation is calculated as follows:
in the formula: h is r (t)、
And
unit pulse displacement, velocity and acceleration response corresponding to the r-th excitation, respectively; gamma-shaped r Is the r column vector of Γ, the elements only in the r degree of freedom of the excitation are 1, the other elements are 0, then by solving the above equation using the Newmark- β method, these unit impulse responses are calculated from the finite element model of the analysis structure;
furthermore, the unit impulse response of the structure is identified by an impulse excitation method, taking into account an n × 1 order impulse excitation vector
Where T denotes transpose, T j Which represents the time of the j-th instant,
is a discrete time pulse, represented as:
wherein the time step is Deltat and the total is N t At discrete time points, the structure is first calculated at this pulse excitation X using the Newmark-beta method (0) (t j ) Response under action is Y (0) (t j ) Then, the discrete-time impulse response for the r-th excitation is found as:
discrete time impulse response in the equation
h r (t j ) Structural impulse response for the r-th excitation; in the calculation of the Newmark-beta method, three responses Y of the displacement, the speed and the acceleration of the structure (0) (t j )，
And
can be found at the same time, therefore, similar to the calculation of equation (4), a discrete time impulse response matrix with respect to the velocity and the acceleration can be further obtained;
in addition, consider another pulsed excitation vector
Wherein
Is composed of
The discrete-time impulse response for the r-th excitation can likewise be determined as:
in the formula Y (1) (t j ) Is a structure pair pulse excitation X (1) (t j ) Is calculated using the Newmark-beta method, and therefore, a discrete-time impulse response matrix of the structure can be identified by applying an impulse excitation to the structure at any time.
Step 2: based on the random vibration theory, a direct convolution formula for calculating the response power spectrum is established.
Non-stationary random excitation process X (t) = { X) for one zero mean n variable 1 (t),…,x r (t),…,x n (t)} T According to the evolutionary spectrum theory proposed by Priestley, the mutual evolutionary power spectrum matrix can be expressed as:
in the formula A * (ω, t) represents the conjugate matrix of the amplitude modulation function, ω is the circular frequency, and by the wiener-cinchona theorem, the random process X (t) 1 ) And X (t) 2 ) The cross-correlation function matrix of (a) may be expressed as:
wherein A (ω, t) is a non-uniform modulation function A r An n-order diagonal matrix of (ω, t), A T* (ω,t 2 ) A conjugate transpose matrix representing an amplitude modulation function;
is an n-order stationary power spectral matrix corresponding to a non-stationary random process; i units of imaginary numbers, d ω denotes a differential, and e denotes a natural exponent.
Then, based on the random vibration theory, under the action of the non-stationary random excitation, the matrix of the evolutionary power spectrum of the structure displacement vector Y is:
in the formula I T* (ω, t) is a conjugate transpose matrix of the evolving frequency response matrix I (ω, t);
in the formula: h (t) = [ h r (t)](ii) a A (ω, τ) represents the non-uniform modulation function A at any one time r An n-order diagonal matrix consisting of (omega, t);
further, the formula (10) is rewritten as a representation form of the corresponding matrix element as follows:
in the formula I kr (ω, t) is the element of the kth row r column of the evolving frequency response matrix I (ω, t), h kr (t- τ) is the Duhami integral term.
And 3, step 3: and (3) carrying out POD characteristic orthogonal decomposition on the non-uniform modulation function excited by each point.
After the characteristic orthogonal decomposition is carried out on the non-uniform modulation function excited by each point, the characteristic orthogonal decomposition is approximately expressed as the sum of the products of a plurality of time functions and frequency functions:
in the formula
Is a frequency dependent matrix R r The q-th feature vector of (2),
as a function of the decomposed frequency;
is obtained by
(iv) the calculated qth principal coordinate;
is the number of significant terms that contain the vast majority of the energy.
And 4, step 4: and calculating an evolution frequency response matrix by using FFT (fast Fourier transform), and further determining an evolution power spectrum matrix and a variance matrix of the structural response.
Because when tau is more than or equal to t, h kr (t- τ) =0; when t < 0, A r (ω, t) =0, so equation (11) is rewritten as:
substituting equation (12) into equation (13), the frequency term in the modulation function is separated from the integral equation, and equation (13) can be converted into a form directly usable by FFT:
fast calculation is carried out by adopting FFT technology for each given time t, and then an evolution power spectrum matrix S of the displacement response can be obtained according to the formula (9) y (ω, t), and thus the time-varying variance of the displacement response, can be:
further, based on an equivalent linearization method, the algorithm expansion is applied to the non-stationary random response high-efficiency analysis of the nonlinear structure, and the method comprises the following steps:
step A: establishing a motion equation of a nonlinear system under any non-stationary excitation;
consider a hysteresis system with N degrees of freedom that is subjected to a non-stationary random excitation X (t), where the number of unit excitations is N = N, and whose equation of motion is expressed as:
in the formula m i ，c i ，y i And x i Mass, damping, displacement and excitation of the ith layer respectively; relative displacement z between layers i ＝y i -y i-1 (i =1,2, \8230;, N) where y 0 =0; the restorative force based on the Bouc-Wen model can be expressed as:
f i (z i ,u i )＝γ i k i z i +(1-γ i )k i u i (i＝1,2,…,N) (17)
in the formula k i Is the initial stiffness; gamma ray i Is the stiffness reduction factor of the ith layer; retardation displacement u i And z i In this regard, the following nonlinear differential equation is satisfied:
in the formula i ,
β i And theta i Is a parameter of the retardation displacement of the ith layer;
and B: converting the nonlinear system into an equivalent linear system at a specific moment;
based on the equivalent linearization criterion, the motion equation of the equivalent linear system of the hysteresis system at any moment α is:
wherein Y (t) = [ Y = 1 ,y 2 ,…,y N ] T Is a displacement vector; u (t) = [ U = [ ] 1 ,u 2 ,…,u N ] T Is a hysteresis displacement vector; m, C, K and K h Respectively, a mass matrix, a damping matrix, an elastic stiffness matrix and a hysteresis stiffness matrix, expressed as:
q (α) and G (α) are equivalent coefficient matrices, expressed as follows:
when assuming theta i =1, equivalent coefficient q i (. Alpha.) and g i (α) can be obtained by the following formula:
in the formula
And
are respectively
And u i (α) standard deviation;
and C: iteratively solving the non-stationary response of each equivalent linear system at a specific moment, wherein the steps are as follows;
step (1) at a given time α, give
E[U(α)U(α) T ]And
assigning an initial value, and generally taking a convergence result of the previous moment; then, calculating initial equivalent coefficient matrixes Q (alpha) and G (alpha), and substituting the initial equivalent coefficient matrixes Q (alpha) and G (alpha) into a formula (19) to obtain an initial equivalent linear system at the alpha moment;
step (2) exciting x for each element i Constructing a unit pulse excitation and applying it to an equivalent linear system represented by formula (19); discrete time impulse response matrix of velocity and hysteresis displacement is obtained through Newmark-beta method
And
step (3) calculating a matrix for impulse response using FFT
And
is convolved to find
And I u (ω,α)；
Step (4) calculating new response statistics
E[U(α)U(α) T ]And
step (5) substituting the new response statistic into expressions (21) to (24) to obtain new equivalent coefficient matrixes Q (alpha) and G (alpha), and updating the equivalent linear system represented by expression (19);
step (6) repeating steps (2) to (5) until the response statistic
E[U(α)U(α) T ]And
converging to obtain a real equivalent linear system at the alpha moment;
step (7) of solving a displacement discrete time impulse response matrix of the real equivalent linear system
Further calculate the displacement response statistic E [ Y (alpha) T ]；
Step (8) moves to the next moment, and steps (1) to (7) are repeated until all the response statistics of all the moments of interest are determined.
The invention has the beneficial effects that: the efficient algorithm for analyzing the random response of the non-proportional damping structure under the earthquake action explicitly expresses the response statistic as a convolution form related to the impulse response, and utilizes the FFT technology to calculate the convolution, thereby greatly improving the response calculation efficiency. The method is suitable for linear structures, nonlinear structures and linear systems subjected to order reduction by a modal superposition method. It may calculate only the degrees of freedom of interest and the response at the moment. When the method is used for an equivalent linearization method for solving the response of a nonlinear system under any seismic excitation, each iteration only needs to calculate the response at a specific moment and on a nonlinear degree of freedom, thereby avoiding a large amount of redundant calculation in the iterative solution process of the traditional method. Therefore, the method can be used as an effective method for analyzing the non-stationary response of the large-scale non-proportional damping structure.
Drawings
FIG. 1 is a simplified model of a seismic-excited high-rise building.
Fig. 2 shows the displacement time-varying rms of the 10 th and 20 th layers of the linear structure.
FIG. 3 is a graph showing the relationship between the time taken for non-stationary response analysis and the structural degrees of freedom.
FIG. 4 is a hysteresis system with N degrees of freedom.
FIG. 5 shows the displacement-time-varying root-mean-square complete non-linearity of the hysteresis system.
FIG. 6 is a graph of the displacement time-varying root mean square local non-linearity of the hysteresis system.
Detailed Description
The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and specific embodiments. The method can be applied to the non-stationary random response analysis of a non-proportional damping structure, wherein the non-proportional damping structure comprises a bottom concrete structure-upper steel structure mixed structure system, a structure provided with a damper, a shock insulation structure and the like.
Example 1: and the linear structure is used for analyzing the response of the high-rise building excited by the earthquake.
Consider a 20-story non-proportionally damped seismic structure, a simplified model of which is shown in figure 1. The structural parameter is m i =10000kg and k i =16000kN/m (1 ≦ i ≦ 20). Its equation of motion is given as follows:
in the formula: m = diag [ M ] i ]；K＝diag[k i ]；
Is the seismic acceleration process; e is a unit vector. Damping matrix of the assumed structure is C = C s +C r In which C is s ＝0.3M+0.002K，C r (1,1)＝20×C s (1,1)，C r All other elements of (1) are 0.
Assuming seismic acceleration process
Is a non-stationary random process whose EPSD is given below
In the formula: f is the Hertz frequency; s. the 0 ＝1/690m 2 s -3 (ii) a c =2; d =0.2. In order to evaluate the accuracy and efficiency of the fast convolution method, the response of the structure under the action of the completely non-stationary seismic excitation is calculated by using the fast convolution method and a traditional frequency domain method respectively. Calculating the cut-off frequency of the taking excitation EPSD to be f u =8Hz; frequency increment is Δ f =1/64Hz; the total time length is 32s; the time step is Δ t =1/64s. The number of approximate terms of the decomposition of EPSD using POD in the fast convolution method is taken to be 10.
The method provided by the invention is used for obtaining the root mean square of the displacement of the building structure under the action of the earthquake. The time-varying root mean square of the 10 th and 20 th layers of the structure is shown in figure 2. Due to the evolution of the response over time, the fast convolution method only computes the response at times of 0.5s time interval, which is sufficient to reflect the statistical nature of the response. As can be seen from FIG. 2, the numerical results obtained by the fast convolution method and the traditional frequency domain method are well matched, which shows that the fast convolution method has better precision on the analysis of the non-stationary response of the linear structure.
In order to fully compare the computational efficiency of the fast convolution method with the conventional frequency domain method, the non-stationary response of 50, 100, 150, 200, 250 and 300 stories of high-rise buildings under the seismic excitation described above was further calculated. In these configurations, the other calculation parameters are kept constant except for the different number of degrees of freedom. The response in all degrees of freedom is calculated for each structure. Fig. 3 shows a relation curve between the computation time and the degree of freedom, and it is obvious that the computation efficiency of the fast convolution method is far higher than that of the traditional frequency domain method in the non-stationary response analysis. Moreover, as the number of degrees of freedom increases, the improvement in efficiency becomes more pronounced. For 300 degrees of freedom, the time consumed by the fast convolution method and the conventional frequency domain method is 11.7s and 276s respectively, and the former accounts for only 4.24% of the latter. In addition, the fast convolution method can also compute only the response of interest. Assuming that only responses at the top level of the structure are needed, the corresponding computation time is also plotted in fig. 3. It can be seen that the efficiency of the fast convolution method is further improved. For 300 degrees of freedom, the time consumed by the fast convolution method is 0.32s, which is only 0.12% of the conventional frequency domain method. For comparison, we have defined an alternative method to replace the impulse excitation method of identifying the impulse response matrix in the fast convolution method with the frequency response matrix method. Its time consumption in the above calculation is also compared, as shown in fig. 3. Clearly, the fast convolution method is more efficient than the alternative method, which indicates that the impulse excitation method is more efficient than the frequency response matrix method.
Example 2: and the nonlinear structure is used for response analysis of the hysteresis system affected by the earthquake.
Consider a 100 degree of freedom hysteresis system as shown in fig. 4. The concentrated mass and the initial linear rigidity between layers of the system are respectively m i =3000kg and k i ＝8×10 7 N/m(1≤i≤50)、m i =2500kg and k i ＝7.5×10 7 N/m (51. Ltoreq. I. Ltoreq.100). The stiffness reduction coefficient of each layer is gamma i =0.6 (1 ≦ i ≦ 100). The retardation displacement of each layer has a parameter of phi i ＝1，
β i ＝300m -1 And theta i 1 (1. Ltoreq. I. Ltoreq.100). The damping matrix C is defined by a rayleigh damping model, and the proportionality coefficient thereof is determined by a damping ratio of 0.05 of the 1 st order mode and the 100 th order mode of the initial linear system. The system is excited by non-steady ground acceleration
The EPSD is defined by the formula, wherein S 0 ＝0.02m 2 s -3 . The cut-off frequency was selected to be 8Hz in the response calculation, and the frequency increment was Δ f =1/32Hz. Approximate number of terms when POD is performed on EPSD
Taken as 10.
The non-stationary response of the hysteresis system is calculated by an equivalent linearization method based on a fast convolution method and an equivalent linearization method based on a traditional frequency domain method. To verify the accuracy of the equivalent linearization method, the calculation of Monte Carlo Simulation (MCS) of 4000 samples was also used as a comparison. The total time length of the calculation of the three methods is taken to be 32s, and the time step is Δ t =1/32s. Fig. 5 shows the root mean square of the time-varying displacement of the 50 th layer and the 100 th layer of the hysteresis system obtained by the three methods. Obviously, the three results are very consistent, and the equivalent linearization method based on the fast convolution method is effective in the analysis of the non-stationary response of the nonlinear structure and has better precision. Table 1 compares the total computation time of two equivalent linearization methods. It can be seen that the equivalent linearization method based on the conventional frequency domain method consumes 154 times longer time than the equivalent linearization method based on the fast convolution method. Therefore, for the equivalent linearization solution of the nonlinear structure, the fast convolution method has much higher computational efficiency than the conventional frequency domain method.
Further, three methods are used for the local nonlinear system. Consider a hysteresis system with non-linearity only in layers 1 to 10, the parameters of which are taken to be
And gamma i =1 (11 ≦ i ≦ 100), and the other parameters are the same as in the previous hysteresis system. Under the same non-stationary seismic excitation effect, the root mean square of the time-varying displacement of the 50 th layer and the 100 th layer of the hysteresis system is shown in fig. 6, and the calculation results of the three methods are well matched. Table 1 also compares the computational efficiency of the two equivalent linearization methods. It can be seen that the equivalent linearization method based on the fast convolution method has more obvious effect on the local nonlinear problemThe rate advantage.
TABLE 1 Total calculated time comparison of two equivalent linearization methods
While the present invention has been described in detail with reference to the embodiments, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.
Claims (4)
1. A method for analyzing the random response of a non-proportional damping structure under the action of earthquake is characterized by comprising the following specific steps:
step 1: the unit impulse response of the structure about any excitation is obtained, specifically:
under the action of N-point seismic excitation X (t), the dynamic equation of a structural system with N degrees of freedom is as follows:
in the formula: y (t) is a linear chain,
and
respectively representing an Nx 1 order displacement vector, a velocity vector and an acceleration vector of the structure; m, C and K are respectively an NxN order mass matrix, a non-proportional damping matrix and a stiffness matrix; Γ is a load distribution matrix of Nxn order containing only elements 0 and 1, which expands N-dimensional load vectors into N-dimensional load vectors;
based on the equivalent initial condition method for solving the causal system unit impulse response function, the unit impulse response of the structure with respect to any excitation is calculated as follows:
in the formula: h is r (t)、
And
respectively, unit pulse displacement, velocity and acceleration response corresponding to the r-th excitation; gamma-shaped r The r column vector of Γ, the elements only in the r degree of freedom of the excitation are 1, the other elements are 0, and then by solving the above equation using the Newmark- β method, these unit impulse responses are calculated from the finite element model of the analysis structure;
furthermore, the unit impulse response of the structure is identified by an impulse excitation method, taking into account an n × 1 order impulse excitation vector
Where T denotes transpose, T j It is indicated that the time of the j-th instant,
is a discrete time pulse, represented as:
wherein the time step is Deltat and the total is N t At discrete time points, the structure is first calculated at this pulse excitation X using the Newmark-beta method (0) (t j ) Response under action is Y (0) (t j ) Then, the discrete-time impulse response for the r-th excitation is found as:
where the discrete time impulse response
h r (t j ) Structural impulse response for the r-th excitation; in the calculation of the Newmark-beta method, three responses Y of the displacement, the speed and the acceleration of the structure (0) (t j )，
And
can be found simultaneously, so that a discrete-time impulse response matrix with respect to speed and acceleration can be further obtained;
in addition, consider another pulsed excitation vector
Wherein
Is composed of
The discrete-time impulse response for the r-th excitation can likewise be determined as:
in the formula Y (1) (t j ) Is a structure pair pulse excitation X (1) (t j ) The response of the structure is calculated by using a Newmark-beta method, so that a discrete-time impulse response matrix of the structure can be identified by applying impulse excitation to the structure at any time;
and 2, step: establishing a direct convolution formula for calculating a response power spectrum based on a random vibration theory, wherein the direct convolution formula specifically comprises the following steps:
non-stationary random excitation process X (t) = { X for a zero mean n variable 1 (t),…,x r (t),…,x n (t)} T According to the evolutionary spectrum theory, its inter-evolutionary power spectrum matrix can be expressed as:
in the formula A * (ω, t) represents the conjugate matrix of the amplitude modulation function, ω is the circular frequency, and by the wiener-cinchona theorem, the random process X (t) 1 ) And X (t) 2 ) The cross-correlation function matrix of (a) may be expressed as:
wherein A (ω, t) is a non-uniform modulation function A r An n-th order diagonal matrix of (ω, t), A T* (ω,t 2 ) A conjugate transpose matrix representing an amplitude modulation function;
is an n-order stationary power spectrum matrix corresponding to a non-stationary random process;
then, based on the random vibration theory, under the action of the non-stationary random excitation, the matrix of the evolution power spectrum of the structure displacement vector Y is:
in the formula I T* (ω, t) is a conjugate transpose matrix of the evolving frequency response matrix I (ω, t);
in the formula: h (t) = [ h r (t)](ii) a A (ω, τ) represents the non-uniform modulation function A at any one time r An nth order diagonal matrix composed of (omega, t);
further, the formula (10) is rewritten as a representation form of the corresponding matrix element as follows:
in the formula I kr (ω, t) is the element of the k-th row r column of the evolving frequency response matrix I (ω, t), h kr (t- τ) is the Duhami integral term;
and step 3: carrying out POD characteristic orthogonal decomposition on the non-uniform modulation function excited by each point;
and 4, step 4: and calculating an evolution frequency response matrix by using FFT (fast Fourier transform), and further determining an evolution power spectrum matrix and a variance matrix of the structural response.
2. The method for analyzing the stochastic response of the non-proportional damping structure under the action of the earthquake as recited in claim 1, wherein the step 3 specifically comprises:
after the characteristic orthogonal decomposition is carried out on the non-uniform modulation function excited by each point, the characteristic orthogonal decomposition is approximately expressed as the sum of the products of a plurality of time functions and frequency functions:
in the formula
Is a frequency-dependent matrix R r The qth feature vector of (1);
is obtained by
(ii) the calculated qth principal coordinate;
is the number of significant terms that contain the vast majority of the energy.
3. The method for analyzing the stochastic response of the non-proportional damping structure under the action of the earthquake as recited in claim 2, wherein the step 4 specifically comprises:
since when t is greater than or equal to t, h kr (t- τ) =0; when t < 0, A r (ω, t) =0, so equation (11) is rewritten as:
substituting equation (12) into equation (13), the frequency term in the modulation function is separated from the integral equation, and equation (13) can be converted into a form directly usable by the FFT:
fast calculating by adopting FFT technology for each given time t, and then obtaining an evolution power spectrum matrix S of the displacement response according to the formula (9) y (ω, t), and thus the time-varying variance of the displacement response, can be:
4. a method for analyzing a non-stationary random response of a non-linear structure, which is characterized in that the method for analyzing the random response of the non-proportional damping structure according to claim 1 is applied to the efficient analysis of the non-stationary random response of the non-linear structure in an extended manner based on an equivalent linearization method, and comprises the following steps:
step A: establishing a motion equation of a nonlinear system under any non-stationary excitation;
consider a hysteresis system with N degrees of freedom that is subjected to a non-stationary random excitation X (t), where the number of unit excitations is N = N, and whose equation of motion is expressed as:
in the formula, m i ，c i ，y i And x i Mass, damping, displacement and excitation of the ith layer respectively; relative displacement z between layers i ＝y i -y i-1 I =1,2, \ 8230;, N, wherein y 0 =0; the restorative force based on the Bouc-Wen model can be expressed as:
f i (z i ,u i )＝γ i k i z i +(1-γ i )k i u i i＝1,2,…,N (17)
in the formula k i Is the initial stiffness; gamma ray i Is the stiffness reduction factor of the ith layer; retardation displacement u i And z i In relation, the following nonlinear differential equation is satisfied:
in the formula i ,
β i And theta i Is a parameter of the retardation displacement of the ith layer;
and B: converting the nonlinear system into an equivalent linear system at a specific moment;
based on the equivalent linearization criterion, the motion equation of the equivalent linear system of the hysteresis system at any moment α is:
wherein Y (t) = [ Y = 1 ,y 2 ,…,y N ] T Is a displacement vector; u (t) = [ U = [ ] 1 ,u 2 ,…,u N ] T Is a hysteresis displacement vector; m, C, K and K h Respectively a mass matrix, a damping matrix, an elastic stiffness matrix and a hysteresis stiffness matrix,
q (α) and G (α) are equivalent coefficient matrices, expressed as follows:
when assuming theta i =1, equivalent coefficient q i (. Alpha.) and g i The (. Alpha.) can be obtained by the following formula:
in the formula
And
are respectively
And u i A standard deviation of (. Alpha.);
and C: iteratively solving the non-stationary response of each equivalent linear system at a specific moment, wherein the steps are as follows;
step (1) at a given time α, give
E[U(α)U(α) T ]And
assigning an initial value, and generally taking a convergence result of the previous moment; then, calculating initial equivalent coefficient matrixes Q (alpha) and G (alpha), and substituting the initial equivalent coefficient matrixes Q (alpha) and G (alpha) into a formula (19) to obtain an initial equivalent linear system at the alpha moment;
step (2) exciting x for each element i Constructing a unit pulse excitation and applying it to an equivalent linear system represented by formula (19); discrete time impulse response matrix of velocity and hysteresis displacement is obtained through Newmark-beta method
And
step (3) of calculating a matrix for impulse response using FFT
And
by convolution of (a) to obtain
And I u (ω,α)；
Step (4) calculating new response statistics
E[U(α)U(α) T ]And
step (5) substituting the new response statistic into expressions (21) to (24) to obtain new equivalent coefficient matrixes Q (alpha) and G (alpha), and updating the equivalent linear system represented by expression (19);
step (6) repeating steps (2) to (5) until the response statistic
E[U(α)U(α) T ]And
converging to obtain a real equivalent linear system at the alpha moment;
step (7) of solving a displacement discrete time impulse response matrix of the real equivalent linear system
Further, a displacement response statistic E [ Y (alpha) is calculated T ]；
Step (8) moves to the next time instant, and steps (1) to (7) are repeated until all the response statistics of all the time instants of interest are determined.
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