CN109885923B - Method for determining optimized parameters of Maxwell damper between symmetrical double-tower structures - Google Patents
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Abstract
The invention discloses a method for determining optimized parameters of a Maxwell damper between symmetrical double-tower structures, which couples the symmetrical double-tower structures into a single 2-DOF system connected with a Maxwell type damper, deduces a frequency response function of tower displacement by taking stable white noise as seismic excitation, and establishes an expression of vibration energy of the symmetrical double-tower structures; the damping coefficient of the Maxwell damper is used as a research parameter, and the vibration energy of the chassis and the tower with the chassis is minimized to be used as a control target respectively, so that an optimized parameter analytical expression of the Maxwell damper is obtained; and verifying the effectiveness of the optimized parameter analytical expression through numerical calculation. The invention provides a control strategy for reducing the dynamic response of a tower structure under the action of an earthquake by arranging a passive coupling unit between symmetrical double tower structures with chassis; under different control targets, a general expression of the optimized damping of the passive control device is deduced by using an optimized design principle; finally, the effectiveness of the control strategy is verified through time domain analysis.
Description
Technical Field
The invention relates to the technical field of anti-seismic and vibration control of engineering structures, in particular to a method for determining optimized parameters of a Maxwell damper between symmetrical double-tower structures.
Background
In modern cities, many high-rise buildings are often designed into a master-slave structure consisting of a plurality of substructures due to requirements on building shapes or structural use functions. When strong shock or strong wind occurs, the structures often generate large response difference due to different vibration frequencies, and the possibility of collision between adjacent structures is high. Some researchers propose that connecting adjacent structures by energy dissipation and shock absorption devices can not only absorb part of seismic energy, but also avoid collision between adjacent structures. Zhu and Xu research the effectiveness of the fluid damper under seismic excitation on connection of adjacent multi-layer buildings, and compared with the dynamic characteristics of the fluid damper defined by a Maxwell model under an earthquake, the research proves that the passive damper has a good control effect on reducing the vibration response of an adjacent asymmetric structure. Luco et al simulated two adjacent structures as shear beams of different heights and studied the optimum distribution of viscous dampers between the two adjacent structures. Based on the energy statistical principle, Juhongping and the like respectively derive analytical expressions of Kelvin type and Maxwell type damper optimization parameters between double-body single-degree-of-freedom systems. Forest sword has studied the adoption attenuator to carry out vibration control to the earthquake response of disjunctor high-rise structure. Kim et al investigated the effectiveness of arranging viscous dampers between two tower structures to mitigate structural response caused by seismic effects. Parametric analytical studies have shown that viscous dampers exist in a certain size to minimize the dynamic response of the turret structure.
In addition, some modern high-rise buildings are often equipped with a chassis structure. Aiming at the structure, a Maxwell damper is arranged between asymmetric turrets on the same floor by Chua vibration and the like, and the zero-frequency optimized damping coefficient and relaxation time of the damper are obtained through parametric analysis. Wuqiaomiyun discusses the influence of passive damper on the system aiming at the adjacent tower structure with corridor, and the results show that the earthquake response of the twin-tower conjoined structure can be well controlled under the action of the passive damper. Patel and Jangid simplify the structure of the double-tower building with the chassis into a symmetrical structure with two degrees of freedom, and connect the linear damper between the top end of each side and the bottom end of the other side, and researches prove the effectiveness of the passive damper in response control on the adjacent symmetrical coupling structure. In the above researches, most of the researches assume that the self-vibration characteristics of adjacent structures are different, the optimized parameters of the damper are mainly suitable for the condition that the self-vibration characteristics of the two adjacent structures are different greatly, and most of double-tower structures with the chassis are designed symmetrically in actual engineering, so that the important practical engineering significance is realized for the vibration control research of the double-tower structures with the chassis.
Disclosure of Invention
The invention aims to solve the technical problem of providing a method for determining optimized parameters of a Maxwell damper between symmetrical double-tower structures aiming at the defects in the prior art.
The technical scheme adopted by the invention for solving the technical problems is as follows:
the invention provides a method for determining optimized parameters of a Maxwell damper between symmetrical double-tower structures, wherein the Maxwell damper is arranged between the symmetrical double-tower structures, each tower structure in the symmetrical double-tower structures comprises a chassis and upper towers, and the Maxwell damper is arranged between the two upper towers; the method comprises the following steps:
step one, coupling a symmetrical double-tower structure into a single 2-DOF system connected with a Maxwell type damper, deducing a frequency response function of tower displacement by taking stable white noise as seismic excitation, and establishing an expression of vibration energy of the symmetrical double-tower structure;
step two, taking the damping coefficient of the Maxwell damper as a research parameter, and respectively taking the vibration energy of the chassis and the tower with the chassis as a control target to obtain an optimized parameter analytical expression of the Maxwell damper;
and step three, verifying the effectiveness of the optimized parameter analytical expression through numerical calculation.
Further, the specific method of the first step of the invention is as follows:
step 1.1, adopting a Maxwell control unit calculation model to simulate a symmetrical double-tower damper, wherein the model is formed by serially connecting a damping element and a spring and calculates a control force generated by the model;
step 1.2, regarding the chassis of the symmetrical double-tower structure as a main structure and the upper tower as a secondary structure, and only considering seismic excitation along the symmetrical plane of the structure in the horizontal direction to construct a calculation model of the symmetrical double-tower structure with the chassis; simplifying the structure of the symmetrical double-tower structure calculation model into a 2-DOF adjacent shearing type tower structure model;
and 1.3, obtaining a motion differential equation of the coupling structure system according to the simplified calculation model, and regarding the earthquake action as a ground acceleration process of stable white noise, so that a vibration energy expression of the symmetrical double-tower structure in a time domain is obtained under the earthquake action of a known power spectral density function.
Further, the vibration energy expression obtained in step 1.3 of the present invention is:
the vibration energy expression of the main structure is as follows:
the vibration energy expression from the structure is:
wherein:
m1、k1、c1the mass, rigidity and damping of the main structure; m is2、k2、c2Mass, stiffness, damping of the secondary structure; k is a radical ofd、cdThe rigidity coefficient and the damping coefficient of the Maxwell damper are shown.
Further, the specific method of the second step of the invention is as follows:
carrying out parameter analysis on the obtained vibration energy expression of the symmetrical double-tower structure, establishing a relationship curve of respective maximum energy and frequency ratio of the master-slave structures under different delay coefficients, wherein the smaller the delay coefficient is, the smaller the maximum energy difference of the master-slave structures is, and setting the delay coefficient to be 0 under the premise of neglecting the damping of the structure, thereby obtaining the optimal damping coefficient; based on the energy minimum principle, the parameters of the passive damper are adjusted to reduce the relative displacement of the structure, and two strategies are adopted: and respectively taking the vibration energy of the chassis and the tower with the chassis as the minimum control target to obtain an optimized parameter analytical expression of the Maxwell damper.
Further, the specific method for optimizing the parameters in the second step of the invention is as follows:
when the minimum energy of the main control structure is taken as an optimization target:
the optimized damping coefficient is as follows:
wherein:
A01＝β2μ3ω1 2+β2μ2ω1 2-μ2-2μ-1,B01＝-μ3ω1β-4μω1β-4μ2ω1β-ω1β
further, the specific method for optimizing the parameters in the second step of the invention is as follows:
when the minimum total structural energy is controlled as an optimization target:
the optimized damping coefficient is as follows:
wherein:
A02＝-2β4μ3ω1-8β4μ2ω1-8β4μω1-4β4μ3ω1-2β4ω1-6β2μ2ω1-μ4ω1-2β2μω1-2μ3ω1-μ2ω1
further, the method for verifying the validity of the optimized parameter analysis expression through numerical calculation in the third step of the present invention includes: numerical examples based on 2-DOF models; numerical example based on MDOF model.
The invention has the following beneficial effects: the method for determining the optimized parameters of the Maxwell damper between the symmetrical double-tower structures simplifies the symmetrical double-tower structures with the chassis and the dampers into a single 2-DOF model, deduces a general expression of average relative vibration energy of the 2-DOF structure under the excitation of stable white noise, and obtains an analytical expression of the optimized parameters of a passive controller by utilizing an optimization design principle under different control targets.
The invention provides a control strategy for reducing the dynamic response of a tower structure under the action of an earthquake by arranging a passive coupling unit between symmetrical double tower structures with chassis; under different control targets, a general expression of the optimized damping of the passive control device is deduced by using an optimized design principle; finally, the effectiveness of the control strategy is verified through time domain analysis.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a Maxwell model of viscous fluid damper according to an embodiment of the present invention;
FIG. 2(a) is a structural model of a symmetric double tower with a Maxwell-type damper according to an embodiment of the present invention;
FIG. 2(b) is a simplified model of a symmetric double tower with Maxwell-type dampers according to an embodiment of the present invention;
FIG. 3(a) is a graph of the maximum energy to frequency ratio of the main structure at different delay factors according to the embodiment of the present invention;
FIG. 3(b) is a graph of maximum energy to frequency ratio from a structure for different delay factors according to an embodiment of the present invention;
FIG. 4(a) is a top displacement time course curve of an exemplary timing example of the control strategy under El-Centro wave in the embodiment of the present invention;
FIG. 4(b) is a graph of top layer displacement time course calculated by two time calculations for the El-Centro wave lower control strategy according to the embodiment of the present invention;
FIG. 5 is an MDOF double tower structure with Maxwell dampers attached according to an embodiment of the present invention;
FIG. 6(a) is a top displacement time course curve of an exemplary timing example of the control strategy under El-Centro wave in the embodiment of the present invention;
FIG. 6(b) is a graph of top layer displacement time course calculated by two time calculations for the El-Centro wave lower control strategy according to the embodiment of the present invention;
FIG. 7 is a top-level energy time course plot of two exemplary control strategies in accordance with an embodiment of the present invention;
fig. 8 is a time course curve of the maximum interlayer displacement calculated under two control strategies according to the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The method for determining the optimized parameters of the Maxwell damper between the symmetrical double-tower structures comprises the following steps of arranging the Maxwell damper between the symmetrical double-tower structures, wherein each tower structure in the symmetrical double-tower structures comprises a chassis and upper towers, and the Maxwell damper is arranged between the two upper towers; the method comprises the following steps:
step one, coupling a symmetrical double-tower structure into a single 2-DOF system connected with a Maxwell type damper, deducing a frequency response function of tower displacement by taking stable white noise as seismic excitation, and establishing an expression of vibration energy of the symmetrical double-tower structure;
step two, taking the damping coefficient of the Maxwell damper as a research parameter, and respectively taking the vibration energy of the chassis and the tower with the chassis as a control target to obtain an optimized parameter analytical expression of the Maxwell damper;
and step three, verifying the effectiveness of the optimized parameter analytical expression through numerical calculation.
In one embodiment of the present invention, the method comprises the following steps:
1 calculation model and equation of motion
1.1 control Unit calculation model
The symmetrical double-tower inter-tower damper is simulated by adopting a Maxwell model, the model is formed by connecting damping elements and springs in series, and the control force F generated by the modeldThe calculation formula is as follows:
in the formula, c0A linear damping constant at zero frequency; k is a stiffness coefficient; λ is a relaxation time coefficient, and λ ═ c0K is the sum of the values of k and k. When the structure adopts Maxwell type damper connection, the output force f of the ith damperiCan be expressed as:
in the formula, c0i、λiZero frequency damping coefficient and relaxation time, d, of the ith damper, respectivelyiIs the indicator vector for the ith damper,
is the velocity difference across the damper.
1.2 calculation model of symmetric double-tower structure with chassis
The symmetric double tower structure is generally provided with a chassis, so that the mass and rigidity of the bottom of the structure are different from those of the upper structure, so that the adjacent symmetric double tower structure is simplified into a 2-DOF adjacent shear type tower, the bottom structure (chassis) is regarded as a main structure, and the top structure (upper tower) is regarded as a secondary structure. The 1 st vibration mode of the control tower structure is taken as a main part, a passive vibration damping device connected between adjacent structures is simulated to be a Maxwell damper and is arranged between the adjacent structure layers, the two tower structures are symmetrical in plane, only the seismic excitation along the symmetrical plane of the structures in the horizontal direction is considered, and a calculation model is shown in figure 2 (a).
Because the two towers are completely symmetrical, the two towers can be regarded as a single 2-dof (degree of freedom) system connected with passive dampers, as shown in fig. 2 (b).
Wherein m is1、k1、c1The mass, rigidity and damping of a main structure (chassis); m is2、k2、c2The mass, stiffness, damping of the secondary structure (upper tower); k is a radical ofd、cdThe rigidity coefficient and the damping coefficient of the Maxwell damper are shown.
1.3 equation of motion
The differential equation of motion of the coupled structure system can be obtained according to the coupled structure calculation model of fig. 2 (b):
constructing virtual incentives
Comprises the following steps:
fΓ+(iω)λfΓ＝(iω)c0(x2-x1) (8)
let the mass ratio mu of the master-slave structure be m1/m2The tower frequency ratio β ═ ω2/ω1The damping coefficient of the Maxwell damper and the tower mass ratio are respectively delta0＝c0/m1And is and
ξ1＝c1/2m1ω1，ξ21＝c2/2m1ω1，ξ2＝c2/2m2ω2，
the united vertical type (6), (7) and (8) are as follows:
relative displacement response x in master-slave structural frequency domain1And x2Can be obtained by solving equations of motion (9), (10):
wherein:
the relative vibration energy of the tower structure is defined as follows:
considering seismic effects as a stationary white noise ground acceleration process, the power spectral density function is SggUnder the action of the earthquake, the average relative vibration energy of a certain structure in a time domain is as follows:
taking equation (11) into equation (15), the average relative vibration energy of the master and slave structures can be expressed as:
(17) the formula can be solved as follows:
wherein:
since the damping of the master and slave structures is much smaller than the damping coefficient of the damper, the influence of the damping coefficient on the calculation result is negligible[12]. Therefore, to simplify the calculation process, the damping ratio of the structure itself, i.e., ξ, is ignored here1＝ξ2When 0, the following parametric expression is obtained:
a0＝λ2 (22)
a1＝2λ (23)
b0＝-λ4 (29)
b5＝0 (34)
b'0＝-λ4 (35)
b5'＝0 (40)
optimization design of Maxwell damper
The Maxwell type damper is a speed-related energy consumption device, the damping coefficient of the damper generally seriously influences the control effect of the structure, and the influence on the control effect is small when the rigidity is changed in a large range[12]Carrying out parametric analysis on the obtained energy expression (18), and showing a relation curve of respective maximum energy and frequency ratio of the master-slave structure under different delay coefficients in the graph of FIG. 3, wherein the smaller the delay coefficient is, the smaller the master-slave structureThe smaller the maximum energy difference of the structure is, especially when the delay coefficient is less than 0.001, the coefficient has almost no influence on the maximum energy of the structure, and the conclusion is consistent with that of the predecessor, so that the delay coefficient can be further set to be 0 under the precondition of neglecting the damping of the structure, and the optimal damping coefficient can be obtained.
Based on the energy minimum principle, the reasonable adjustment of the parameters of the passive damper can reduce the relative displacement of the structure, thereby ensuring the safety of the structure. The analysis is performed with the optimization goals of controlling master structure energy minimum and controlling master-slave structure energy and minimum respectively.
In the first strategy, when the minimum energy of the main control structure is taken as an optimization target, the passive coupling unit optimization design equation is as follows:
therefore, an optimal parameter analytic solution of the Maxwell type damper can be obtained:
optimizing a damping coefficient:
wherein
A01＝β2μ3ω1 2+β2μ2ω1 2-μ2-2μ-1,B01＝-μ3ω1β-4μω1β-4μ2ω1β-ω1β,
And in the second strategy, when the minimum total structural energy is controlled as an optimization target, the passive coupling unit optimization design equation is as follows:
therefore, an optimal parameter analytic solution of the Maxwell type damper can be obtained:
optimizing a damping coefficient:
3. numerical example based on 2-DOF model
In order to observe the influence of the obtained two passive damper optimization parameter analytical solutions on the structural vibration control effect more intuitively, a certain 2-DOF model under EL-Centro wave excitation is firstly subjected to numerical analysis.
2-DOF example: assuming that the main structure mass is 2.58 x 105kg, having a shear stiffness of 4X 109N/m; the mass of the secondary structure is 1.29 multiplied by 105kg, shear stiffness all 4X 108N/m. The structure adopts a Rayleigh damping model, and the damping ratio of the first order mode and the second order mode is 0.02. The seismic waves are El-Centro waves. The 2-DOF structure mass ratio μ is 2 and the frequency ratio β is 0.447 according to the above definition. The optimal damping coefficient of the Maxwell damper is c according to the strategy obtained by the formula (41)dopt＝2.24×107rad/s; the optimal damping coefficient of the Maxwell damper under the second strategy is obtained by the formula (43) and is cdopt＝2.37×107rad/s。
And (3) programming a time-course analysis program of the coupling structure, and obtaining a top-layer displacement time-course curve of the Maxwell calculation example under the seismic waves under two control strategies by using the graph of FIG. 4. As can be seen from the two graphs, the displacement response of the top layer of the structure under the two strategies is effectively controlled, and the effectiveness of the theoretical expression is verified.
4. Numerical example based on MDOF model
In order to verify that the analytical solution of the optimized parameters of the Maxwell damper obtained by 2-SDOF is also applicable to the MDOF model, taking a certain high-rise symmetric double-tower building with a chassis as an example, the part of the chassis (main structure) is 3 floors, the part of the symmetric tower (secondary structure) is 17 floors, and the concentrated mass of each floor of the chassis is 2.5 multiplied by 106kg, shear stiffness all 8.0X 109N/m; the concentrated mass of each floor of the symmetrical tower is 1.0 multiplied by 106kg, shear stiffness all 4.0X 109N/m. And a Rayleigh damping model is adopted, and the first-order damping ratio and the second-order damping ratio of the tower are both 0.02. The structural model is shown in fig. 5. And modal analysis is adopted to obtain that the first-order natural vibration circle frequency of each tower is 5.67rad/s, and the first-order natural vibration circle frequency of the main structure is 25.18 rad/s. When the control strategy of controlling the minimum energy of the main structure is adopted, the optimal coupling damping coefficient is cdopt＝1.24×108rad/s; when the control strategy of controlling the minimum total energy of the structure is adopted, the optimal damping coefficient is cdopt＝1.37×108rad/s。
The time course analysis program of the coupled MDOF structure under the El-Centro wave is compiled, and the time course of the top layer displacement of the structure connected with Maxwell under the two control strategies is obtained and is shown in FIG. 6. The displacement response of the top layer of the structure under the two strategies is effectively controlled. The Maxwell type damper optimization parameter analytical solution obtained based on the single 2-DOF model is proved to be also applicable to the MDOF structure with the chassis and the symmetric double towers.
Fig. 7 shows the energy time course curves of the MDOF structure under different control strategies. From fig. 7, it is found that the vibration energy of the structure is obviously reduced after the damper is connected and the control strategy provided by the text is adopted, and this also shows that the damper fully exerts the energy consumption function thereof and effectively reduces the vibration energy of the structure.
Fig. 8 shows the maximum interlayer displacement of each layer of the MDOF structure without dampers and with different control strategies. As can be seen from fig. 8, the three curves have approximately the same change law: the rigidity of the front three layers of main structures (chassis) is higher than that of the secondary structures, so the maximum interlayer displacement generated is far smaller than that of the upper tower (secondary structure). The maximum inter-floor displacement of the upper tower (from the structure) increases gradually as the floors increase, reaches a maximum at the fifth level and then decreases as the floors increase. From the maximum interlayer displacement per layer, it can be seen that the maximum interlayer displacement under both control strategies is significantly smaller than that under the uncontrolled condition. This is consistent with the numerical example conclusions in two degrees of freedom, again indicating that the optimized parameters are also applicable to multi-degree of freedom systems.
It will be understood that modifications and variations can be made by persons skilled in the art in light of the above teachings and all such modifications and variations are intended to be included within the scope of the invention as defined in the appended claims.
Claims (5)
1. A method for determining optimized parameters of a Maxwell damper between symmetrical double-tower structures is characterized in that the Maxwell damper is arranged between the symmetrical double-tower structures, each tower structure in the symmetrical double-tower structures comprises a chassis and upper towers, and the Maxwell damper is arranged between the two upper towers; the method comprises the following steps:
step one, coupling a symmetrical double-tower structure into a single 2-DOF system connected with a Maxwell type damper, deducing a frequency response function of tower displacement by taking stable white noise as seismic excitation, and establishing an expression of vibration energy of the symmetrical double-tower structure;
step two, taking the damping coefficient of the Maxwell damper as a research parameter, and respectively taking the vibration energy of the chassis and the tower with the chassis as a control target to obtain an optimized parameter analytical expression of the Maxwell damper;
step three, verifying the validity of the optimized parameter analytical expression through numerical calculation;
the specific method of the first step comprises the following steps:
step 1.1, adopting a Maxwell control unit calculation model to simulate a symmetrical double-tower damper, wherein the model is formed by serially connecting a damping element and a spring and calculates a control force generated by the model;
step 1.2, regarding the chassis of the symmetrical double-tower structure as a main structure and the upper tower as a secondary structure, and only considering seismic excitation along the symmetrical plane of the structure in the horizontal direction to construct a calculation model of the symmetrical double-tower structure with the chassis; simplifying the structure of the symmetrical double-tower structure calculation model into a 2-DOF adjacent shearing type tower structure model;
step 1.3, obtaining a motion differential equation of the coupling structure system according to the simplified calculation model, and regarding the earthquake action as a ground acceleration process of stable white noise, so that a vibration energy expression of the symmetrical double-tower structure in a time domain is obtained under the earthquake action of a known power spectral density function;
the vibration energy expression obtained in step 1.3 is as follows:
the vibration energy expression of the main structure is as follows:
the vibration energy expression from the structure is:
wherein:
power spectral density function of Sgg，a0-a6An expression consisting of a set of master and slave structural parameters, b0-b5Is an expression formed by another group of main and auxiliary structure parameters.
2. The method for determining the optimized parameters of the Maxwell damper between the symmetrical double-tower structures as claimed in claim 1, wherein the specific method in the second step is as follows:
carrying out parameter analysis on the obtained vibration energy expression of the symmetrical double-tower structure, establishing a relationship curve of respective maximum energy and frequency ratio of the master-slave structures under different delay coefficients, wherein the smaller the delay coefficient is, the smaller the maximum energy difference of the master-slave structures is, and setting the delay coefficient to be 0 under the premise of neglecting the damping of the structure, thereby obtaining the optimal damping coefficient; based on the energy minimum principle, the parameters of the passive damper are adjusted to reduce the relative displacement of the structure, and two strategies are adopted: and respectively taking the vibration energy of the chassis and the tower with the chassis as the minimum control target to obtain an optimized parameter analytical expression of the Maxwell damper.
3. The method for determining the optimized parameters of the Maxwell damper between the symmetrical double-tower structures as claimed in claim 2, wherein the specific method for optimizing the parameters in the second step is as follows:
when the control of the energy minimum of the main structure is taken as an optimization target, the main structure is a chassis:
the optimized damping coefficient is as follows:
wherein:
A01＝β2μ3ω1 2+β2μ2ω1 2-μ2-2μ-1,B01＝-μ3ω1β-4μω1β-4μ2ω1β-ω1β
4. the method for determining the optimized parameters of the Maxwell damper between the symmetrical double-tower structures as claimed in claim 2, wherein the specific method for optimizing the parameters in the second step is as follows:
when the minimum energy of the total structure is controlled as an optimization target, the total structure is a chassis and a tower with the chassis:
the optimized damping coefficient is as follows:
wherein:
A02＝-2β4μ3ω1-8β4μ2ω1-8β4μω1-4β4μ3ω1-2β4ω1-6β2μ2ω1-μ4ω1-2β2μω1-2μ3ω1-μ2ω1
mass ratio mu m of master-slave structure1/m2The tower frequency ratio β ═ ω2/ω1。
5. The method for determining the optimized parameters of the Maxwell damper between the symmetrical double-tower structures as claimed in claim 1, wherein the method for verifying the validity of the analytical expression of the optimized parameters by numerical example in the third step comprises: numerical examples based on 2-DOF models; numerical example based on MDOF model.
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