In this paper, the frequency domain analysis method is used to consider the position of TMD in the multi-degree-of-freedom structure and the modal characteristics of the structure, and the frequency response equation of the controlled generalized coordinates of NDOF structure with MTMD is derived. On this basis, the parameters of MTMD are optimized. The example shows that as long as the MTMD design is correct, the dynamic response of seismic wave of the control structure can be effectively reduced.
I. Overview
As early as 1909, Frahm of the United States put forward the concept of controlling or weakening the motion of another mass with the motion of one mass, and applied for a patent. After that, people did a lot of research. At that time, Frahm proposed tuned mass vibration absorber to reduce the dynamic response of ships, and then it was slowly applied to building construction in the field of civil engineering to reduce the vibration of buildings under the action of wind and earthquake and the vibration caused by using loads (such as galloping). For a long time, the research on tuned mass dampers has mostly focused on single tuned mass dampers.
Because the premise of TMD's function is accurate frequency modulation, and STMD is only a frequency value, it is difficult to know the frequency of the controlled vibration mode accurately for various reasons, and the frequency of the structure will change during the vibration process, so it is almost impossible to realize accurate frequency modulation of STMD, and it is also difficult to apply it in practice. It is also difficult to set a large mass at a certain position from the perspective of design and construction, so other forms of TMD are needed. In recent years, in order to improve the robustness of TMD to the uncertainties of the main system and TMD itself, some scholars began to study MTMD. However, the above-mentioned researches on STMD and MTMD are actually aimed at single-degree-of-freedom main structures, and the real seismic analysis of TMD bridges is rare. In this paper, the frequency domain analysis method is adopted, considering the position of TMD in a multi-degree-of-freedom structure and the characteristics of structural vibration modes, the generalized coordinates of controlled vibration modes of MDOF structure and the frequency response equation of MTMD are derived, on this basis, the parameters of MTMD are optimized, and an example of a bridge is given.
Secondly, the frequency and amplitude equation of MTMD control report in MDOF structure.
Let the degree of freedom of the structure be m, the number of TMD be n (set as odd number), and the frequency of MTMD is distributed around the frequency of the control mode at a certain interval. In order to facilitate processing and manufacturing, each TMD adopts the same stiffness and damping constant, and only the mass changes.
Third, the parameter analysis and design of MTMD
The parameters that should be determined when designing MTMD include: the number of TMDs of MTMD, the stiffness constant of each TMD, the damping constant of each TMD and the frequency interval of TMD.
A large number of calculations show that the optimization parameters of MTMD will be different for different structures or different vibration modes of the same structure, so it is necessary to carry out parameter optimization analysis for specific vibration modes of specific structures. Due to the limitation of space, the optimization parameters of longitudinal and transverse MTMD (respectively controlling the seismic response of longitudinal modes) of Guanjiagou Bridge are given below.
Fourthly, MTMD controls earthquake time history analysis.
Guanjiagou Bridge is a simply supported beam bridge with a total length of 464 meters. It is a viaduct on Wanxian-Liangping Expressway in Sichuan Province. The whole bridge adopts 1 1 40m prestressed concrete simply supported beam, double-column thin-walled pier and U-shaped gravity abutment. The bridge crosses a valley with a relative height difference of 100 meters, and has several high piers, the highest of which is 97 meters higher than the natural ground. In order to analyze the effect of MTMD in detail, this paper uses 19 different seismic waves to analyze the seismic response time history of Guanjiagou Bridge before and after setting MTMD. The calculation shows that MTMD has a good vibration suppression effect on most seismic waves. Due to the limitation of space, only two seismic waves are given.
MTMD obviously changes the time-history response of the structure and reduces the dynamic response. In the first few seconds, the time-history response of MTMD is basically unchanged. This is because MTMD is still in its infancy and has not moved completely. Although some seismic waves have an increase in response at individual moments, they are all at non-strong moments, and the response is small and irrelevant.
In addition, we should pay attention to a very important situation, that is, the frequency components of ground excitation and the energy (or amplitude) carried by each frequency component. If the seismic wave has a large acceleration peak and the frequency component carrying the main energy is close to the main control frequency of the structural dynamic response, the seismic wave will control the structure. On the other hand, if the frequency component carrying the main energy deviates from the main control frequency of the dynamic response of the structure, the seismic wave has no control effect on the structure. A prerequisite for TMD to absorb structural dynamic response energy is that TMD must move fully. If TMD does not move relative to the main structure, it will be counterproductive, leading to an increase in structural response (equivalent to an increase in the mass of the original structure, and n is the total number of MTMD TMD); If it starts to move, However, the larger the scale of the movement, the smaller its role. As for the seismic wave that plays a controlling role, it will become the dynamic response of the main vibration mode of the large structure, and correspondingly it will make the TMD move greatly, so it has a good control effect on the structure. The seismic wave that has no control over the structure deviates from the main control frequency of the dynamic response of the structure, so it will not amplify the dynamic response of the main control mode of the structure, and correspondingly it will not make the TMD move greatly, so it will not have a good control over the structure, but it doesn't matter, because it has no control over the dynamic response of the structure, and this seismic wave does not belong to the controlled object.
In another case, the occurrence time of the maximum peak response is peaceful, and the period of strong vibration has passed before TMD can move completely. At this time, even if the frequency component of the main energy carried by the seismic wave is close to the frequency of the controlled vibration mode of the structure, the vibration suppression effect of TMD will be slightly worse. Judging from a large number of examples, this is just an example.
Verb (abbreviation of verb) conclusion
In this paper, the frequency domain analysis method is used to consider the position of TMD in multi-degree-of-freedom structures and the characteristics of structural modes, and the frequency response equation of generalized coordinates of MTMD controlled modes of MDOF structures is derived. On this basis, the parameters of MTMD are optimized. The example shows that as long as the MTMD design is correct, the dynamic response of seismic wave of the control structure can be effectively reduced.
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