CN104895210B - Double layer FM mass damper Optimization Design based on coupling stiffness - Google Patents

Abstract

The invention discloses a kind of dual tuned mass damper Optimization Design based on coupling stiffness.The present invention uses following technical proposals：1）Set up structure-SDTMD system models；2）Set up structure-SDTMD systems and kinetics equation；3）Parameter optimization is carried out to the dual tuned mass damper based on coupling stiffness with Gene hepatitis B vaccine；4）By comparing, a kind of dual tuned mass damper based on coupling stiffness of optimization of optimum combination parameter designing is chosen.The innovation of the present invention be to design it is a kind of be applied to the structured new dual tuned mass damper based on coupling stiffness of institute, will be appreciated that the dynamic respond of Structures under Earthquake can be efficiently controlled, and be better than TMD, DTMD, DDTMD.

Description

Double layer FM mass damper Optimization Design based on coupling stiffness

Technical field

The present invention relates to a kind of double layer FM mass damper (Stiffness based double based on coupling stiffness Tuned mass dampers, SDTMD) Optimization Design.

Background technology

Earthquake causes huge disaster to the mankind, is as the natural calamity of serious threat human life's property safety One of numerous multiple states of earthquake, the Wenchuan earthquake for example occurred in recent years, Earthquakes in Japan, Yaan earthquake.These violent earthquakes are not Great economic loss is only caused, also gives people to bring huge grief and serious injury at heart.21 century, with generation Boundary's rapid development of economy, people propose higher and higher requirement to the security and taking precautions against natural calamities property of engineering structure, it is desirable to engineering Structure can not be destroyed when natural calamity (such as macroseism and typhoon) happens suddenly, and require that engineering structure can under its effect Nothing is damaged.Then we propose some revolutionary requirements to preventing and reducing natural disasters for engineering structure, and structural vibration control skill Art is expected to be to realize this revolutionary desired fundamental way of preventing and reducing natural disasters.Traditional seismic design of structures is generally by enhancing The intensity of building structure itself resists the effect of external load with rigidity, so as to reach the effect of antidetonation.So we are being carried out During Aseismic Design, it is necessary first to accurate to estimate external loads, the characteristic of assurance material therefor to be born of structure, and need The rational design of selection and analysis method.But the height of earthquake load is uncertain, material non-linear and performance when using Variation and existing structure analysis and design method limitation cause structure exist be unsatisfactory for using function and safety will The possibility asked.In view of the limitation of traditional structure Seismic Design Method, industry scholars start constantly to seek new to this Method, the design method of structural vibration control is exactly to produce and develop in this case.

Structural vibration control be by taking certain control measure with the dynamic characteristics that adjusts building structure itself or External load effect is offset by applying external energy, so as to reach quake evaluation performance.According to whether outside resources, knot Structure control generally can be divided into following four classes：(1) passive control system, it is a kind of do not need extra power structure control technique, one As refer to add a subsystem, or the processing for constructing some components of structure itself at some position of structure to change The dynamic characteristics (e.g., tuned mass damper (TMD) and multiple tuned mass dampers (MTMD)) of structure changes system；(2) it is main Autocontrol system, it is a kind of to need the structure control technique of extra power, by applying the controling power opposite with direction of vibration come real Existing structure control, controling power is determined by the dynamic response of feedforward external excitation and (or) feedback arrangement；(3) semi-active control aystem, Typically based on passive control, when stature dynamic-load response starts more to prescribe a time limit, using controlling organization come inside active adjustment structure Parameter, makes structural parameters be in optimum state, and required external energy is smaller；(4) hybrid control system, active control and passive The use in conjunction of control, makes it coordinate the cooperation that gets up, and this control system takes full advantage of passive control and active control Respective advantage, both can largely be dissipated vibrational energy by passive control system, can be protected again using active control system Demonstrate,prove control effect, such as main passive tuned mass damper (active-passive tuned mass damper, APTMD).

The content of the invention

The defect existed for prior art, it is an object of the invention to provide a kind of double layer FM matter based on coupling stiffness Measure damper (SDTMD) Optimization Design

To reach above-mentioned purpose, the present invention is using such as following technical proposals：

A kind of double layer FM mass damper Optimization Design based on coupling stiffness, it is characterised in that：Operating procedure It is as follows：

1) structure-double layer FM mass damper SDTMD mechanics of system models based on coupling stiffness are set up：By structure The quality m of itself_s, damping c_sWith rigidity k_s, series connection increases a small matter again on the basis of single tuned mass damper TMD Gauge block, adds an additional springs between architecture quality block and small mass.Then structure-based on coupling stiffness is set up The mechanical model of double layer FM mass damper SDTMD systems；

2) structure-double layer FM mass damper SDTMD system dynamics equations based on coupling stiffness are set up：According to Structural Dynamics principle, is carried out to structure and first tuned mass damper TMD1, second tuned mass damper TMD2 Force analysis, sets up structure-double layer FM mass damper SDTMD system equations based on coupling stiffness；

3) parameter optimization is carried out to the double layer FM mass damper SDTMD based on coupling stiffness；

4) the double layer FM mass damper SDTMD based on coupling stiffness of optimization is designed：By comparing, choose optimal Combination parameter, designs the double layer FM mass damper based on coupling stiffness of optimization, for carrying out vibration control to structure.

The step 1) in set up the mechanical models of structure-SDTMD systems：Using structure as a single-degree-of-freedom particle, Determine that it damps c according to its material characteristics_sWith rigidity k_s, by TMD1 devices in structure, then by TMD2 devices on TMD1, It is k that a rigidity is added between TMD2 and structure_LAdditional springs；Structure-SDTMD systems are constituted with this.

The step 2) in set up the kinetic equations of structure-SDTMD systems：Respectively to structure, TMD1, TMD2 carry out by Power is analyzed, and according to theory of structural dynamics, listing its system equation is：

In formula,For earthquake ground motion acceleration；y_sDisplacement for structure relative to substrate；y_TFor TMD1 (i.e. big matter Gauge block) relative to the displacement of structure；y_tIt is TMD2 (i.e. small mass) relative to the displacement of structure；m_s、c_sAnd k_sRespectively structure Controlled vibration shape quality, damping and rigidity；m_T、c_TAnd k_TRespectively TMD1 mass, damping and rigidity；m_t、c_tAnd k_tRespectively TMD2 mass, damping and rigidity；k_LFor the rigidity of additional springs.

The step 3) in based on coupling stiffness double layer FM mass damper carry out Parameters Optimal Design be：

Displacement (the y of structure-SDTMD systems_s) dynamic magnification factor：

TMD1 strokes (y_T) dynamic magnification factor be：

TMD2 strokes (y_t) dynamic magnification factor be：

In formula：

R_eT(λ)=D₁₁R_e(λ)-D₁₂I_m(λ)

I_mT(λ)=D₁₂R_e(λ)+D₁₁I_m(λ)

R_et(λ)=D₂₁R_e(λ)-D₂₂I_m(λ)

I_mt(λ)=D₂₂R_e(λ)+D₂₁I_m(λ)

C₁₂=-2 (1+ η) ξ_tf_tλ

C₂₂=2 ξ_Tf_Tλ

In formula：λ is the frequency ratio of main structure；f_TFor TMD1 frequency ratio；f_tFor TMD2 frequency ratio；f_LFor additional springs Frequency ratio；ξ_sFor the damping ratio of main structure；ξ_TFor TMD1 damping ratio；ξ_tFor TMD2 damping ratio；μ_TFor TMD1 and structure Mass ratio；μ_tFor TMD2 and the mass ratio of structure；η is TMD2 and TMD1 mass ratio.

In optimization process, according to Practical Project, λ, μ are set_T, η value, to f_T、f_t、f_L、ξ_T、ξ_tCarry out parameter optimization.

The step 4) optimum combination parameter is chosen by comparing, design the SDTMD of optimization：Define optimized parameter evaluation Criterion：The minimum of the minimum value of the structure maximum power amplification coefficient of double layer FM mass damper based on coupling stiffness is set Change, i.e., Smaller, then device vibration control validity is just about good；Profit Parameter optimization is carried out with Gene hepatitis B vaccine, and is compared with TMD, DTMD, DDTMD.

Compared with prior art, the present invention has prominent substantive distinguishing features and significant advantage as follows：

The inventive method design it is a kind of be applied to the structured double layer FM mass damper based on coupling stiffness, it is excellent More part is that the dynamic respond of Structures under Earthquake can be efficiently controlled, and better than TMD, DTMD, DDTMD.

Brief description of the drawings

Fig. 1 is double layer FM mass damper (SDTMD) Optimization Design flow chart based on coupling stiffness.

Fig. 2 is double layer FM mass damper (SDTMD) system model structural representation based on coupling stiffness.

Fig. 3 is TMD, DTMD, SDTMD f_TWith η variation relation curve maps.

Fig. 4 is DTMD, SDTMD f_tWith η variation relation curve maps.

Fig. 5 is SDTMD f_LWith η variation relation curve maps.

Fig. 6 is TMD, DTMD, SDTMD ξ_TWith η variation relation curve maps.

Fig. 7 is DTMD, SDTMD ξ_tWith η variation relation curve maps.

Fig. 8 is TMD, DTMD, DDTMD, SDTMDWith η variation relation curve maps.

Fig. 9 is TMD, DTMD, SDTMDWith η variation relation curve maps.

Figure 10 is DTMD, base SDTMDWith η variation relation curve maps.

Embodiment

Below in conjunction with the accompanying drawings, the specific embodiment of the present invention is elaborated.

Embodiment one：

As shown in figure 1, this double layer FM mass damper Optimization Design based on coupling stiffness, including following step Suddenly：

1) structure-double layer FM mass damper SDTMD mechanics of system models based on coupling stiffness are set up：By structure The quality m of itself_s, damping c_sWith rigidity k_s, series connection increases a small matter again on the basis of single tuned mass damper TMD Gauge block, adds an additional springs between architecture quality block and small mass.Then structure-based on coupling stiffness is set up The mechanical model of double layer FM mass damper SDTMD systems；

2) structure-double layer FM mass damper SDTMD system dynamics equations based on coupling stiffness are set up：According to Structural Dynamics principle, is carried out to structure and first tuned mass damper TMD1, second tuned mass damper TMD2 Force analysis, sets up structure-double layer FM mass damper SDTMD system equations based on coupling stiffness；

3) parameter optimization is carried out to the double layer FM mass damper SDTMD based on coupling stiffness；

4) the double layer FM mass damper SDTMD based on coupling stiffness of optimization is designed：By comparing, choose optimal Combination parameter, designs the double layer FM mass damper based on coupling stiffness of optimization, for carrying out vibration control to structure.

As shown in Fig. 2 the step 1) in set up the mechanical models of structure-SDTMD systems：It regard structure as a list Free degree particle, determines that it damps c according to its material characteristics_sWith rigidity k_s, by TMD1 devices in structure, then by TMD2 devices On TMD1, it is k that a rigidity is added between TMD2 and structure_LAdditional springs；Structure-SDTMD systems are constituted with this.

The step 2) in set up the kinetics equations of structure-SDTMD systems：Structure, TMD1, TMD2 are carried out respectively Force analysis, according to theory of structural dynamics, listing its system equation is：

In formula,For earthquake ground motion acceleration；y_sDisplacement for structure relative to substrate；y_TFor TMD1 (i.e. big matter Gauge block) relative to the displacement of structure；y_tIt is TMD2 (i.e. small mass) relative to the displacement of structure；m_s、c_sAnd k_sRespectively structure Controlled vibration shape quality, damping and rigidity；m_T、c_TAnd k_TRespectively TMD1 mass, damping and rigidity；m_t、c_tAnd k_tRespectively TMD2 mass, damping and rigidity；k_LFor the rigidity of additional springs.

The step 3) in based on coupling stiffness double layer FM mass damper carry out Parameters Optimal Design be：

Displacement (the y of structure-SDTMD systems_s) dynamic magnification factor：

TMD1 strokes (y_T) dynamic magnification factor be：

TMD2 strokes (y_t) dynamic magnification factor be：

In formula：

R_eT(λ)=D₁₁R_e(λ)-D₁₂I_m(λ)

I_mT(λ)=D₁₂R_e(λ)+D₁₁I_m(λ)

R_et(λ)=D₂₁R_e(λ)-D₂₂I_m(λ)

I_mt(λ)=D₂₂R_e(λ)+D₂₁I_m(λ)

C₁₂=-2 (1+ η) ξ_tf_tλ

C₂₂=2 ξ_Tf_Tλ

In formula：λ is the frequency ratio of main structure；f_TFor TMD1 frequency ratio；f_tFor TMD2 frequency ratio；f_LFor additional springs Frequency ratio；ξ_sFor the damping ratio of main structure；ξ_TFor TMD1 damping ratio；ξ_tFor TMD2 damping ratio；μ_TFor TMD1 and structure Mass ratio；μ_tFor TMD2 and the mass ratio of structure；η is TMD2 and TMD1 mass ratio.

In optimization process, according to Practical Project, λ, μ are set_T, η value, to f_T、f_t、f_L、ξ_T、ξ_tCarry out parameter optimization.

Calculating is optimized with Gene hepatitis B vaccine, the double layer FM quality resistance based on coupling stiffness is equipped in the structure Drawn during Buddhist nun's device when equipping the double layer FM mass damper based on coupling stiffness in the structure, big mass and original structure frequency f_T, small mass and original structure frequency compare f_t, TMD1 damping ratio ξ_T, TMD2 damping ratio ξ_t, additional springs frequency compare f_L, position Mobile force amplifying coefficientThe dynamic magnification factor of TMD1 strokesThe dynamic magnification factor of TMD2 strokesWith η variation relation curve, as shown in Fig. 3 to 10.

As seen from Figure 8, the effective sex ratio list of the vibration control of the double layer FM mass damper based on coupling stiffness Individual tuned mass damper (TMD), double layer FM mass damper (DTMD) and the double layer FM quality resistance based on damping connection Buddhist nun's device (DDTMD) is good, and with η increase, validity is become better and better.

As seen from Figure 9, the TMD1's of the damping system equipped with the double layer FM mass damper based on coupling stiffness Stroke is relative to being obviously reduced equipped with single tuned mass damper (TMD) and double layer FM mass damper (DTMD).Work as η When >=0.75 with TMD2 and TMD1 mass ratio η increase, stroke reduce it is unobvious.

Integrated and found out by Fig. 3 to Figure 10, the damping system equipped with the double layer FM mass damper based on coupling stiffness ξ_tIt is obviously reduced relative to double layer FM mass damper (DTMD), less than 0.6；Double layer FM quality based on coupling stiffness The f of the damping of damper_tReduce with TMD2 and TMD1 mass ratio η increase, be less than or close to 0.35, be unsuitable for reality Border engineering application；And as η >=0.75 with TMD2 and TMD1 mass ratio η increase, validity improves unobvious, and TMD1 Stroke differ too big with TMD2 stroke, therefore it is contemplated that 0.25 in practice<η<0.75 situation.

Compare Fig. 3 to Figure 10, it is considered to the factor of validity, choose μ_T=0.01, η=0.5, f_T=1.021, f_t= 0.429, f_L=0.751, ξ_T=0, ξ_t=0.219,This One group of design data SDTMD device, the validity of the SDATMD devices is good compared with TMD, DTMD, DDTMD, and parameter is in reasonable model In enclosing, it can preferably control to reduce damage of the vibration to structure.

Claims (4)

1. a kind of double layer FM mass damper Optimization Design based on coupling stiffness, it is characterised in that including following step Suddenly：

1) structure-double layer FM mass damper SDTMD mechanics of system models based on coupling stiffness are set up：By structure itself Quality m_s, damping c_sWith rigidity k_s, series connection increases a small mass again on the basis of single tuned mass damper TMD, An additional springs are added between architecture quality block and small mass, structure-double-deck tune based on coupling stiffness is then set up The mechanical model of humorous mass damper SDTMD systems；

2) structure-double layer FM mass damper SDTMD system dynamics equations based on coupling stiffness are set up：According to structure Principle of dynamics, stress is carried out to structure and first tuned mass damper TMD1, second tuned mass damper TMD2 Analysis, sets up structure-double layer FM mass damper SDTMD system equations based on coupling stiffness, the equation is expressed as down Formula：

$m_s\&lsqb;{\stackrel{\&CenterDot;\&CenterDot;}{x}}_g\left(t\right)+{\stackrel{\&CenterDot;\&CenterDot;}{y}}_S\&rsqb;+c_s{\stackrel{\&CenterDot;}{y}}_s+k_s{y}_s-c_T{\stackrel{\&CenterDot;}{y}}_T-k_T{y}_T-k_L(y_T+y_t)=0---\left(1\right)$

$m_T\&lsqb;{\stackrel{\&CenterDot;\&CenterDot;}{x}}_g\left(t\right)+{\stackrel{\&CenterDot;\&CenterDot;}{y}}_s+{\stackrel{\&CenterDot;\&CenterDot;}{y}}_T\&rsqb;+c_T{\stackrel{\&CenterDot;}{y}}_T+k_T{y}_T-c_t{\stackrel{\&CenterDot;}{y}}_t-k_t{y}_t=0---\left(2\right)$

$m_t\&lsqb;{\stackrel{\&CenterDot;\&CenterDot;}{x}}_g\left(t\right)+{\stackrel{\&CenterDot;\&CenterDot;}{y}}_s+{\stackrel{\&CenterDot;\&CenterDot;}{y}}_T+{\stackrel{\&CenterDot;\&CenterDot;}{y}}_t\&rsqb;+c_t{\stackrel{\&CenterDot;}{y}}_t+k_t{y}_t+k_L(y_T+y_t)=0---\left(3\right)$

In formula,For earthquake ground motion acceleration；y_sDisplacement for structure relative to substrate；y_TFor TMD1, i.e., big quality Block, relative to the displacement of structure；y_tFor TMD2, i.e., small mass, relative to the displacement of structure；m_s、c_sAnd k_sRespectively structure Controlled vibration shape quality, damping and rigidity；m_T、c_TAnd k_TRespectively TMD1 mass, damping and rigidity；m_t、c_tAnd k_tRespectively TMD2 Quality, damping and rigidity；k_LFor the rigidity of additional springs；

3) parameter optimization is carried out to the double layer FM mass damper SDTMD based on coupling stiffness；

4) the double layer FM mass damper SDTMD based on coupling stiffness of optimization is designed：By comparing, optimum combination is chosen Parameter, designs the double layer FM mass damper based on coupling stiffness of optimization, for carrying out vibration control to structure.

2. the double layer FM mass damper Optimization Design according to claim 1 based on coupling stiffness, its feature It is, the step 1), set up the mechanical model of structure-SDTMD systems：Using structure as a single-degree-of-freedom particle, according to Its material characteristics determines that it damps c_sWith rigidity k_s, TMD1 is arranged in structure, then TMD2 is arranged on TMD1, in TMD2 It is k that a rigidity is added between structure_LAdditional springs；Structure-SDTMD systems are constituted with this.

3. the double layer FM mass damper Optimization Design according to claim 1 based on coupling stiffness, its feature It is, the step 3), carrying out Parameters Optimal Design to the double layer FM mass damper SDTMD based on coupling stiffness is：

Displacement structure (y_s) dynamic magnification factor：

$\begin{array}{c}{\mathrm{DMF}}_{H_s}=\left|\frac{{\mathrm{\&omega;}}_s^2{H}_s(-i\mathrm{\&omega;})}{{\stackrel{\&CenterDot;\&CenterDot;}{X}}_g}\right|\\ =\left|\frac{{\stackrel{\&OverBar;}{R}}_e\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{I}}_m\left(\mathrm{\&lambda;}\right)i}{R_e\left(\mathrm{\&lambda;}\right)+I_m\left(\mathrm{\&lambda;}\right)i}\right|\\ =\frac{\sqrt{{\&lsqb;{\stackrel{\&OverBar;}{R}}_e\left(\mathrm{\&lambda;}\right)\&rsqb;}^2+{\&lsqb;{\stackrel{\&OverBar;}{I}}_m\left(\mathrm{\&lambda;}\right)\&rsqb;}^2}}{\sqrt{{\&lsqb;R_e\left(\mathrm{\&lambda;}\right)\&rsqb;}^2+{\&lsqb;I_m\left(\mathrm{\&lambda;}\right)\&rsqb;}^2}}\end{array}---\left(6\right)$

TMD1 strokes (y_T) dynamic magnification factor be：

$\begin{array}{c}{\mathrm{DMF}}_{H_T}=\left|\frac{{\mathrm{\&omega;}}_s^2{H}_T(-i\mathrm{\&omega;})}{{\stackrel{\&CenterDot;\&CenterDot;}{X}}_g}\right|\\ =\left|\frac{{\stackrel{\&OverBar;}{R}}_{eT}\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{I}}_{mT}\left(\mathrm{\&lambda;}\right)i}{R_{eT}\left(\mathrm{\&lambda;}\right)+I_{mT}\left(\mathrm{\&lambda;}\right)i}\right|\\ =\frac{\sqrt{{\&lsqb;{\stackrel{\&OverBar;}{R}}_{eT}\left(\mathrm{\&lambda;}\right)\&rsqb;}^2+{\&lsqb;{\stackrel{\&OverBar;}{I}}_{mT}\left(\mathrm{\&lambda;}\right)\&rsqb;}^2}}{\sqrt{{\&lsqb;R_{eT}\left(\mathrm{\&lambda;}\right)\&rsqb;}^2+{\&lsqb;I_{mT}\left(\mathrm{\&lambda;}\right)\&rsqb;}^2}}\end{array}---\left(7\right)$

TMD2 strokes (y_t) dynamic magnification factor be：

$\begin{array}{c}{\mathrm{DMF}}_{H_t}=\left|\frac{{\mathrm{\&omega;}}_s^2{H}_t(-i\mathrm{\&omega;})}{{\stackrel{\&CenterDot;\&CenterDot;}{X}}_g}\right|\\ =\left|\frac{{\stackrel{\&OverBar;}{R}}_{et}\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{I}}_{mt}\left(\mathrm{\&lambda;}\right)i}{R_{et}\left(\mathrm{\&lambda;}\right)+I_{mt}\left(\mathrm{\&lambda;}\right)i}\right|\\ =\frac{\sqrt{{\&lsqb;{\stackrel{\&OverBar;}{R}}_{et}\left(\mathrm{\&lambda;}\right)\&rsqb;}^2+{\&lsqb;{\stackrel{\&OverBar;}{I}}_{mt}\left(\mathrm{\&lambda;}\right)\&rsqb;}^2}}{\sqrt{{\&lsqb;R_{et}\left(\mathrm{\&lambda;}\right)\&rsqb;}^2+{\&lsqb;I_{mt}\left(\mathrm{\&lambda;}\right)\&rsqb;}^2}}\end{array}---\left(8\right)$

In formula：

${\stackrel{\&OverBar;}{R}}_e\left(\mathrm{\&lambda;}\right)=D_{11}+\&lsqb;C_{11}({\mathrm{\&mu;}}_T{f}_T^2+{\mathrm{\&mu;}}_t{f}_L^2)+2C_{12}{\mathrm{\&mu;}}_T{\mathrm{\&xi;}}_T{f}_T\mathrm{\&lambda;}\&rsqb;+C_{21}{\mathrm{\&mu;}}_t{f}_L^2$

${\stackrel{\&OverBar;}{I}}_m\left(\mathrm{\&lambda;}\right)=D_{12}-\&lsqb;-C_{12}({\mathrm{\&mu;}}_T{f}_T^2+{\mathrm{\&mu;}}_t{f}_L^2)+2C_{11}{\mathrm{\&mu;}}_T{\mathrm{\&xi;}}_T{f}_T\mathrm{\&lambda;}\&rsqb;+C_{22}{\mathrm{\&mu;}}_t{f}_L^2$

$R_e\left(\mathrm{\&lambda;}\right)=\&lsqb;D_{11}(-{\mathrm{\&lambda;}}^2+1)+2D_{12}{\mathrm{\&xi;}}_s\mathrm{\&lambda;}\&rsqb;-\&lsqb;C_{11}({\mathrm{\&mu;}}_T{f}_T^2+{\mathrm{\&mu;}}_t{f}_L^2)+2C_{12}{\mathrm{\&mu;}}_T{\mathrm{\&xi;}}_T{f}_T\mathrm{\&lambda;}\&rsqb;{\mathrm{\&lambda;}}^2-C_{21}{\mathrm{\&mu;}}_t{f}_L^2{\mathrm{\&lambda;}}^2$

$I_m\left(\mathrm{\&lambda;}\right)=\&lsqb;D_{12}(-{\mathrm{\&lambda;}}^2+1)-2D_{11}{\mathrm{\&xi;}}_s\mathrm{\&lambda;}\&rsqb;+\{\&lsqb;-C_{12}({\mathrm{\&mu;}}_T{f}_T^2+{\mathrm{\&mu;}}_t{f}_L^2)+2C_{11}{\mathrm{\&mu;}}_T{\mathrm{\&xi;}}_T{f}_T\mathrm{\&lambda;}\&rsqb;-2C_{21}{\mathrm{\&mu;}}_t{f}_L^2\}{\mathrm{\&lambda;}}^2$

${\stackrel{\&OverBar;}{R}}_{eT}\left(\mathrm{\&lambda;}\right)=C_{11}\&lsqb;R_e\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{R}}_e\left(\mathrm{\&lambda;}\right){\mathrm{\&lambda;}}^2\&rsqb;-C_{12}\&lsqb;I_m\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{I}}_m\left(\mathrm{\&lambda;}\right){\mathrm{\&lambda;}}^2\&rsqb;$

${\stackrel{\&OverBar;}{I}}_{mT}\left(\mathrm{\&lambda;}\right)=C_{12}\&lsqb;R_e\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{R}}_e\left(\mathrm{\&lambda;}\right){\mathrm{\&lambda;}}^2\&rsqb;+C_{11}\&lsqb;I_m\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{I}}_m\left(\mathrm{\&lambda;}\right){\mathrm{\&lambda;}}^2\&rsqb;$

R_eT(λ)=D₁₁R_e(λ)-D₁₂I_m(λ)

I_mT(λ)=D₁₂R_e(λ)+D₁₁I_m(λ)

${\stackrel{\&OverBar;}{R}}_{et}\left(\mathrm{\&lambda;}\right)=C_{21}\&lsqb;R_e\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{R}}_e\left(\mathrm{\&lambda;}\right){\mathrm{\&lambda;}}^2\&rsqb;-C_{22}\&lsqb;I_m\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{I}}_m\left(\mathrm{\&lambda;}\right){\mathrm{\&lambda;}}^2\&rsqb;$

${\stackrel{\&OverBar;}{I}}_{mt}\left(\mathrm{\&lambda;}\right)=C_{22}\&lsqb;R_e\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{R}}_e\left(\mathrm{\&lambda;}\right){\mathrm{\&lambda;}}^2\&rsqb;+C_{21}\&lsqb;I_m\left(\mathrm{\&lambda;}\right)+{\stackrel{\&OverBar;}{I}}_m\left(\mathrm{\&lambda;}\right){\mathrm{\&lambda;}}^2\&rsqb;$

R_et(λ)=D₂₁R_e(λ)-D₂₂I_m(λ)

I_mt(λ)=D₂₂R_e(λ)+D₂₁I_m(λ)

$C_{11}=-{\mathrm{\&lambda;}}^2+f_L^2+(1+\mathrm{\&eta;})f_t^2$

C₁₂=-2 (1+ η) ξ_tf_tλ

$D_{11}=(-{\mathrm{\&lambda;}}^2+f_T^2)(-{\mathrm{\&lambda;}}^2+f_t^2+f_L^2)-4{\mathrm{\&xi;}}_T{\mathrm{\&xi;}}_t{f}_T{f}_t{\mathrm{\&lambda;}}^2+{\mathrm{\&eta;f}}_t^2(-{\mathrm{\&lambda;}}^2+f_L^2)$

$D_{12}=-2{\mathrm{\&xi;}}_t{f}_t\mathrm{\&lambda;}(-{\mathrm{\&lambda;}}^2+f_T^2)-2{\mathrm{\&xi;}}_T{f}_T\mathrm{\&lambda;}(-{\mathrm{\&lambda;}}^2+f_t^2+f_L^2)-2{\mathrm{\&eta;\&xi;}}_t{f}_t\mathrm{\&lambda;}(-{\mathrm{\&lambda;}}^2+f_L^2)$

$C_{21}=f_L^2-f_T^2$

C₂₂=2 ξ_Tf_Tλ

$D_{21}=-{\mathrm{\&eta;f}}_t^2(-{\mathrm{\&lambda;}}^2+f_L^2)-(-{\mathrm{\&lambda;}}^2+f_L^2)(-{\mathrm{\&lambda;}}^2+f_t^2+f_L^2)+4{\mathrm{\&xi;}}_T{\mathrm{\&xi;}}_t{f}_T{f}_t{\mathrm{\&lambda;}}^2$

$D_{22}=2{\mathrm{\&eta;\&xi;}}_t{f}_t\mathrm{\&lambda;}(-{\mathrm{\&lambda;}}^2+f_L^2)+2{\mathrm{\&xi;}}_t{f}_t\mathrm{\&lambda;}(-{\mathrm{\&lambda;}}^2+f_T^2)+2{\mathrm{\&xi;}}_T{f}_T\mathrm{\&lambda;}(-{\mathrm{\&lambda;}}^2+f_t^2+f_L^2)$

In formula：λ is the frequency ratio of main structure；f_TFor TMD1 frequency ratio；f_tFor TMD2 frequency ratio；f_LFor the frequency of additional springs Rate ratio；ξ_sFor the damping ratio of main structure；ξ_TFor TMD1 damping ratio；ξ_tFor TMD2 damping ratio；μ_TFor TMD1 and the quality of structure Than；μ_tFor TMD2 and the mass ratio of structure；η is TMD2 and TMD1 mass ratio；In optimization process, according to Practical Project, setting λ、μ_T, η value, to f_T、f_t、f_L、ξ_T、ξ_tCarry out parameter optimization.

4. the double layer FM mass damper Optimization Design according to claim 1 based on coupling stiffness, its feature It is, the step 4), design the double layer FM mass damper based on coupling stiffness of optimization：Define optimized parameter evaluation Criterion：The minimum of the minimum value of the structure maximum power amplification coefficient of double layer FM mass damper based on coupling stiffness is set Change, i.e.,Smaller, then device vibration control validity is better；Profit Parameter optimization is carried out with genetic algorithm, and is compared with TMD, DTMD, DDTMD.