Research on TMD collision damping based on two-way dense ribbed cavity building cover

The proposed plate method is to simplify the anisotropic concrete hollow-core building cover according to the principle of equal bending and shear stiffness of equivalent cross-section, equivalently simplify it into an orthogonal isotropic, solid concrete building cover with the same thickness, and use the traditional theory of solid slab and shell to carry out the computational analysis and structural design of stress bending moment and deflection of the hollow-core building cover [23].

In the analysis process, the proposed plate method needs to satisfy the following conditions:

1) The spacing of rib beams in the middle mezzanine of the hollow-core building cover is not more than 2.2 times of the total thickness of the building cover, and not more than 900mm.

When the bidirectional dense-ribbed cavity floor cover is equivalent to an isotropic plate, its modulus of elasticity is expressed as: (1) E=II0Ec \[E=\frac{I}{{{I}_{0}}}{{E}_{c}}\]

Where, E is the modulus of elasticity of concrete when the bidirectional dense-ribbed cavity floor cover is equivalent to a solid slab, E_c is the modulus of elasticity of concrete in the bidirectional dense-ribbed cavity floor cover, I is the moment of inertia of the cross-section of the bidirectional dense-ribbed cavity floor cover, and I₀ is the moment of inertia of the cross-section of the bidirectional dense-ribbed cavity floor cover when it is equivalent to a solid slab.

When equivalent to an isotropic slab, the formula is: (2) Ex=IxIoxEc \[{{E}_{x}}=\frac{{{I}_{x}}}{{{I}_{ox}}}{{E}_{c}}\] (3) Ey=IyIoyEc \[{{E}_{y}}=\frac{{{I}_{y}}}{{{I}_{oy}}}{{E}_{c}}\] (4) Exvy=Eyvx \[{{E}_{x}}{{v}_{y}}={{E}_{y}}{{v}_{x}}\] (5) max(vx,vy)=vc \[\max \left( {{v}_{x}},{{v}_{y}} \right)={{v}_{c}}\] (6) Gxy=ExEy2(1+vxvy) \[{{G}_{xy}}=\frac{\sqrt{{{E}_{x}}{{E}_{y}}}}{2\left( 1+\sqrt{{{v}_{x}}{{v}_{y}}} \right)}\]

Where E_x and E_y are the modulus of elasticity in the x and y directions, respectively, of the bidirectional dense-ribbed cavity floor cover equivalent to a solid slab, and I_x and I_y are the moments of inertia in the x and y directions, respectively, of the cross-section of the bidirectional dense-ribbed cavity floor cover. I_ox and I_oy are the moments of inertia of the equivalent anisotropic slab in the x and y directions, respectively, v_x and v_y are the Poisson’s ratios of the equivalent anisotropic slab in the x and y directions, respectively, and G_xy is the shear modulus of the equivalent anisotropic slab. v_c is the Poisson’s ratio of the concrete.

The differential equation for the elastic surface of an orthotropic anisotropic plate under transverse loading is: (7) D1∂4f∂x4+D2∂4f∂y4+2D3∂4f∂x2∂y2=q \[{{D}_{1}}\frac{{{\partial }^{4}}f}{\partial {{x}^{4}}}+{{D}_{2}}\frac{{{\partial }^{4}}f}{\partial {{y}^{4}}}+2{{D}_{3}}\frac{{{\partial }^{4}}f}{\partial {{x}^{2}}\partial {{y}^{2}}}=q\]

Where D₁, D₂ is the flexural stiffness and D₃ is the torsional stiffness. Then: (8) D1=E1h312(1−v1v2)D2=E2h312(1−v1v2)D3=v2D1+2DK=v1D2+2DKDK=Gh312 \[\begin{matrix} {{D}_{1}}=\frac{{{E}_{1}}{{h}^{3}}}{12}\left( 1-{{v}_{1}}{{v}_{2}} \right) \\ {{D}_{2}}=\frac{{{E}_{2}}{{h}^{3}}}{12}\left( 1-{{v}_{1}}{{v}_{2}} \right) \\ {{D}_{3}}={{v}_{2}}{{D}_{1}}+2{{D}_{K}}={{v}_{1}}{{D}_{2}}+2{{D}_{K}} \\ {{D}_{K}}=\frac{G{{h}^{3}}}{12} \\ \end{matrix}\]

Where E₁, E₂ for the two main axis direction of the modulus of elasticity, v₁, v₂ for the corresponding Poisson’s ratio, G for the shear modulus, h for the plate thickness.

Equation (7) in the appropriate boundary conditions in the theoretical solution or numerical solution can be derived from the expression of the deflection of the thin plate f, and then use the following formula can be derived from the bending moment and torque. i.e: (9) Mi=−D1(∂2f∂x2+v2∂2f∂y2)My=−D2(∂2f∂y2+v1∂2f∂x2)Mxy=−2Dk∂2f∂x∂y \[\begin{matrix} {{M}_{i}}=-{{D}_{1}}\left( \frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}+{{v}_{2}}\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}} \right) \\ {{M}_{y}}=-{{D}_{2}}\left( \frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}+{{v}_{1}}\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}} \right) \\ {{M}_{xy}}=-2{{D}_{k}}\frac{{{\partial }^{2}}f}{\partial x\partial y} \\ \end{matrix}\]

where M_x, M_y and M_xy are the bending moments and torsion moments per unit plate width. Strength and deflection checks of this orthotropic anisotropic plate are performed using deflection f and the above equation.

Figure 1 shows a schematic diagram of a bidirectional dense-ribbed cavity floor slab [24]. There is a bi-directional dense-ribbed cavity floor slab of plan dimension a·b, simply supported on all four sides in an air field with elastic upper and lower surface slabs. The upper and lower slab thicknesses are d and the rib height is h.

Figure 1.

Schematic diagram of two-way tight floor

where t₁ and t₂ are time integration domains, T is the kinetic energy of the system; U is the strain energy of the system, and W_ex is the work done by the external force.

Kinetic energy T is: (13) T=12∫0a∫0bm¯w˙2dxdy \[T=\frac{1}{2}\int_{0}^{a}{\int_{0}^{b}{{\bar{m}}}}{{\dot{w}}^{2}}dxdy\]

Where m¯$\bar{m}$ is the mass per unit of the sandwich panel and ẇ is the speed of vibration at the surface of the panel. The strain energy (bending strain energy of the sandwich plate and shear strain energy of the core layer) is: (14) U=12∫0c∫0b( σxεx+σyεy+τxyγyy+τyzcγyxc+ τxxcγzxc )dxdy \[U=\frac{1}{2}\int_{0}^{c}{\int_{0}^{b}{\left( {{\sigma }_{x}}{{\varepsilon }_{x}}+{{\sigma }_{y}}{{\varepsilon }_{y}}+{{\tau }_{xy}}{{\gamma }_{yy}}+\tau _{yz}^{c}\gamma _{yx}^{c}+ \right.}}\left. \tau _{xx}^{c}\gamma _{zx}^{c} \right)dxdy\]

The work done by the external force W_es is: (15) Wes=∫0a∫0b[ F(x,y)−Pi(x,y)−Pb(x,y) ]w(x,y)dxdy \[{{W}_{es}}=\int_{0}^{a}{\int_{0}^{b}{\left[ F(x,y)-{{P}_{i}}(x,y)-{{P}_{b}}(x,y) \right]}}w(x,y)dxdy\]

where the transverse point excitation F(_x,y) = F₀δ(x–x₀)δ(y–y₀), as well as the acoustic pressure load generated by the action of air on the upper surface of the floor slab P_t(x,y), and the acoustic pressure load generated by the action of air on the lower surface P_b(x,y), are obtained from the Rayleigh integral equation: (16) Pt=iρ0f∫0a∫0bw˙e−iL,RRdxdy \[{{P}_{t}}=i{{\rho }_{0}}f\int_{0}^{a}{\int_{0}^{b}{{\dot{w}}}}\frac{{{e}^{-iL,R}}}{R}dxdy\] (17) Pb=−iρ0f∫−0a∫0bw˙e−iLRRRdxdy \[{{P}_{b}}=-i{{\rho }_{0}}f\int_{-0}^{a}{\int_{0}^{b}{{\dot{w}}}}\frac{{{e}^{-iLR}}R}{R}dxdy\]

where, ρ₀ is the fluid density, f is the excitation frequency, k₀ is the number of waves in the acoustic field, k₀ = 2πf/c, c_v is the air density, and R is the distance between the point on the plate surface and the field point.

For rectangular planes, the corresponding boundary conditions for a hollow web sandwich panel with simple perimeter support are as follows:

1) x = 0, x = L_x, ω = 0, θ_y = 0, ∂θx∂x=0\[\frac{\partial {{\theta }_{x}}}{\partial x}=0\], ∂2ω∂x2=0\[\frac{{{\partial }^{2}}\omega }{\partial {{x}^{2}}}=0\].

Under a transverse uniform load q, the q and function ϕ can be expanded into a heavy triangular series, i.e.: (18) q=16qπ2∑m=1,3,...∑n=1,3,...1mnsinmπxLxsinnπyLy \[q=\frac{16q}{{{\pi }^{2}}}\sum\limits_{m=1,3,\cdots }{\sum\limits_{n=1,3,\cdots }{\frac{1}{mn}}}\sin \frac{m\pi x}{{{L}_{x}}}\sin \frac{n\pi y}{{{L}_{y}}}\] (19) ϕ=16qLx8π8∑m=1,3,...∑n=1,3,...AmnsinmπxLxsinnπyLy \[\phi =\frac{16qL_{x}^{8}}{{{\pi }^{8}}}\sum\limits_{m=1,3,\cdots }{\sum\limits_{n=1,3,\cdots }{{{A}_{mn}}}\sin \frac{m\pi x}{{{L}_{x}}}\sin \frac{n\pi y}{{{L}_{y}}}}\]

Clearly, the function ϕ satisfies the boundary conditions and the coefficients A_mn can be determined as: (20) Amn=1Δmn \[{{A}_{mn}}=\frac{1}{{{\Delta }_{mn}}}\] (21) Δmn=π2mn{ 2DfH1m8+2Df(2H1+H2)m6n2λ2 +2Df(H2+2H3)m2n6λ6+2Df(H1+2H2+H3)m4n4λ4+2DfH3n8λ8+β [ (2DfH4+CxH1)m6+(2DfH5+CyH3)n6λ6 +(2DfH5+4DfH4+CxH2+CyH1)m4n2λ2+(4DfH5+2DfH4+CxH3+CyH2)m2n4λ4 ]+β2 [ (CxH4+2Df−CxA2)m4 +(CxH5+4Df+CyH4+CxA1+CyA3)m2n2λ2+(CyH5+2Df−CyA4)n4λ4 ] } \[\begin{align} & {{\Delta }_{mn}}={{\pi }^{2}}mn\left\{ 2{{D}_{f}}{{H}_{1}}{{m}^{8}}+2{{D}_{f}}\left( 2{{H}_{1}}+{{H}_{2}} \right){{m}^{6}}{{n}^{2}}{{\lambda }^{2}} \right. \\ & +2{{D}_{f}}\left( {{H}_{2}}+2{{H}_{3}} \right){{m}^{2}}{{n}^{6}}{{\lambda }^{6}}+2{{D}_{f}}\left( {{H}_{1}}+2{{H}_{2}}+{{H}_{3}} \right){{m}^{4}}{{n}^{4}}{{\lambda }^{4}} \\ & +2{{D}_{f}}{{H}_{3}}{{n}^{8}}{{\lambda }^{8}}+\beta \left[ \left( 2{{D}_{f}}{{H}_{4}}+{{C}_{x}}{{H}_{1}} \right){{m}^{6}}+\left( 2{{D}_{f}}{{H}_{5}}+{{C}_{y}}{{H}_{3}} \right){{n}^{6}}{{\lambda }^{6}} \right. \\ & +\left( 2{{D}_{f}}{{H}_{5}}+4{{D}_{f}}{{H}_{4}}+{{C}_{x}}{{H}_{2}}+{{C}_{y}}{{H}_{1}} \right){{m}^{4}}{{n}^{2}}{{\lambda }^{2}} \\ & +\left. \left( 4{{D}_{f}}{{H}_{5}}+2{{D}_{f}}{{H}_{4}}+{{C}_{x}}{{H}_{3}}+{{C}_{y}}{{H}_{2}} \right){{m}^{2}}{{n}^{4}}{{\lambda }^{4}} \right] \\ & +{{\beta }^{2}}\left[ \left( {{C}_{x}}{{H}_{4}}+2{{D}_{f}}-{{C}_{x}}{{A}_{2}} \right){{m}^{4}} \right. \\ & +\left( {{C}_{x}}{{H}_{5}}+4{{D}_{f}}+{{C}_{y}}{{H}_{4}}+{{C}_{x}}{{A}_{1}}+{{C}_{y}}{{A}_{3}} \right){{m}^{2}}{{n}^{2}}{{\lambda }^{2}} \\ & \left. \left. +\left( {{C}_{y}}{{H}_{5}}+2{{D}_{f}}-{{C}_{y}}{{A}_{4}} \right){{n}^{4}}{{\lambda }^{4}} \right] \right\} \end{align}\]

where λ=LxLy\[\lambda =\frac{{{L}_{x}}}{{{L}_{y}}}\], β=Lx2π2\[\beta =\frac{L_{x}^{2}}{{{\pi }^{2}}}\].

Substituting ϕ gives the angle and deflection: (22) θx=16qLx8π8∑m=1,3,...∑n=1,3,...ΔmnθxΔmncosmπxLxsinnπyLy $${\theta _x} = {{16qL_x^8} \over {{\pi ^8}}}\mathop \sum \limits_{m = 1,3,...} \mathop \sum \limits_{n = 1,3,...} {{\Delta _{mn}^{{\theta _x}}} \over {{\Delta _{mn}}}}\cos {{m\pi x} \over {{L_x}}}\sin {{n\pi y} \over {{L_y}}}$$ (23) θy=16qLx8π8∑m=1,3,...∑n=1,3,...ΔmnθyΔmnsinmπxLxcosnπyLy $${\theta _y} = {{16qL_x^8} \over {{\pi ^8}}}\mathop \sum \limits_{m = 1,3,...} \mathop \sum \limits_{n = 1,3,...} {{\Delta _{mn}^{{\theta _y}}} \over {{\Delta _{mn}}}}\sin {{m\pi x} \over {{L_x}}}\cos {{n\pi y} \over {{L_y}}}$$ (24) ω=16qLx8π8∑m=1,3,...∑n=1,3,...ΔmnωΔmnsinmπxLxsinnπyLy $$\omega = {{16qL_x^8} \over {{\pi ^8}}}\mathop \sum \limits_{m = 1,3,...} \mathop \sum \limits_{n = 1,3,...} {{\Delta _{mn}^\omega } \over {{\Delta _{mn}}}}\sin {{m\pi x} \over {{L_x}}}\sin {{n\pi y} \over {{L_y}}}$$

In the formula: (25) Δmnθx=−A1mn2λ2π3Lx3+A2m3π3Lx3+mπ3Lx \[\Delta _{mn}^{{{\theta }_{x}}}=-{{A}_{1}}m{{n}^{2}}{{\lambda }^{2}}\frac{{{\pi }^{3}}}{L_{x}^{3}}+{{A}_{2}}{{m}^{3}}\frac{{{\pi }^{3}}}{L_{x}^{3}}+m\frac{{{\pi }^{3}}}{{{L}_{x}}}\] (26) Δmnθx=−A3m2nλπ3Lx3+A4n3λ3π3Lx3+nλπLx \[\Delta _{mn}^{{{\theta }_{x}}}=-{{A}_{3}}{{m}^{2}}n\lambda \frac{{{\pi }^{3}}}{L_{x}^{3}}+{{A}_{4}}{{n}^{3}}{{\lambda }^{3}}\frac{{{\pi }^{3}}}{L_{x}^{3}}+n\lambda \frac{\pi }{{{L}_{x}}}\] (27) Δmnω=H1m4π4Lx4+H2m2n2λ2π4Lx4+H3n4λ4π4Lx4+H4m2π2Lx2+H5n2λ2π2Lx2+1 \[\begin{align} & \Delta _{mn}^{\omega }={{H}_{1}}{{m}^{4}}\frac{{{\pi }^{4}}}{L_{x}^{4}}+{{H}_{2}}{{m}^{2}}{{n}^{2}}{{\lambda }^{2}}\frac{{{\pi }^{4}}}{L_{x}^{4}}+{{H}_{3}}{{n}^{4}}{{\lambda }^{4}}\frac{{{\pi }^{4}}}{L_{x}^{4}} \\ & +{{H}_{4}}{{m}^{2}}\frac{{{\pi }^{2}}}{L_{x}^{2}}+{{H}_{5}}{{n}^{2}}{{\lambda }^{2}}\frac{{{\pi }^{2}}}{L_{x}^{2}}+1 \end{align}\]

Where Δmnθx$\Delta _{mn}^{{{\theta }_{x}}}$, Δmnθy$\Delta _{mn}^{{{\theta }_{y}}}$, Δmnω$\Delta _{mn}^{\omega }$ are the corner coefficients and deflection coefficients respectively [25].

After obtaining θ_X, θ_Y and ω, the internal forces M and Q can be obtained by substituting them into the physical equations.

The above formula for calculating the internal force is in the form of a series, we can write a corresponding program to calculate the internal force, in this paper MATLAB software is applied.

A tuned mass damper (TMD) consists of multiple VFPBs and mass blocks exhibiting displacement-dependent linear damped hysteresis, with the VFPBs providing different coefficients of friction by polishing the upper surface of the base plate or by bedding it with a special material [26]. The stiffness of the TMD is determined by the radius of the VFPBs, and, in practice, multiple TMDs can be placed underneath the mass blocks to share the TMD’s stresses and power moments. Meanwhile, the planar arrangement of the VFPBs should maintain geometric stability. Additionally, each VFPB should have the same radius and friction coefficient arrangement to ensure that the mass block body moves in an overall synchronized manner without causing local stress concentration. Because all mechanical behaviors of the TMD exhibit isotropy at the macrostructural analysis level, the TMD can be calculated as a whole during the design process. Concrete, steel blocks, and tanks filled with water can all be used as mass blocks for TMDs.

Based on this, this paper constructs a semi-active TMD vibration damping device applied to a bidirectional dense-ribbed cavity building cover based on TMD, and its specific structure is shown in Fig. 2. This device can effectively improve the vibration damping effect of two-way dense-ribbed cavity building cover, based on the optimization of the parameters of the TMD device to give the two-way dense-ribbed cavity building cover stronger vibration damping performance.The realization of the TMD vibration damping effect relies on the two aspects of the function of the damper system, one is the transfer of kinetic energy, which will be converted into the vibration of the main structure of the kinetic energy of the mass element of the TMD. The second is the ability to dissipate, using the damping element of the TMD to consume and absorb part of the kinetic energy of the TMD. Therefore, the TMD is generally arranged at the larger structural modal vibration displacement to facilitate the realization of energy transfer and dissipation. In this paper, the main vertical vibration patterns of the structure are extracted through structural modal analysis, and the TMD is arranged based on the characteristics of the vibration patterns.

Figure 2.

TMD shock absorber construction

In order to enable the proposed semi-active TMD to simultaneously control the bi-directional vibration of the building structure in the plane, four independent springs are provided in the diamond-shaped spring device, each with a stiffness of k_c. The variable mass m_d(t) of the semi-active TMD can be shifted in two directions, X and Y, with displacements u_xd and u_yd, respectively, and the stiffnesses k_xd(t) and k_ywd(t) for the X -direction and the Y -direction can be expressed as follows: (33) kxd(t)=kecos2[θ(t)] \[{{k}_{xd}}(t)={{k}_{e}}{{\cos }^{2}}[\theta (t)]\] (34) kyd(t)=kesin2[θ(t)] \[{{k}_{yd}}(t)={{k}_{e}}{{\sin }^{2}}[\theta (t)]\]

Since the semi-active TMD can simultaneously adapt to the instantaneous vibration frequencies of the structure in the X - and Y -direction under earthquake, for this reason, it utilizes the wavelet transform to receive and process the displacement signals of the top layer of the main structure in the X - and Y -direction, and then captures its instantaneous frequencies in both directions. The instantaneous frequencies ω_xd(t) and ω_yd(t) in the X and Y directions are obtained by tracking, respectively: (35) ωxd(t)=kxd(t)md(t) \[{{\omega }_{xd}}(t)=\sqrt{\frac{{{k}_{xd}}(t)}{{{m}_{d}}(t)}}\] (36) ωyd(t)=kyd(t)md(t) \[{{\omega }_{yd}}(t)=\sqrt{\frac{{{k}_{yd}}(t)}{{{m}_{d}}(t)}}\]

From equations (33) to (36), the instantaneous pinch angle of the rhombic spring device can be expressed as: (37) θ(t)=arctanωyd(t)ωxd(t) \[\theta (t)=\arctan \frac{{{\omega }_{yd}}(t)}{{{\omega }_{xd}}(t)}\]

From Eq. It can be seen that the instantaneous angle of the rhombic spring device is adjusted according to the instantaneous frequency ratio of the two directions, which can redistribute the stiffness ratio of the two directions, while the mass is further adjusted to match the bi-directional instantaneous vibration frequency. Therefore, its instantaneous mass is: (38) md(t)=kecos2[θ(t)]/[ ωxd(t) ]2 \[{{m}_{d}}(t)={{k}_{e}}{{\cos }^{2}}[\theta (t)]/{{\left[ {{\omega }_{xd}}(t) \right]}^{2}}\]

The variable mass of the semi-active TMD can be water, earth and solid, etc., and is set in an adjustable range of mass considering the efficiency of the variable mass electromechanical system: (39) md(t)={ mdmin(md(t)<mdmin)md(t)(mdmin≤md(t)≤mdmax)mdmax(md(t)>mdmax) \[{{m}_{d}}(t)=\left\{ \begin{array}{*{35}{l}} {{m}_{d\min }} & \left( {{m}_{d}}(t)<{{m}_{d\min }} \right) \\ {{m}_{d}}(t) & \left( {{m}_{d\min }}\le {{m}_{d}}(t)\le {{m}_{d\max }} \right) \\ {{m}_{d\max }} & \left( {{m}_{d}}(t)>{{m}_{d\max }} \right) \\ \end{array} \right.\]

where m_dmin and m_dmax are the preset minimum and maximum values of the variable mass, respectively.

TMD belongs to a kind of dynamic vibration absorbing device, which is composed of solid mass block, vibration dampers and vibration damping spring system, and mainly utilizes the resonance principle of the main structure and sub-structure to suppress the vibration of the main structure. Under the action of broadband random excitation, the corresponding dynamic equations of the semi-active TMD vibration damping system for a two-way dense-ribbed cavity building cover are expressed as:

m₁ and m₂ are the main structure and substructure masses respectively, k₁, c₁ and k₂, c₂ are the elasticity and damping coefficients of the main structure and substructure respectively, and the corresponding dynamic equations are shown below: (40) { m1x¨1+c2(x˙1−x˙2)+k2(x1−x2)+c1x˙1+k1x1=F(t)m2x¨2+c2(x˙2−x˙1)+k2(x2−x1)=0 \[\left\{ \begin{matrix} {{m}_{1}}{{{\ddot{x}}}_{1}}+{{c}_{2}}\left( {{{\dot{x}}}_{1}}-{{{\dot{x}}}_{2}} \right)+{{k}_{2}}\left( {{x}_{1}}-{{x}_{2}} \right)+{{c}_{1}}{{{\dot{x}}}_{1}}+{{k}_{1}}{{x}_{1}}=F(t) \\ {{m}_{2}}{{{\ddot{x}}}_{2}}+{{c}_{2}}\left( {{{\dot{x}}}_{2}}-{{{\dot{x}}}_{1}} \right)+{{k}_{2}}\left( {{x}_{2}}-{{x}_{1}} \right)=0 \\ \end{matrix} \right.\]

where ẍ₁ is the acceleration of mass m₁, ẍ₂ is the acceleration of mass m₂, ẋ₁ is the velocity of mass m₁, ẋ₂ is the velocity of mass m₂, and F(t) is the external excitation of the system.

When the main structure is subjected to external excitation as sinusoidal wave, i.e. F(t) = F₀e^iωt, x₁ = X₁e^iωt, x₂ = X₂e^iωt, it can be obtained by substituting it into the above equation: (41) { eiet⋅ [ m1⋅X1⋅(iω)2+c2⋅(X1⋅iω−X2⋅iω) +k2⋅(X1−X2)+c1⋅X1⋅iω+K1⋅X1 ]=F0⋅eiωteiωt⋅ [ m2⋅X2⋅(iω)2 +c2⋅(X2⋅iω−X1⋅iω)+k2(X2−X1) ]=0 $$\{ \matrix{ {\matrix{ {{e^{iet}} \cdot [{m_1} \cdot {X_1} \cdot {{(i\omega )}^2} + {c_2} \cdot \left( {{X_1} \cdot i\omega - {X_2} \cdot i\omega } \right)} \hfill \cr { + {k_2} \cdot \left( {{X_1} - {X_2}} \right) + {c_1} \cdot {X_1} \cdot i\omega + {K_1} \cdot {X_1}] = {F_0} \cdot {e^{i\omega t}}} \hfill \cr } } \hfill \cr {\matrix{ {{e^{i\omega t}} \cdot [{m_2} \cdot {X_2} \cdot {{(i\omega )}^2}} \hfill \cr { + {c_2} \cdot \left( {{X_2} \cdot i\omega - {X_1} \cdot i\omega } \right) + {k_2}\left( {{X_2} - {X_1}} \right)] = 0} \hfill \cr } } \hfill \cr } $$ (42) [ −ω2m1+(c1+c2)iω+k1+k2−c2iω−k2−c2iω−k2−ω2m2+c2iω+k2 ]⋅[ X1X2 ]=[ F00 ] $\left[ \begin{matrix} -{{\omega }^{2}}{{m}_{1}}+\left( {{c}_{1}}+{{c}_{2}} \right)i\omega +{{k}_{1}}+{{k}_{2}} & -{{c}_{2}}i\omega -{{k}_{2}} \\ -{{c}_{2}}i\omega -{{k}_{2}} & -{{\omega }^{2}}{{m}_{2}}+{{c}_{2}}i\omega +{{k}_{2}} \\ \end{matrix} \right]\cdot \left[ \begin{array}{*{35}{l}} {{X}_{1}} \\ {{X}_{2}} \\ \end{array} \right]=\left[ \begin{matrix} {{F}_{0}} \\ 0 \\ \end{matrix} \right]$

The solution for amplitudes X₁ and X₂ can be expressed as: (43) [ X1X2 ]=[ −ω2m1+(c1+c2)iω+k1+k2−c2iω−k2−c2iω−k2−ω2m2+c2iω+k2 ]−1[ F00 ] $\left[ \begin{array}{*{35}{l}} {{X}_{1}} \\ {{X}_{2}} \\ \end{array} \right]={{\left[ \begin{matrix} -{{\omega }^{2}}{{m}_{1}}+\left( {{c}_{1}}+{{c}_{2}} \right)i\omega +{{k}_{1}}+{{k}_{2}} & -{{c}_{2}}i\omega -{{k}_{2}} \\ -{{c}_{2}}i\omega -{{k}_{2}} & -{{\omega }^{2}}{{m}_{2}}+{{c}_{2}}i\omega +{{k}_{2}} \\ \end{matrix} \right]}^{-1}}\left[ \begin{matrix} {{F}_{0}} \\ 0 \\ \end{matrix} \right]$ (44) | −ω2m1+(c1+c2)iω+k1+k2−c2iω−k2−c2iω−k2−ω2m2+c2iω+k2 |=a−b−c+d $\begin{align} & \left| \begin{matrix} -{{\omega }^{2}}{{m}_{1}}+\left( {{c}_{1}}+{{c}_{2}} \right)i\omega +{{k}_{1}}+{{k}_{2}} & -{{c}_{2}}i\omega -{{k}_{2}} \\ -{{c}_{2}}i\omega -{{k}_{2}} & -{{\omega }^{2}}{{m}_{2}}+{{c}_{2}}i\omega +{{k}_{2}} \\ \end{matrix} \right| \\ & =a-b-c+d \\ \end{align}$

In the formula: (45) { a=ω4m1m2b=ω3(m1c2i+c1im2+c2im2)c=ω2(m1k2+c1c2+k1m2+k2m2+2c22)d=ω(c1ik2+c2ik2+k1c2i−k2c2i)+k1k2 \[\left\{ \begin{array}{*{35}{l}} a={{\omega }^{4}}{{m}_{1}}{{m}_{2}} \\ b={{\omega }^{3}}\left( {{m}_{1}}{{c}_{2}}i+{{c}_{1}}i{{m}_{2}}+{{c}_{2}}i{{m}_{2}} \right) \\ c={{\omega }^{2}}\left( {{m}_{1}}{{k}_{2}}+{{c}_{1}}{{c}_{2}}+{{k}_{1}}{{m}_{2}}+{{k}_{2}}{{m}_{2}}+2c_{2}^{2} \right) \\ d=\omega \left( {{c}_{1}}i{{k}_{2}}+{{c}_{2}}i{{k}_{2}}+{{k}_{1}}{{c}_{2}}i-{{k}_{2}}{{c}_{2}}i \right)+{{k}_{1}}{{k}_{2}} \\ \end{array} \right.\]

From Eqs. (43) and (44), the complex amplitude X₁ about the main system and the complex amplitude X₂ about the damper are: (46) [ X1X2 ]=F0a−b−c+d⋅[ k2−ω2m2+c2iωk2+c2iω ] \[\left[ \begin{matrix} {{X}_{1}} \\ {{X}_{2}} \\ \end{matrix} \right]=\frac{{{F}_{0}}}{a-b-c+d}\cdot \left[ \begin{matrix} {{k}_{2}}-{{\omega }^{2}}{{m}_{2}}+{{c}_{2}}i\omega \\ {{k}_{2}}+{{c}_{2}}i\omega \\ \end{matrix} \right]\] (47) X1=F0⋅(k2−ω2m2+c2iω)a−b−c+d \[{{X}_{1}}=\frac{{{F}_{0}}\cdot \left( {{k}_{2}}-{{\omega }^{2}}{{m}_{2}}+{{c}_{2}}i\omega \right)}{a-b-c+d}\] (48) X2=F0⋅(k2+c2iω)a−b−c+d \[{{X}_{2}}=\frac{{{F}_{0}}\cdot \left( {{k}_{2}}+{{c}_{2}}i\omega \right)}{a-b-c+d}\]

In this paper, only the complex amplitude X₁ of the main system is studied and the results are shown below: (49) | X1 |=F0⋅[ (k2−ω2m2)2+(ωc2)2 ]0.5[ (k1k2+ω4m1m2−k1m2ω2−k2m1ω2−c1c2ω2)2+(k1c2ω−ω3m1c2−ω3m2c2+k2ωc1−ω3m2c1)2]0.5 \[\left| {{X}_{1}} \right|=\frac{{{F}_{0}}\cdot {{\left[ {{\left( {{k}_{2}}-{{\omega }^{2}}{{m}_{2}} \right)}^{2}}+{{\left( \omega {{c}_{2}} \right)}^{2}} \right]}^{0.5}}}{{{\left[ \begin{align} & {{\left( {{k}_{1}}{{k}_{2}}+{{\omega }^{4}}{{m}_{1}}{{m}_{2}}-{{k}_{1}}{{m}_{2}}{{\omega }^{2}}-{{k}_{2}}{{m}_{1}}{{\omega }^{2}}-{{c}_{1}}{{c}_{2}}{{\omega }^{2}} \right)}^{2}} \\ & +{{\left( {{k}_{1}}{{c}_{2}}\omega -{{\omega }^{3}}{{m}_{1}}{{c}_{2}}-{{\omega }^{3}}{{m}_{2}}{{c}_{2}}+{{k}_{2}}\omega {{c}_{1}}-{{\omega }^{3}}{{m}_{2}}{{c}_{1}} \right)}^{2}} \\ \end{align} \right]}^{0.5}}}\]

Introducing the dimensionless parameters, X_st = F₀/K₁, the mass ratio μ = m₂/m₁, the damping ratio of the main system ξ₁ = c₁/2m₁ω₀, the damping ratio of the subsystem ξ₂ = c₂/2m₂ω₀, the intrinsic frequency of the main system ω₀ = (k₁/m₁)^0.5. the intrinsic frequency of the subsystem ω₁ = (k₂/m₂)^0.5, the ratio of the frequency of the external excitation to the frequency of the main system λ = ω/ω₀, and the ratio of the frequency of the subsystem to the frequency of the main system α = ω₁/ω₀. By substituting the above dimensionless coefficients into Eq. (49), the result of the amplitude ratio |X_st/X₁| can be expressed as: (50) | X1Xn |=k1−1μ−1[ (ω2m2−k2)2+(c2ω) ]0.5k1−2μ−1[ (k1k2+ω4m1m2−k1m2ω2−k2m1ω2−c1c2ω2)2+(k1c2ω−ω3m1c2−ω3m2c2+k2ω2c1−ω3m2c1)0.5]=[ (λ2−a2)2+(2ξ2λ)2(μa2λ2+4ξξξ2λ2−λ4+λ2a2+λ2−a2)2+(2ξ2λ3−2ξ2λ+2ξ2μ3+2ξ13k3−2ξ1a2)2 ]0.5 $\begin{align} & \left| \frac{{{X}_{1}}}{{{X}_{n}}} \right|=\frac{k_{1}^{-1}{{\mu }^{-1}}{{\left[ {{\left( {{\omega }^{2}}{{m}_{2}}-{{k}_{2}} \right)}^{2}}+\left( {{c}_{2}}\omega \right) \right]}^{0.5}}}{k_{1}^{-2}{{\mu }^{-1}}\left[ \begin{align} & {{\left( {{k}_{1}}{{k}_{2}}+{{\omega }^{4}}{{m}_{1}}{{m}_{2}}-{{k}_{1}}{{m}_{2}}{{\omega }^{2}}-{{k}_{2}}{{m}_{1}}{{\omega }^{2}}-{{c}_{1}}{{c}_{2}}{{\omega }^{2}} \right)}^{2}} \\ & +{{\left( {{k}_{1}}{{c}_{2}}\omega -{{\omega }^{3}}{{m}_{1}}{{c}_{2}}-{{\omega }^{3}}{{m}_{2}}{{c}_{2}}+{{k}_{2}}{{\omega }_{2}}{{c}_{1}}-{{\omega }^{3}}{{m}_{2}}{{c}_{1}} \right)}^{0.5}} \\ \end{align} \right]} \\ & ={{\left[ \frac{{{\left( {{\lambda }^{2}}-{{a}^{2}} \right)}^{2}}+{{\left( 2{{\xi }_{2}}\lambda \right)}^{2}}}{\begin{align} & {{\left( \mu {{a}^{2}}{{\lambda }^{2}}+4{{\xi }_{\xi }}{{\xi }_{2}}{{\lambda }^{2}}-{{\lambda }^{4}}+{{\lambda }^{2}}{{a}^{2}}+{{\lambda }^{2}}-{{a}^{2}} \right)}^{2}} \\ & +{{\left( 2{{\xi }_{2}}{{\lambda }^{3}}-2{{\xi }_{2}}\lambda +2{{\xi }_{2}}{{\mu }^{3}}+2\xi _{1}^{3}{{k}^{3}}-2{{\xi }_{1}}{{a}^{2}} \right)}^{2}} \\ \end{align}} \right]}^{0.5}} \\ \end{align}$

For TMDs, the selection of a more appropriate damping and stiffness is crucial for its damping effect, and relying only on manual optimization is far from achieving its optimal effect. Based on this problem, in this section, for the determination of the optimal parameters, a genetic algorithm (GA) is chosen to optimize the parameters [27].

The optimized parameters are further analyzed to verify that the damping effect of the damping device is directly proportional to its mass, and considering the actual engineering problems, it is necessary to control its mass, so the impact on the vibration of the building structure is found through the analysis of the vibration of the building structure, and then the optimal parameters of the TMD damping device are obtained.

The previous computational model for a two-way dense-ribbed cavity building cover was constructed using four-side simple support, and the performance of the two-way dense-ribbed cavity building cover will be affected by the cavity hollowing rate. Based on the computational equations and solutions of two-way dense-ribbed cavity building given in the previous paper, the error of the computational results between the proposed plate method and the direct modeling method given in this paper is explored. The calculated results of A and B section moment values of two-way dense ribbed cavity building cover with different hollowing ratio are shown in "Table 1. Where M1 and M² are the cross-section bending moments with vector direction parallel to the y-axis on the AB section, and DNS and ASM are the direct modeling method and the proposed plate method, respectively.

As can be seen from the table, under this boundary condition, the difference between the cross-section bending moment in the mid-span region and the literature reference value is not large, and the error is within 3.5%, while the difference between the cross-section bending moment in the side-span region and the literature value is large, and its maximum error can be up to about 10.26%. There may be two reasons for the error in the cross-section bending moment in the plate edge region, one is due to the simplified assumption conditions of the elastic thin plate theory, which leads to the discrepancy between the analytical solution in the boundary region and the actual one. Secondly, considering that the cross-section of the two-way dense-ribbed cavity floor cover can be regarded as a multi-chambered continuous cylindrical cross-section, when the four sides are constrained, the cylindrical cross-section close to the boundary is subjected to torsional constraints, which makes the cross-sectional bending moments change, leading to an increase in the relative error. When two materials are used in the calculation of the proposed plate method, there is a sudden change in the calculation results at 45% hollow ratio compared to 40%, which may be due to the fact that the beam stiffness is larger in the case of the two materials, which has an effect on the deflection and bending moments of the plate, and produces a sudden change with the increase in the hollow ratio. Overall, the difference between the calculation results obtained by the proposed plate method and the direct modeling method is small, and the fluctuation of the data in the mid-span region is smaller than that in the side-span region. However, similarly, there is a sudden change in the calculation of the proposed plate method when using two materials as opposed to one material when the hollow core ratio of 40% rises to 50%, which may be attributed to the fact that in this case, the stiffness of the edge beams is greater than that of the plate, and the increase in the stiffness of the edge members affects the deflections and bending moments during the calculation and analysis of the plate.

Based on the computational model designed in the previous section, the corresponding finite element model is constructed in the finite element software, and the load is applied to the model, and then the deformation and deflection results of the bidirectional dense-ribbed cavity building cover are shown in Fig. 3. Figure 3(a)~(b) shows the deflection cloud and the comparison results of the calculated and tested load-deflection curves of the slab center, respectively.

Figure 3.

Deformation deflection

The calculated value of plate center deflection is 9.59 mm, which is slightly smaller than the measured value, which is related to the relative idealization of the calculation model. It can also be seen in the figure that the displacement contour is almost a concentric circle centered on the center of the test plate, and the displacement value decreases sequentially to the four sides, which is fully in line with the characteristics of the bidirectional plate, and the overall performance is good.It can be seen from the comparison results of the plate core calculation and the test load-deflection curve that the two curves are roughly consistent, and the change trend is similar, the deformation of the test plate is small at the initial loading stage, and the result of finite element numerical simulation is large, which is due to the idealization of the model, and the model is based on the assumption proposed above, and the error is basically controlled within 2% when the load is loaded to 12kN/m².

The deflection maps of the bidirectional dense-ribbed cavity floor cover and the ordinary solid slab are shown in Fig. 4. Bidirectional dense-ribbed cavity floor cover and solid plate are completely symmetric bidirectional plate, from the deflection cloud diagram can also be seen that the displacement contour is concentric circle form, the deflection value is close to 0 at the plate’s support, which is very close to the results obtained by the experimental plate model, and the compression deformation along the thickness direction of the model is very small and can be ignored. Analyzing the deflection results alone, the maximum deflection value of the bidirectional dense-ribbed cavity floor cover model in this paper is 9.59 mm which is between the solid slab 7.95 mm and the dense-ribbed floor cover 14.26 mm, close to 67.25% of the mid-span deflection value of the dense-ribbed slab, and closer to the maximum deflection value of the solid slab. This shows that the two-way dense ribbed cavity floor cover not only has the same deformation characteristics as the ordinary solid slab, but also has good bi-directional force characteristics and overall force performance. The bidirectional dense-ribbed hollow-cavity floor cover has better integrity and greater stiffness with similar material dosage, and the deflection deformation is nearly 32.75% lower than that of the dense-ribbed floor cover, which is about 20.63% larger than that of the solid slab with the same cross-sectional height, but it saves 45%-60% of the concrete dosage.

Figure 4.

Deflection cloud map

After clarifying the variation of deflection and bending moment values of the bidirectional dense-ribbed cavity building cover, this paper designs a semi-active TMD vibration damping control device, which aims to enhance the collision damping performance of the bidirectional dense-ribbed cavity building cover. To this end, this paper implements the optimization of TMD damping parameters through GA to provide a basis for the parameter setting of the TMD semi-active damping device after TMD dynamics modeling. Based on the principle and calculation method of genetic algorithm given in the previous section, a program is written in MATLAB to search for the optimal damping parameters of the undamped system of two-way dense-ribbed cavity building cover. The numerical and analytical solutions of the TMD optimal frequency ratios and optimal damping ratios of the two-way dense-ribbed cavity building cover without damping under different mass ratios are shown in Table 2, where AS and NS are the analytical and numerical solutions, respectively. As can be seen from the table, under the trend of increasing mass ratio from 0.5% to 9.5%, the error between the analytical and numerical solutions of the optimal frequency ratio of the collision damping dynamics equations of the bidirectional dense-ribbed cavity building cover with no damping of the main structure is in the range of -0.03% to 0.01%. For the optimal damping ratio, the error between the analytical and numerical solutions is between -0.11% and 0.10%. Overall, the optimal parameters of the TMD semi-active vibration damping device obtained by using genetic algorithm optimization can meet the collision damping requirements of two-way dense-ribbed cavity building covers.

In addition, for the influence of the optimal parameters of the main structure damping ratio on the structural displacement-power amplification factor of the bidirectional dense-ribbed cavity building cover, the mass ratio of the TMD semiactive vibration damping device is set to 9.5% when the main structure damping ratio is considered to be 1.5%, 2.5%, 3.5%, and 4.5%, respectively, to obtain the curves of structural displacement-power amplification write-up with the change of the external load frequency shown in Fig. 5. Based on the trend of the curve in the figure, it can be seen that with the main structural damping ratio of the bidirectional dense-ribbed cavity floor cover increased from 1.5% to 4.5%, the structural displacement power amplification factor of the bidirectional dense-ribbed cavity floor cover will be gradually reduced, and the trend of the development trend of “M” is shown.And the maximum value of structural displacement power amplification coefficient can be obtained when the external load frequency ratio is around 0.8 and 1.1, therefore, when the external load frequency ratio is between 0.8 and 1.1 to obtain a better structural displacement power amplification coefficient, and continue to increase the external load frequency on the structural vibration damping of bidirectional dense-ribbed hollow-cavity building cover is not obvious to improve the structural vibration damping effect.

Figure 5.

Structural displacement power amplification coefficient

In order to further analyze the damping control effect of the TMD semi-active control damping device set up in this paper, the following two types of linear TMDs with different optimization strategies are introduced for comprehensive consideration.The first type of LTMD1 is optimized based on the traditional fixed-point theory. Considering the existence of system damping, it is still necessary to determine the optimal parameters using numerical search method. The optimization objective is to reduce the maximum response amplitude of the main structure under different excitation frequencies to the minimum, which is when the system parameters are the optimal ones. The second type of LTMD2 is based on the above energy index optimization, which makes the wind speed-energy map area minimum, and the optimal parameters are finally obtained. Fig. 6 shows the comparison results of energy response of two-way dense-ribbed cavity floor structure without control and with three different types of TMD devices, namely, LTMD1, LTMD2 and TMD semi-active vibration damping control device in this paper.

Figure 6.

The average energy response of the main structure is compared

From the figure, it can be seen that the uncontrolled structure shows an obvious resonance response near the critical wind speed of 55m/s~67m/s, and its maximum value reaches 1.69×109kJ. And the structures fitted with three different types of damping devices show good resonance response suppression, but their control performance is not the same. the LTMD1, the LTMD2, and the TMD semiactive damping device in this paper The corresponding energy response results are 5.89×10⁷kJ·m/s, 5.26×10⁷kJ·m/s, and 3.92×10⁷kJ·m/s, respectively, from which it can be seen that the energy response values of the two types of linear TMDs are relatively close to each other, while they are all significantly larger than those of the TMD semi-active vibration damping device set in this paper. TMD semi-active damping device optimized values. This result implies that the TMD semi-active damping device set up in this paper can provide better damping control performance than the linear TMD in the given wind speed interval.

When the wind speed is located in the low wind speed interval on [40,64] m/s, it can be clearly observed that the TMD semi-active vibration damping device set up in this paper has the best performance in vibration damping control for the main structure of the bidirectional dense-ribbed cavity building cover, which is much lower than the energy values of the remaining two types of linear TMDs in the whole wind speed interval. Meanwhile, the maximum value of the average energy of the main structure is 0.33 times that of LTMD1 and 0.26 times that of LTMD2. And in the high wind speed interval [64,80] m/s, the TMD semi-active vibration damping device set up in this paper has a similar maximum value with LTMD1, but it is about 14.79% higher than the maximum value of LTMD2. Therefore, the application of TMD device in bidirectional dense-ribbed cavity building cover in this paper can significantly improve the collision damping performance of bidirectional dense-ribbed cavity building cover, and lay the foundation for enhancing the building safety and stability of bidirectional dense-ribbed cavity building cover.