Vertical seismic protection of structures with inerter-based negative stiffness absorbers

Abstract

One of the most effective approaches for seismic protection of structural systems is isolation. The majority of existing seismic protection systems are related to horizontal ground motion and only few focus on vertical seismic components, as this would require vertical flexibility, a feature conflicting with the need for sufficient support of the isolated structure. In order to overcome this difficulty, a novel vertical Stiff dynamic absorber (SDA) is proposed that combines a Quasi-Zero Stiffness design including negative stiffness elements, an enhanced Tuned Mass Damper, and Inerter elements. The present work deals with the optimization problem for the determination of the various design parameters of the proposed configuration, aiming at minimizing the vertical accelerations induced on the structure while keeping the displacement within reasonable limits. The values of the optimized design parameters depend only on the characteristics of the isolated structure irrespectively of the excitation input. This feature differentiates the proposed SDA from similar versions found in the literature and renders the design applicable for use across a wide range of vertical vibration control applications. The proposed configuration is designed to achieve great acceleration isolation without compromising the weight-bearing capacity of the structure. This is realized through the appropriate determination of the static and dynamic stiffnesses via a novel calculation approach that is based on the initial static equilibrium point. The role of various design factors is investigated based on time-history responses of a reference structure to artificial accelerograms matching EC8 criteria, and the effect of the optimized SDA is evaluated on real earthquake records, confirming its capacity to greatly reduce the response to structure accelerations. The selection of the design parameters is based on engineering criteria and an indicative implementation of the design elements is included, demonstrating the feasibility of the proposed configuration.

Appendices

Appendix 1: Analysis of the proposed indicative implementation of the assembly

The potential energy of the assembly in Fig. 2 is:

$$U_{H} = \frac{1}{2}2k_{H} (l_{HI} - l_{H} )^{2}$$

(A.1)

where ${k}_{H}$ is the stiffness of the inclined spring, ${l}_{H}$ its length at position u and ${l}_{HI}$ its initial (uncompressed) length. The spring compression is denoted by V(u):

$$v(u) = l_{HI} - l_{H}$$

(A.2)

Equation (A.1) becomes:

$$U_{H} = \frac{1}{2}2k_{H} v^{2}$$

(A.3)

The negative stiffness force of the assembly results as

$$N = \frac{{\partial U_{H} }}{\partial u} = 2k_{H} v\frac{\partial v}{{\partial u}}$$

(A.4)

and the negative stiffness ${\text{k}}_{\text{N}}$ is:

$$k_{N} = \frac{\partial N}{{\partial u}} = 2k_{H} \left[ {\left( {\frac{\partial v}{{\partial u}}} \right)^{2} + v\left( {\frac{{\partial^{2} v}}{{\partial u^{2} }}} \right)} \right]$$

(A.5)

Moreover, the compression force Q on the spring k_H is:

$$Q(u) = 2k_{H} v$$

(A.6)

In view of Fig. 2 the following equations hold:

$$l_{H} = b - w$$

(A.7a)

$$w = [a^{2} - u^{2} ]^{1/2}$$

(A.7b)

The combination of Eqs. (A.2) and (A.7) leads to:

$$\frac{\partial v}{{\partial u}} = - \frac{{\partial l_{H} }}{\partial u} = \frac{\partial w}{{\partial u}} = - \frac{u}{w}$$

(A.8)

$$\frac{{\partial^{2} v}}{{\partial u^{2} }} = \frac{{\partial^{2} w}}{{\partial u^{2} }} = - \frac{{a^{2} }}{{w^{3} }}$$

(A.9)

As a result, Eqs. (A.4) and (A.5) become:

$$N(u) = 2k_{H} (l_{HI} - l_{H} )( - \frac{u}{w}) = - 2k_{H} \left[ {1 - \frac{{b - l_{H1} }}{a}\frac{a}{w}} \right]u$$

(A.10)

$$k_{N} (u) = 2k_{H} \left[ {\left( \frac{u}{w} \right)^{2} - (l_{HI} - l_{H} )\left( { - \frac{{a^{2} }}{{w^{3} }}} \right)} \right] = 2k_{H} \left[ {1 - \frac{{b - l_{H1} }}{a}\left( \frac{a}{w} \right)^{3} } \right]u$$

(A.11)

Introducing the notations

$$c_{I} = \frac{{b - l_{HI} }}{a}$$

(A.12a)

$$c = w/a = [1 - (u/a)^{2} ]^{1/2}$$

(A.12b)

Equations (A.11) and (A.12) become:

$$N(u) = - 2k_{H} [1 - c_{I} /c]u$$

(A.13)

$$k_{N} (u) = - 2k_{H} [1 - c_{I} /c^{3} ]$$

(A.14)

Appendix 2: SDA under external force excitation

Considering an external force $f(t) = \tilde{F}e^{j\omega t}$ acting on the structure of Fig. 1c (while X_G = 0), the equations of motions of the two masses take the form:

$$(m_{S} + b_{R} )\ddot{x}_{S} + c_{P} (\dot{x}_{S} - \dot{x}_{D} ) + k_{R} x_{S} + k_{P} (x_{S} - x_{D} ) = \tilde{F}e^{j\omega t}$$

(B.1)

$$m_{D} \ddot{x}_{D} + c_{P} (\dot{x}_{D} - \dot{x}_{S} ) + c_{N} \dot{x}_{D} + k_{N} x_{D} + k_{P} (x_{D} - x_{S} ) = 0$$

(B.2)

The steady state responses of the system are

$$x(t) = \left[ {\begin{array}{*{20}c} {\tilde{X}_{S} } \\ {\tilde{X}_{D} } \\ \end{array} } \right]e^{j\omega t}$$

(B.3)

where $\tilde{X}_{S} , \tilde{X}_{D}$, denote the response complex amplitudes. The above-mentioned equations of motion of the system hence become:

$$- \omega^{2} (m_{S} + b_{R} )\tilde{X}_{S} + j\omega c_{P} (\tilde{X}_{S} - \tilde{X}_{D} ) + k_{R} \tilde{X}_{S} + k_{P} (\tilde{X}_{S} - \tilde{X}_{D} ) = \tilde{F}$$

(B.4)

$$- \omega^{2} m_{D} \tilde{X}_{D} + j\omega c_{P} (\tilde{X}_{D} - \tilde{X}_{S} ) + j\omega c_{N} \tilde{X}_{D} + k_{N} \tilde{X}_{D} + k_{P} (\tilde{X}_{D} - \tilde{X}_{S} ) = 0$$

(B.5)

The complex amplitude ${{\tilde{\text{F}}}}$ of the acting force can be conveniently written in terms of the complex amplitude of the displacement ${{\tilde{\text{X}}}}_{{ST}}$ and total stiffness ${k}_{T}$:

$$\tilde{F}(t) = \left[ {\begin{array}{*{20}c} {k_{T} } \\ 0 \\ \end{array} } \right]\tilde{X}_{ST}$$

(B.6)

The resulting transfer functions (TFs) of the displacement are

$$\left[ {\begin{array}{*{20}c} {\tilde{H}_{XS} } \\ {\tilde{H}_{XD} } \\ \end{array} } \right] = \left[ {\frac{{{{\tilde{X}_{S} } \mathord{\left/ {\vphantom {{\tilde{X}_{S} } {X_{ST} }}} \right. \kern-\nulldelimiterspace} {X_{ST} }}}}{{{{\tilde{X}_{D} } \mathord{\left/ {\vphantom {{\tilde{X}_{D} } {X_{ST} }}} \right. \kern-\nulldelimiterspace} {X_{ST} }}}}} \right] = H^{ - 1} \left[ {\begin{array}{*{20}c} {k_{T} } \\ 0 \\ \end{array} } \right]$$

(B.7)

where $\tilde{H}$ has been defined in (31) as

$$\tilde{H} = \left[ {\begin{array}{*{20}l} { - \omega^{2} (m_{S} + b_{R} ) + j\omega c_{P} + (k_{R} + k_{P} )} \hfill & { - (j\omega c_{P} + k_{P} )} \hfill \\ { - (j\omega c_{P} + k_{P} )} \hfill & { - \omega^{2} m_{D} + j\omega (c_{P} + c_{N} ) + (k_{N} + k_{P} )} \hfill \\ \end{array} } \right]$$

(B.8)

The resulting force $f_{B }$ at the base is calculated as:

$$f_{B} = f(t) - m_{S} \ddot{x}_{S} - m_{D} \ddot{x}_{D}$$

(B.9)

Hence the TF of the force at the base is:

$$\tilde{H}_{FB} = \frac{{\tilde{F}_{B} }}{{\tilde{F}}} = \frac{{\tilde{F} + \omega^{2} m_{S} \tilde{X}_{S} + \omega^{2} m_{D} \tilde{X}_{D} }}{{\tilde{F}}}$$

(B.10)

In view of (B.6) the above reads

$$\tilde{H}_{FB} = 1 + \frac{{\omega^{2} }}{{k_{T} }}(m_{S} \tilde{H}_{XS} + m_{D} \tilde{H}_{XD} )$$

(B.11)

where

$$\tilde{H}_{XS} = \frac{{k_{T} \tilde{H}_{22} }}{D}$$

(B.12a)

$$\tilde{H}_{XD} = \frac{{k_{T} \tilde{H}_{12} }}{D}$$

(B.12b)

and $D = \tilde{H}_{11} \tilde{H}_{22} - \tilde{H}_{12}^{2}$ is the determinant of the matrix $\tilde{H}$. It follows that:

$$\tilde{H}_{FB} = 1 + \omega^{2} \frac{{m_{S} \tilde{H}_{22} + m_{D} \tilde{H}_{12} }}{D} = \frac{{\tilde{H}_{22} (\tilde{H}_{11} + \omega^{2} m_{S} ) - \tilde{H}_{12} (\tilde{H}_{12} + \omega^{2} m_{D} )}}{D}$$

(B.13)

Turning to paragraph 2.3 and in view of Eq. (31.a), the transfer function of the structure acceleration a_s for a ground acceleration excitation of amplitude ${\text{A}}_{\text{G}}$ is calculated:

$$\tilde{H}_{AS} = 1 - \omega^{2} \tilde{H}_{US} = 1 - \frac{{\omega^{2} }}{D}(m_{S} \tilde{H}_{22} + m_{D} \tilde{H}_{12} )$$

(B.14)

Rearranging the above

$$\tilde{H}_{AS} = \frac{{\tilde{H}_{22} (\tilde{H}_{11} + \omega^{2} m_{S} ) - \tilde{H}_{12} (\tilde{H}_{12} + \omega^{2} m_{D} )}}{D}$$

(B.15)

In view of the above and (B-13) it follows that:

$$\tilde{H}_{FB} = \tilde{H}_{AS}$$

(B.16)

The transfer function of the response structure acceleration for a base excitation is the same as the transfer function of the response force at the base in the case of a force excitation.

Appendix 3: Selection of ground motion acceleration excitation power spectrum

Expressions (34) and (35) call for a specific ground excitation profile. The relevant excitation power spectrum is selected following the same procedure used in Kapasakalis et al. (2021d). The vertical component of the ground motion is described in EN1998-1 (2004) by an elastic ground acceleration response spectrum. For each seismic zone the vertical acceleration is given by the ratio avg/ag, the recommended values of which are 0.9 for seismic action Type 1 (magnitude Mw > 5.5) and 0.45 for Type 2. The basic shape of the spectrum is similar to the recommended horizontal. However, the spectral amplification in the vertical case is 3.0 instead of 2.5. The vertical acceleration response spectrum with spectral acceleration ag = 0.36 g and Type 1 is can be found in Fig. 1b of Kapasakalis et al. (2021d).

A database of 50 artificial accelerograms was generated matching the aforementioned spectrum using the SeismoArtif Software (Seismosoft 2018). The method used for the generation of each accelerogram was the “Artificial Accelerogram Generation & Adjustment”, an iterative procedure based on the adaptation of a random process (Gasparini and Vanmarcke 1976) to a target spectrum, which in this case is that of EC8 described above. The process can be summarized as follows:

A random process is used for the generation of a large enough set of phase angles for a series of sinusoidal waves, the sum of which comprise a periodic function. To simulate the transient nature of the earthquakes, the steady state motions are multiplied by a relevant envelope shape (or intensity function), forming an initial artificial accelerogram. In each iteration, this is transferred to the frequency domain using the Fourier Transformation Method, and it is corrected to match the target spectrum. Subsequently, the Inverse Fourier Transform is utilized for the appropriate PGA and Baseline corrections in the time domain. The Fourier Transformation Method is again applied on the corrected accelerogram and the same process is repeated until convergence is achieved. An example of such generated accelerogram can be found in Fig. 1c of Kapasakalis et al. (2021d).

The mean acceleration response spectrum of the 50 artificial accelerograms of the database compared to the EC8 design vertical acceleration response spectrum has been shown in Fig. 1b of Kapasakalis et al. (2021d). An accurate match is observed with a percentage deviation under 10% in all the range of the natural periods.

Τhe mean power spectral density S_AM of the 50 accelerograms is calculated (Fig. 

Fig. 11

Power Spectral Density (PSD) of one of the 50 artificial accelerograms (green), mean PSD of the 50 artificial accelerograms, S_AM, (red) and Least Square Fitting (LSF), S_A, of the mean PSD (blue) (Fig. 3 of Kapasakalis et al. 2021d)

11). Finally, the PSD of the ground motion acceleration, S_A, which is used for the calculation of R_AS in Eq. 35.c is calculated as the least-square fitting curve of the mean power spectral density S_AM. S_AM together with the PSD of a random artificial accelerogram are also shown in Fig. 11.

Appendix 4: Effect of key SDA design factors to real earthquakes records

4.1 Effect of objective function

Table 8 below presents the results of the SDA optimized with the two different objective functions, Q_AS and R_AS, defined in Eqs. (33.c) and (35.c) respectively and discussed in paragraph 3.4. Peak response quantities (structure acceleration, a_S, structure relative displacement, u_S, oscillating mass relative displacement, u_D) are shown for the SDA optimized under each method, and for the conventional damping system. The results verify the equivalency of the two methods in terms of the resulting response.

Table 8 Comparison of the peak response quantities (structure acceleration, a_S, structure relative displacement, u_S, oscillating mass relative displacement, u_D) of the stiff dynamic absorber optimized with two different objective functions, Q_AS, R_AS, and corresponding quantities of the conventional damping system (CD) for real earthquakes

4.2 Effect of inerter between structure-base

Table

Table 9 Comparison of the peak response quantities (structure acceleration, a_S, structure relative displacement, u_S, oscillating mass relative displacement, u_D) of the stiff dynamic absorber optimized (i) with an inerter between the structure and the base (SDA), and (ii) without an inerter (SDA-NI), and corresponding quantities of the conventional damping system (CD) for real earthquakes

9 below presents the results of the system optimized (i) including the inerter b_R placed between the structure and the base (SDA as shown in Fig. 1c), and (ii) without the inerter b_R (SDA-NI). Peak response quantities (structure acceleration, a_S, structure relative displacement, u_S, oscillating mass relative displacement, u_D) are shown for the SDA optimized under each method, and for the conventional damping system. The results verify the beneficial impact of the inerter for all response quantities in almost all the earthquake cases.

4.3 Effect of additional internal inerters

Table

Table 10 Comparison of the peak response quantities (structure acceleration, a_S, structure relative displacement, u_S, oscillating mass relative displacement, u_D) of the stiff dynamic absorber optimized (i) with a single inerter between the structure and the base (SDA), and (ii) with two additional inerters placed in parallel to the k_P, k_N spring elements (SDA-II), and corresponding quantities of the conventional damping system (CD) for real earthquakes

10 below presents the results of the system optimized (i) including a single inerter b_R (SDA as shown in Fig. 1c), and (ii) including two additional inerers, b_N, b_P, b_R (SDA-II as shown in Fig. 1d). Peak response quantities (structure acceleration, a_S, structure relative displacement, u_S, oscillating mass relative displacement, u_D) are shown for the SDA optimized under each method, and for the conventional damping system. The results confirm that the additional inerters do not add a significant advantage in the efficiency of the system.

4.4 Effect of static stiffness k _S

Tables 

Table 11 Peak structure acceleration, a_S, for the SDA optimized for different values of static displacement (X_VSD = 1, 2, 3 cm), compared to the conventional damping system (CD) of corresponding static stiffness

11 and

Table 12 Structure peak relative displacement, u_S and oscillating mass peak relative displacement, u_D, for the SDA optimized for different values of static displacement (X_VSD = 1,2,3 cm)

12 below present the results of the SDA system optimized for different values of static displacement (X_VSD = 1, 2, 3 cm), compared to the conventional damping system (CD) of corresponding static stiffness.). Peak response quantities (structure acceleration, a_S, structure relative displacement, u_S, oscillating mass relative displacement, u_D) are shown for each earthquake. Structure acceleration, a_S, and structure relative displacement, u_S, are shown in comparison to those of the conventional damping system of corresponding static stiffness.

Appendix 5: Indicative implementation of key elements of the SDA configuration—negative stiffness element and inerter

5.1 Detailed design of conventional spring and stress analysis

An indicative example regarding the realization of the NS element having constant NS (c_I = 0), with the proposed configuration presented in Fig. 2 and Appendix 1, is presented below. The value (constant) of the NS element is obtained from Table 2 which corresponds to the optimized SDA with an objective function that of Q_AS (k_NS = − 86.45 kN/m). The fixed values for this set of optimized parameters are presented in Table 1.

In order for the generated NS to be constant, the dimensionless parameter c_I (Eq. A.12.a) is set equal to zero. Thus, the length of the undeformed conventional stiffness element k_H (l_HI) is equal to the parameter b. In addition, in the static equilibrium position, when we consider c_I = 0, the rod of length a is placed horizontally. As a consequence, the maximum absolute NS element stroke (u_D) is equal to the rod length a.

In Appendix 4, the dynamic responses of the SDA are presented for all the considered real earthquake records. It is observed that the maximum (absolute) value of the additional mass relative displacement is below 8 cm, for all real and artificial seismic records. Having this value as an upper limit of the u_D (and thus the NS stroke) the rod length is set equal to 8 cm.

The value of the two horizontal conventional stiffness elements (k_H) is obtained from Eq. (A.14) (k_H = 43.225 kN/m). In this example, the k_H is realized as a conventional spiral spring. The design parameters for this spiral spring are presented in Table 

Table 13 Design parameters for the spiral spring that realizes the positive stiffness element k_H

13, and are selected with analytical expressions (Compression Spring Calculator (Load Based Design) 2021) in order for the spring to have a factor of safety against torsional yielding at its solid length greater than 1.3, and a factor of safety against buckling greater than 1.2. The material used is chrome-vanadium (ASTM A232 Alloy Steel Wire).

In addition, a stress analysis for the spiral spring is performed by employing the FE code ABAQUS (2021) in order to evaluate the maximum stresses that occur when the spring is fully deformed (8 cm). The mechanical properties used in order to model the spiral spring are obtained from MatWeb (2021). In Fig. 

Fig. 12

a Spiral spring deflections along the spring length, and b maximum von Mises stresses along the spring length

12, the deflections along the spring length, and the maximum von Mises stresses are presented when the spring with the characteristics presented above, is in its fully deformed state. It is observed that the stresses are retained 25% (safety factor 1.32) below the maximum allowable stress that this material allows.

5.2 Realization of inerter element

The inerter can be realized by an easy to construct mechanical device proposed in Smith (2002), which satisfies certain practical conditions. More specifically, a plunger sliding is inserted in a cylinder which drives a flywheel through a rack, pinion, and gears (see Fig. 3 of Smith 2002). This configuration does not have the limitation that one of the two terminals of the inerter needs to be grounded. To model the dynamics of this device, let r₁ be the radius of the rack pinion, r₂ the radius of the gear wheel, r₃ the radius of the flywheel pinion, γ the radius of gyration of the flywheel, m the mass of the flywheel, and assume the mass of all other components is negligible. The equivalent inertance generated by this mechanical device is:

$$b = \frac{1}{2}m_{2} a_{1}^{2} a_{2}^{2}$$

(E.1)

where m₂ is the mass of the flywheel, α₁ = γ/r₃ and α₂ = r₂/r₁. The required inertance ratio for the SDA device is μ_b = 43%, thus for a reference structure mass of 1000 kg, the generated inertance is 430 kg. For α₁ = α₁ = 5, and a radius of the flywheel R₂ = 7.5 cm with a thickness of 1 cm, the mass of the flywheel (steel) is m₂ = 1.376 kg, leading to a generated inertance of 430 kg.