JP7312343B2 - How to set the specifications of the damping device - Google Patents

Description

The present invention relates to a method for setting specifications of a vibration damping device.

Patent Literature 1 discloses a technique related to a vibration reduction mechanism for reducing vibration of a multi-layered structure such as a high-rise building and a specification setting method thereof. In this prior art, a rotary inertia mass damper that operates by interlayer deformation to generate a rotary inertia mass by rotation of a weight is installed in an arbitrary layer of a multilayer structure, and an additional spring is installed in series with the rotary inertia mass damper. Then, the natural frequency determined by the rotational inertia mass and the additional spring is tuned to the natural frequency of the multi-layered structure or the specific frequency at which resonance becomes a problem.

Other related techniques are described in Patent Documents 2 to 4.

Japanese Patent No. 4968682 JP 2008-133947 A JP 2009-180346 A Japanese Unexamined Patent Application Publication No. 2010-070908

The tuning of the natural period of the vibration reduction mechanism in Patent Document 1 is to match the natural period of the vibration reduction mechanism with the natural period of the structure, and is not tuning based on the fixed point theory. The applicant asserted in the trial for invalidation of Patent Document 1 that "synchronization of natural periods" is "matching of natural periods."

In view of the above facts, it is an object of the present invention to provide a method for setting the specifications of a vibration damping device so that one vibration mode or any one of a plurality of vibration modes is optimally tuned or optimally damped in a structure having one or more vibration modes.

A method for setting specifications of a vibration damping device according to a first aspect is a method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertia mass element, a damping element, and a rigidity element, wherein any one of the one vibration mode or a plurality of the vibration modes is optimally tuned and optimally damped by a new vibration mode generated by the vibration damping device coupled with the structure, an optimal tuning formula, an optimal damping formula, an additional stiffness ratio or an additional weight ratio, and an optimum. A viscous damping constant is used to set the specifications of the inertial mass element, the damping element, and the rigid element that constitute the vibration damping device.

A method for setting specifications of a vibration damping device according to a second aspect is a method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element, and a stiffness element, wherein the vibration damping device is adjusted using an optimum tuning formula and one of an additional stiffness ratio and an additional weight ratio so that any one of the one vibration mode or a plurality of the vibration modes is optimally tuned by a new vibration mode generated by the vibration damping device interacting with the structure. The specifications of the inertia mass element and the rigid element to be constructed are set.

A method for setting specifications of a vibration damping device according to a third aspect is a method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element, and a stiffness element, wherein the one vibration mode or any one of the plurality of vibration modes is optimally damped by a new vibration mode generated by the vibration damping device interacting with the structure, using an optimum damping formula, one of an added stiffness ratio or an added weight ratio, and an optimum viscous damping constant, The specifications of the damping element that constitutes the vibration damping device are set.

A method for setting the specifications of a vibration damping device according to a fourth aspect includes the steps of setting a target performance for the magnitude of shaking of the structure during an earthquake, and setting the installation positions and the number of installations of the vibration damping devices so as to satisfy the target performance.

In the method for setting the specifications of the vibration damping device of the fifth aspect, the vibration damping device has the inertia mass element and the damping element arranged in parallel in the numerical analysis model, and the rigid element is arranged in series with the inertia mass element and the damping element arranged in parallel.

In the method of setting the specifications of the vibration damping device of the sixth aspect, the vibration damping device has the inertial mass element, the damping element, and the rigid element arranged in series in the numerical analysis model. The order of arrangement of the inertial mass element, damping element and rigid element is not limited.

In the method for setting the specifications of the vibration damping device of the seventh aspect, in the numerical analysis model of the vibration damping device, the damping element and the rigid element are arranged in parallel, and the inertia mass element is arranged in series with the damping element and the rigid element arranged in parallel.

In the method for setting the specifications of the vibration damping device of the eighth aspect, the vibration damping device has a numerical analysis model in which the damping element has a first damping element and a second damping element, and the first element in which the inertia mass element and the first damping element are arranged in parallel, and the second element in which the rigid element and the second damping element are arranged in parallel are arranged in series. Note that the order of arrangement of the inertial mass element and the rigid element is not limited.

According to the present invention, in a structure having one or more vibration modes, the specifications of the vibration damping device can be set so that one vibration mode or any one of a plurality of vibration modes is optimally tuned or optimally damped.

The first D. M. Numerical analysis model of tuning system. The second D. M. Numerical analysis model of tuning system. The third D. M. Numerical analysis model of tuning system. Fourth D. M. Numerical analysis model of tuning system. The first D. M. This is a multi-mass system numerical analysis model of a multi-story building with a tuning system installed. The second D. M. This is a multi-mass system numerical analysis model of a multi-story building with a tuning system installed. The third D. M. This is a multi-mass system numerical analysis model of a multi-story building with a tuning system installed. Fourth D. M. This is a multi-mass system numerical analysis model of a multi-story building with a tuning system installed. In each numerical analysis model shown in FIGS. 5 to 8, the controlled object mode and D. M. FIG. 10 is a table summarizing the optimum tuning formula, additional stiffness ratio, additional weight ratio and optimum damping formula for matching or substantially matching the viscous damping constant with the mode, (A) being the first D.D. M. A tuning system and a second D.I. M. Table of tuning systems, (B) a third D.C. M. A tuning system and a fourth D.I. M. 2 is a table of tuning systems; The first D. M. Numerical model of a one-story building with a tuning system installed. (A) is a graph showing the relationship between the natural period and the damping coefficient at the time of optimum tuning, and (B) is a graph showing the relationship between the viscous damping constant and the damping coefficient at the time of optimum tuning. (A) is a graph showing a resonance curve of absolute acceleration response magnification, and (B) is a graph showing a resonance curve of relative displacement response magnification. 4 is a flow chart showing an example of the procedure of a specification setting method; (A) is a table summarizing the analysis conditions of the bilayer shear model, and (B) is the eigenvalue results. (A) is the first D. M. 14 is a table showing the setting of the stiffness element for first-order mode control in step 53 of the flowchart of FIG. 13 when using the tuning system, and (B) is the eigenvalue result at ₁ T _∞ obtained in step 54 of the flowchart of FIG. 13. (A) is the first D. M. D.1 for first order mode control in step 57 of the flow chart of FIG. 13 when using a tuning system. M. 14 is a table showing the setting of elements, and (B) is the eigenvalue result that satisfies the optimum tuning equation in step 57 of the flow chart of FIG. 13; (A) is the first D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the second D. M. 14 is a table showing the setting of stiffness elements for first-order mode control in step 53 of the flowchart of FIG. 13 when using the tuning system, and (B) is the eigenvalue result at ₁ T _mc∞ obtained in step 54 of the flowchart of FIG. 13. FIG. (A) is the second D. M. D.1 for first order mode control in step 57 of the flow chart of FIG. 13 when using a tuning system. M. 14 is a table showing the setting of elements, and (B) is the eigenvalue result that satisfies the optimum tuning equation in step 57 of the flow chart of FIG. 13; (A) is the second D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the third D. M. In step 53 of the flow chart of FIG. M. 14 is a table showing the setting of elements, and (B) is the eigenvalue result at ₁ T _∞ obtained in step 54 of the flowchart of FIG. 13 . (A) is the third D. M. FIG. 14B is a table showing the setting of stiffness elements for first-order mode control in step 57 of the flowchart of FIG. 13 when using a tuning system, and (B) shows the eigenvalue results that satisfy the optimal tuning equation in step 57 of the flowchart of FIG. (A) is the third D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the fourth D. M. 14 is a table showing the setting of the stiffness element for first-order mode control in step 53 of the flowchart of FIG. 13 when using the tuning system, and (B) is the eigenvalue result at ₁ T _mc∞ obtained in step 54 of the flowchart of FIG. 13. (A) is the fourth D. M. D.1 for first order mode control in step 57 of the flow chart of FIG. 13 when using a tuning system. M. 14 is a table showing the setting of elements, and (B) is the eigenvalue result that satisfies the optimum tuning equation in step 57 of the flow chart of FIG. 13; (A) is the fourth D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the first D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the second D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the third D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the fourth D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the first D. M. FIG. 10 is a table showing the settings of each element for second-order mode control of the tuning system, (B) the eigenvalue results; (A) is the second D. M. FIG. 10 is a table showing the settings of each element for second-order mode control of the tuning system, (B) the eigenvalue results; (A) is the third D. M. FIG. 10 is a table showing the settings of each element for second-order mode control of the tuning system, (B) the eigenvalue results; (A) is the fourth D. M. FIG. 10 is a table showing the settings of each element for second-order mode control of the tuning system, (B) the eigenvalue results; The horizontal axis is the resonance curve at the time of primary mode control with an equal interval scale. The horizontal axis is the resonance curve under first-order mode control on a logarithmic scale. The horizontal axis is the resonance curve at the time of second-order mode control with an equal interval scale. The horizontal axis is the resonance curve under second-order mode control in a logarithmic scale. (A) is a table summarizing the analysis conditions of the five-layer shear model, and (B) is the eigenvalue results. (A) is the first D. M. FIG. 11 is a table showing the settings of each element for first and second order mode control of the tuning system, (B) Eigenvalue results; (A) is the second D. M. FIG. 11 is a table showing the settings of each element for first and second order mode control of the tuning system, (B) Eigenvalue results; (A) is the third D. M. FIG. 11 is a table showing the settings of each element for first and second order mode control of the tuning system, (B) Eigenvalue results; (A) is the fourth D. M. FIG. 11 is a table showing the settings of each element for first and second order mode control of the tuning system, (B) Eigenvalue results; The horizontal axis is the resonance curve at the time of primary mode and secondary mode control with an equal interval scale. The horizontal axis is the resonance curve in the primary mode and secondary mode control on a logarithmic scale. (A) is a table summarizing the analysis conditions of the eight-layer shear model, and (B) is the eigenvalue results. (A) is the first D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the second D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the third D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the fourth D. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. The horizontal axis is the resonance curve at the time of primary mode control with an equal interval scale. The horizontal axis is the resonance curve under first-order mode control on a logarithmic scale. (A) is a graph of maximum response acceleration acting on each layer, and (B) is a graph of maximum response displacement. The third D. M. It is a numerical analysis model considering the mounting rigidity element in the tuning system 3 . Fourth D. M. It is a numerical analysis model in which the tuning system 4 considers the mounting rigidity element. (A) is the third D.D. in consideration of mounting rigidity. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. (A) is the fourth D.D. in consideration of mounting rigidity. M. FIG. 4 is a table showing the settings of each element for first-order mode control of the tuning system, and (B) is the eigenvalue result. The horizontal axis is the resonance curve at the time of primary mode control with an equal interval scale. The horizontal axis is the resonance curve under first-order mode control on a logarithmic scale. (A) is a graph of maximum response acceleration acting on each layer, and (B) is a graph of maximum response displacement. It is the response spectrum of the input seismic motion. D. M. 1 is a model diagram of a building in which a tuning system is installed using a shear link mechanism; FIG. The horizontal axis is the resonance curve at the time of primary mode control with an equal interval scale. The first D. M. tuning system, the second D.I. M. tuning system, the third D.C. M. A tuning system and a fourth D.I. M. 4 is a table showing specifications and basic performance of a tuning system and an oil damper as a comparative example; 3 is a graph showing (A) the maximum response displacement, (B) the maximum response acceleration, (C) the maximum response story shear force, (D) the maximum response story shear force coefficient, (E) the maximum response story drift angle, and (F) the maximum response overturning moment. 3 is a graph showing (A) the reduction rate of the maximum response displacement, (B) the reduction rate of the maximum response acceleration, (C) the reduction rate of the maximum response story shear force, (D) the reduction rate of the maximum response story shear force coefficient, (E) the reduction rate of the maximum response story drift angle, and (F) the reduction rate of the maximum response overturning moment. It is the response spectrum of the input seismic motion. D. M. 1 is a model diagram of a building in which a tuning system is installed using a toggle mechanism; FIG. The horizontal axis is the resonance curve at the time of primary mode control with an equal interval scale. The first D. M. tuning system, the second D.I. M. tuning system, the third D.C. M. A tuning system and a fourth D.I. M. 4 is a table showing specifications and basic performance of a tuning system and an oil damper as a comparative example; 3 is a graph showing (A) the maximum response displacement, (B) the maximum response acceleration, (C) the maximum response story shear force, (D) the maximum response story shear force coefficient, (E) the maximum response story drift angle, and (F) the maximum response overturning moment. 3 is a graph showing (A) the reduction rate of the maximum response displacement, (B) the reduction rate of the maximum response acceleration, (C) the reduction rate of the maximum response story shear force, (D) the reduction rate of the maximum response story shear force coefficient, (E) the reduction rate of the maximum response story drift angle, and (F) the reduction rate of the maximum response overturning moment. D. M. 1 is a model diagram of a multi-story building with a tuning system installed on the first floor; FIG. In each numerical analysis model shown in FIGS. 5 to 8, the controlled object mode and D. M. FIG. 10 is a table summarizing the optimum tuning formula, the added stiffness ratio, the added weight ratio and the optimum damping formula for matching or substantially matching the response magnification of the fixed point of the relative displacement response magnification curve with the mode, (A) is the first D.D. M. A tuning system and a second D.I. M. Table of tuning systems, (B) a third D.C. M. A tuning system and a fourth D.I. M. 2 is a table of tuning systems; In each numerical analysis model shown in FIGS. 5 to 8, the controlled object mode and D. M. FIG. 10 is a table summarizing the optimum tuning formula, the added stiffness ratio, the added weight ratio and the optimum damping formula for matching or substantially matching the response magnification of the fixed point of the absolute acceleration response magnification curve with the mode; M. A tuning system and a second D.I. M. Table of tuning systems, (B) a third D.C. M. A tuning system and a fourth D.I. M. 2 is a table of tuning systems;

<Embodiment>
Tuning system 1, second D.M. tuning system 2, third D.M. tuning system 3, and fourth D.M. In addition, when it is not necessary to distinguish between them or when they are collectively referred to, they are simply described as "DM tuning system". Also, the inertial mass element will be described and explained using the abbreviation "DM element".

(First D.M. tuning system 1)
As in the numerical analysis model shown in FIG. M. Tuning system 1 is D.I. M. An element 10 and a damping element 20 are arranged in parallel, and a rigid element 30 is arranged in series with them.

(Second DM tuning system 2)
As in the numerical analysis model shown in FIG. M. Tuning system 2 is D.I. M. Element 10, damping element 20 and stiffening element 30 are arranged in series.

(Third D.M. tuning system 3)
As in the numerical analysis model shown in FIG. M. The tuning system 3 comprises a damping element 20 and a stiffening element 30 arranged in parallel, and D. M. Elements 10 are arranged in series with these.

(Fourth DM Tuning System 4)
As in the numerical analysis model shown in FIG. M. Tuning system 4 is D.I. M. A first element 11 in which an element 10 and a first damping element 22 are arranged in parallel, and a second element 12 in which a rigid element 30 and a second damping element 24 are arranged in parallel are arranged in series.

(D.M. element, damping element, first damping element, second damping element and rigid element)
Next, D. M. Element 10, damping element 20, first damping element 22, second damping element 24 and rigid element 30 will be described.

D. M. Element 10 is an inertial mass damper. The inertial mass damper is a damper having a reaction force (inertia force) proportional to relative acceleration, such as a rotary inertial mass damper or a liquid flow damper having an inertial mass effect. In addition, the inertial mass damper may have a limiter mechanism that stops the increase of the inertial force. Further, the mechanism for rotating the rotating body of the rotary inertia mass damper described above may be any mechanism. For example, it may be a mechanism using a hydraulically driven gear motor or the like.

The damping element 20, the first damping element 22 and the second damping element 24 are damping dampers having a damping force such as viscous dampers and viscoelastic dampers. Also, the damping damper may have a limiter mechanism that stops increasing the damping force when the acting force exceeds a threshold value.

Some of the rotary inertia mass dampers described above have damping elements integrated in parallel. Therefore, D. M. A first D.D. element 10 and a damping element 20 arranged in parallel. M. tuning system 1 and D. M. A fourth D.D. element 10 and a first damping element 22 arranged in parallel. M. The tuning system 4 may use a rotating inertial mass damper with parallel integrated damping elements.

The rigid element 30 is a spring such as a coil spring, a rigid part of a viscoelastic damper, a mounting member (such as a gusset plate, a brace, and an arm of a shear link mechanism 210 (FIG. 62) and a toggle mechanism 310 (FIG. 68) to be described later) for mounting various dampers such as the inertial mass damper, viscous damper, and viscoelastic damper. From another point of view, the rigid element is a rigid member with a restoring force.

Therefore, from another point of view, D.I. M. A tuned system is a combination of an inertial mass damper, a damping damper and a rigid member in the configuration of the numerical analysis model described above.

Also, D.I. M. The dimensions of the element 10 are D.D. M. It is the size of the inertial mass (DM) generated in the element 10 (inertial mass damper), and its unit is (ton), for example. The dimensions of the damping element 20, the first damping element 22, and the second damping element 24 are the magnitudes of damping coefficients generated in the damping element 20, the first damping element 22, and the second damping element 24 (damping dampers) during vibration, and the unit is, for example, (kN s/m). Further, the specification of the rigid element 30 is the magnitude of rigidity generated in the rigid element 30 (rigid member) during vibration, and its unit is, for example, (kN/m).

[Example of installation of structures]
Next, the first D.I. M. Tuning system 1 (see FIG. 1), the second D.C. M. Tuning system 2 (see FIG. 2), the third D.C. M. A tuning system 3 (see FIG. 3) and a fourth D.I. M. An installation example in which the tuning system 4 (see FIG. 4) is installed in a structure having a plurality of vibration modes will be described.

A building 100 as an example of a structure shown in FIG. 73 is composed of a plurality of layers (three layers in the illustrated example) and has a plurality of vibration modes. At least one floor of this building 100 has a first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. A tuning system 4 is installed. In addition, in FIG. 73, D.I. M. Although the tuning system is installed, it may be installed in another layer, may be installed in a plurality of layers, or may be installed across multiple layers. It should be noted that D.I. M. The structural form in which the tuning system is installed may consider a rigid element (e.g., a mounting rigid element), as in the case of installing a damping device (such as a shear link mechanism 210 (FIG. 62) and a toggle mechanism 310 (FIG. 68), which will be described later).

In addition, the first D.C. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. The floor where the tuning system 4 is installed and the number (amount) of installation are adjusted according to the target performance of the magnitude of the shaking of the building 100, as will be described later.

[How to set specifications]
Next, the first D.I. M. Tuning system 1 (see FIG. 1), the second D.C. M. Tuning system 2 (see FIG. 2), the third D.C. M. A tuning system 3 (see FIG. 3) and a fourth D.I. M. The D.C. forming the tuning system 4 (see FIG. 4). M. A method for setting the specifications of the element 10, the damping element 20, the first damping element 22, the second damping element 24, and the rigid element 30 so as to optimally tune or optimally damp any vibration mode will be described.

Here, the building 100 has a plurality of vibration modes (primary mode, secondary mode, . . . j-th order mode . . . ). An arbitrary j-order mode of the building 100 is a vibration mode to be controlled (hereinafter referred to as a "controlled mode").

The first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. The tuning system 4 couples with the building 100 to generate a new vibration mode (hereinafter referred to as "DM mode"). Then, this "DM mode" can be optimally tuned to the controlled object mode (arbitrary vibration mode) of the building 100 . Alternatively, the controlled object mode or D. By optimally damping the M.mode, the vibration of the control object mode can be effectively reduced.

The definition of the optimum tuning of the design formula, which is the relational expression of the natural period shown in FIG. M. It is to make the viscous damping constant of "DM mode", which is a new vibration mode generated when the tuning system is coupled with the building 100, equal or substantially equal.

Also, the definition of the optimum damping in the design formula, which is the relational expression of the natural period shown in FIG. By giving the M. mode the optimum viscous damping constant, the vibration of the control object mode is effectively reduced. A description of the "optimum viscous damping constant" will be provided later.

Note that the damping force proportional to the speed is the "viscous damping". Also, the ratio of viscous damping to critical damping, which is the boundary between vibration and non-vibration, is the "viscous damping constant." A viscous damping constant of 1.0 is a state of critical damping.

Here, design formulas other than those shown in FIG. 9 are established by changing the definition of optimum tuning.

In the design formula, which is the relational expression of the natural period shown in FIG. M. It is to match or substantially match the response magnification of a fixed point in the relative displacement response magnification curve of the "DM mode", which is a new vibration mode generated when the tuning system is coupled with the building 100 .

In the design formula, which is the relational expression of the natural period shown in FIG. M. It is to match or substantially match the response magnification of a fixed point in the absolute acceleration response magnification curve of the "DM mode", which is a new vibration mode generated when the tuning system is coupled with the building 100 .

9, 74, and 75 are all relational expressions of the natural period, which are mathematical formulas created by the present inventors for setting these specifications, which have never been used before. Similarly, the method of setting each specification using these design formulas is also newly developed by the present inventors, and is an unprecedented method of setting each specification.

First, the case of using the design formula shown in FIG. 9 will be described, and the case of using the design formulas of FIGS. 74 and 75 will be described later.

<For Fig. 9>
A case using the design formula shown in FIG. 9 will be described.

In this specification setting method, the controlled object mode of the building 100 is the first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. The D.M. mode of the tuning system 4 uses the optimum tuning formula, the optimum damping formula, one of the added stiffness ratio or the added weight ratio, and the optimum viscous damping constant so as to achieve optimum tuning or optimum damping. M. The specifications of the element 10, the damping element 20, the first damping element 22, the second damping element 24 and the rigid element 30 are set.

In other words, the controlled object mode (arbitrary vibration mode) of the building 100 and the D. M. The viscous damping constant of the D.M. M. The specifications of the element 10, the damping element 20, the first damping element 22, the second damping element 24 and the rigid element 30 are set.

It should be noted that, as described above, the D. M. The dimension of the element 10 is the size of the inertial mass (DM) generated during vibration, and its unit is, for example, (ton). The dimensions of the damping element 20, the first damping element 22, and the second damping element 24 set here are the magnitudes of the damping coefficients generated during vibration, and the units are, for example, (kN·s/m). Further, the specification of the rigid element 30 set here is the magnitude of rigidity generated during vibration, and the unit thereof is, for example, (kN/m).

Details of the optimum tuning formula, optimum damping formula, added stiffness ratio, added weight ratio, optimum viscous damping constant and setting method will be described later.

[Details of how to set specifications]
FIG. 5 shows the first D.C. M. It is a multi-mass system numerical analysis model of a multilayer building 100 (see FIG. 73) in which the tuning system 1 (see FIG. 1) is installed. FIG. 6 shows the second D.C. M. It is a multi-mass system numerical analysis model of a multi-story building 100 in which the tuning system 2 (see FIG. 2) is installed. FIG. 7 shows the third D.C. M. It is a multi-mass point system numerical analysis model of a multi-story building 100 in which the tuning system 3 (see FIG. 3) is installed. FIG. 8 shows the fourth D.C. M. It is a multi-mass system numerical analysis model of a multi-story building 100 in which the tuning system 4 (see FIG. 4) is installed.

5 to 8, m _n is the mass of any n layers in the building 100, and k _n is the stiffness of any n layers. Also, _j m _d is D. M. element (D.M. element for j-th order mode control), _jkd is a stiffness element (stiffness element for j-th order mode _control ) when controlling the j-th order mode, and _{jcd is} a damping element (damping element for j-th mode control) when controlling the jth mode.

In addition, in the numerical analysis model diagrams of FIGS. 5 to 8, the first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. Although any one of the tuning systems 4 is installed, it is not limited to this. It can be installed across other layers, multiple layers, and multiple layers, and the numerical analysis model can be appropriately changed according to the installation position.

FIG. 9 is a table summarizing the optimum tuning formula, added stiffness ratio, added weight ratio, and optimum damping formula, which are design formulas for optimum tuning or optimum damping in each of the numerical analysis models shown in FIGS. These optimum tuning formula, added stiffness ratio, added weight ratio and optimum damping formula are mathematical formulas created by the present inventors for setting these specifications, and are unprecedented formulas. In addition, the process of deriving the optimum tuning formula, added stiffness ratio, added weight ratio, and optimum damping formula, and using these formulas, D.D. M. The reason why the specifications of the tuning system can be set will be described later.

It should be noted that, in the setting of these specifications, the first D.1. M. Tuning system 1 (see FIG. 1), the second D.C. M. Tuning system 2 (see FIG. 2), the third D.C. M. A tuning system 3 (see FIG. 3) and a fourth D.I. M. “Optimum tuning” is to set the specifications of the tuning system 4 (see FIG. 4) and make the controlled vibration mode (jth mode) of the building 100 (see FIGS. 73 and 5 to 8) match or substantially match the viscous damping constant of the “DM mode.”

Also, in the setting of these specifications, the first D.I. M. Tuning system 1 (see FIG. 1), the second D.C. M. Tuning system 2 (see FIG. 2), the third D.C. M. A tuning system 3 (see FIG. 3) and a fourth D.I. M. The specifications of the tuning system 4 (see FIG. 4) are set, and the controlled object mode or D. "Optimum damping" is to effectively reduce the vibration of the controlled mode by giving the M. mode the optimum viscous damping constant.

Further, since the "optimal tuning formula" and the "optimal damping formula" are relational expressions of the natural period, the first D.3. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. This holds true regardless of the position where the tuning system 4 is installed (placement position (horizontal position) in each layer and installation layer (vertical position).

Here, the second D.I. M. Tuning system 2 and the fourth D.I. M. In the tuning system 4, either the added stiffness ratio or the added weight ratio is determined, and the optimum damping formula is calculated using the determined added stiffness ratio or added weight ratio. When setting the specifications of the rigid element 30 first, the additional stiffness ratio is used. M. When setting the specifications of the element 10, the additional weight ratio is used.

(Optimal viscous damping constant)
Next, the optimum viscous damping constant calculated by the optimum damping formula in the table of FIG. A building 101 of a one-mass system model using the M. tuning system 1 (see FIG. 1) will be described as an example of a structure. Specifically, complex eigenvalue analysis is used to obtain changes in the “natural period” and the “viscous damping constant” due to the damping coefficient _cd at the time of optimum tuning, and the results are used to explain the “optimal viscous damping constant,” which is the optimum viscous damping constant.

FIG. 11A is a graph showing the relationship between the natural period and the damping coefficient at optimum tuning, and FIG. 11B is a graph showing the relationship between the viscous damping constant and the damping coefficient at optimum tuning.

Note that T ₁ is the natural period of the primary mode of the building 101 (see FIG. 10), and T ₁ ,D. M. is D. M. Eigenperiod of the first mode of the tuned system 1 (see FIG. 1). h ₁ is the first mode viscous damping constant of the building 101 (see FIG. 10), and h ₁ ,D. M. is D. M. Viscous damping constant for the first mode of Tuned System 1 (see FIG. 1).

Also, h ₁ = _0.5√1 κ _k /1 is when the damping adjustment coefficient α of the “optimum damping formula” (see FIG. 9) is 1 (α=1), h ₁ = _0.5√1 κ _k /2 is when the damping adjustment coefficient α of the “optimum damping formula” is 2 (α=2), and h ₁ = _0.5√1 κ _k /3 is when the damping adjustment coefficient α of the “optimum damping formula” is 1 (α =3).

From the graph of FIG. 11(A), when the damping coefficient _cd is 0, the natural period of _T1 and _{T1,D. M.} It can be seen that the natural period of Furthermore, when the damping coefficient c _d or more satisfies h ₁ = _0.5√1 κ _k /1 where α=1, it can be seen that the natural periods of both coincide.

From the graph of FIG. 11(B), when the damping coefficient _cd is 0, the viscous damping constant of _h1 and _{h1,D. M.} The viscous damping constant of is zero. When h ₁ =0.5 √ ₁ κ _k /1 where α = 1 is satisfied, the viscous damping constants of both coincide and become large below the damping coefficient _cd , and when the damping coefficient _cd is exceeded, the viscous damping constants of both diverge.

The aforementioned h ₁ = _0.5√1 κ _k /1 is obtained by obtaining h ₁ and h _{1,D . M.} become equal, and the viscous damping constant _h1 peaks (maximum value) when the damping adjustment coefficient α is 1 (α=1). That is, if α is 1 or more, the viscous damping constant of the "controlled object mode (arbitrary vibration mode)" and the viscous damping constant of the "DM mode" match or substantially match.

FIG. 12A is a graph showing a resonance curve of absolute acceleration response magnification, and FIG. 12B is a graph showing a resonance curve of relative displacement response magnification. A fixed point P and _a fixed point _Q in FIG.

From these graphs, if the damping adjustment coefficient α of the optimum damping formula (see FIG. 9) is set to 1 (α=1), both the absolute acceleration response magnification and the relative displacement response magnification have peaks (maximum values) between fixed points P and Q.

If the damping adjustment coefficient α of the optimum damping formula is assumed to be 3 (α=3), there are peaks in the absolute acceleration response magnification and the relative displacement response magnification at the natural period below the fixed point P and at the natural period above the fixed point Q, respectively.

If the damping adjustment coefficient α in the optimum damping formula (see FIG. 9) is set to 2 (α=2), the area between fixed points P and Q is substantially smooth and there is substantially no peak.

Therefore, since disturbances such as wind and earthquakes are not always steady vibrations, the optimum damping formula (see FIG. 9) that can reduce vibrations in the best balance has a damping adjustment coefficient α of 2 (α=2). Therefore, in the present embodiment, the viscous damping constant calculated by setting the damping adjustment coefficient α of the optimum damping formula to 2 (α=2) is referred to as the “optimal viscous damping constant”.

In other words, in steady vibration, matching or substantially matching the viscous damping constants of the "controlled mode (arbitrary vibration mode)" and the "D.M. Further, in steady-state vibration, effectively reducing vibration in the controlled object mode by giving the optimum viscous damping constant to the “controlled object mode or D.M.

[Flowchart of how to set specifications]
FIG. 13 is a flow chart showing an example of the procedure of the specification setting method. This setting method is an example of setting the specifications so as to satisfy both the optimum tuning and the optimum attenuation. In addition, it is a case where the optimum viscous damping constant is given to the controlled object mode, and the vibration of the controlled object mode is optimally damped.

First, in step 51, the target performance of the magnitude of shaking tolerable by the building 100 (FIG. 73) at the time of disturbance such as an earthquake, for example, the maximum response inter-story drift angle and the maximum response absolute response acceleration is set.

It should be noted that this target performance is a comprehensive judgment based on laws, building restrictions, and the like. For example, in time history response analysis that requires ministerial approval, it is common to set the story drift angle to 1/100 or less.

Next, in step 52, the control order (jth mode), that is, the above-described controlled object mode is set. Note that there may be a plurality of controlled object modes. For example, two modes, primary and secondary, may be controlled.

Next, in step 53, the D.I. M. Consider tuning systems.

Specifically, the first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. A tuning system 4 is determined. Also, the determined D. M. Set the layer, position and cardinality to install the tuning system. In addition, these are examined and set according to the structure and conditions of the building 100 . For example, D.I. M. If the conditions (space, etc.) for installing the tuning system are good, they will be installed intensively on the first layer (lowest layer). Alternatively, if the installation conditions of the first layer and the second layer are good, both the first layer and the second layer have D.I. M. Install a tuning system.

Also, the first D.I. M. For the tuning system 1, the specifications of the rigid element 30 are set. The second D. M. For tuning system 2, D. M. Set the specifications of the element 10 or rigid element 30 . The third D. M. For tuning system 3, D. M. Set the specifications of the element 10. Fourth D. M. For tuning system 4, D. M. Set the specifications of the element 10 or rigid element 30 .

Next, in step 54, complex eigenvalue analysis is performed to obtain the natural period of the j-th mode.

Specifically, the first D.I. M. For tuned system 1, determine _j T ₀ and _j T _∞ . The second D. M. For tuned system 2, determine _j T ₀ and _j T _mc∞ or _j T _ck∞ . The third D. M. For tuned system 3, determine _j T ₀ and _j T _∞ . Fourth D. M. For tuned system 4, determine _j T ₀ and _j T _mc∞ or _j T _ck∞ .

Next, in step 55, the added stiffness ratio _j κ _k (see FIG. 9) or the added weight ratio _j κ _m (see FIG. 9) of the j-th mode is calculated.

Specifically, the first D.I. M. For tuning system 1, the added stiffness ratio _j κ _k is determined. The second D. M. For tuning system 2, the added stiffness ratio _j κ _k or the added weight ratio _j κ _m is determined. The third D. M. For tuning system 3, the added weight ratio _j κ _k is determined. Fourth D. M. For the tuning system 4, the added stiffness ratio _j κ _k or the added weight ratio _j κ _m is determined.

Next, in step 56, the optimum viscous damping constant _hj of the j-th mode is calculated using the optimum damping formula (see FIG. 9). In this embodiment, 2 (α=2) is recommended for the damping adjustment coefficient α of the optimum damping formula. However, the damping adjustment coefficient α only needs to be 1 or more, and the "viscous damping constant of the controlled object mode (arbitrary vibration mode)" and the "viscous damping constant of the D.M. mode" match or substantially match.

Next, in step 57, the optimum tuning formula (see FIG. 9) of the j-th mode is satisfied by the result (eigenperiod) of the complex eigenvalue analysis.

Specifically, the first D.I. M. For tuning system 1, D. M. Determine the element 10 ( _j m _d ). The second D. M. For tuning system 2, D. M. Determine element 10 ( _j m _d ) or rigid element 30 ( _j k _d ). The third D. M. For tuned system 3, determine the stiffness element 30 ( _j k _d) . Fourth D. M. For tuning system 4, D. M. Determine element 10 ( _j m _d ) or rigid element 30 ( _j k _d ).

Next, in step 58, the optimal viscous damping constant _hj of the j-th mode is satisfied by the result of the complex eigenvalue analysis (viscous damping coefficient).

Specifically, the first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2 and a third D.I. M. For the tuned system 3, determine the damping element 20 ( _{j cd} ). Fourth D. M. For the tuned system 4, determine the first damping element 22 ( _{j cmd} ) and the second damping element 24 ( _{j ckd} ).

Next, in step 59, it is confirmed whether or not the target performance set in step 51 is satisfied by time history response analysis.

If the target performance is satisfied, the process ends.

If the target performance is not satisfied, the process returns to step 53 , step 52 or step 51 .

When returning to step 53, the D.I. M. Reconsider the entrainment system. Specifically, the D.I. M. Change the tuning system, or change the installed layer, position and cardinal number.

When returning to step 52, the degree of the controlled object mode is reset or the degree of the controlled object mode is increased (if there was only the first order, it is changed to two, the first order and the second order, etc.).

When returning to step 51, the target performance is reset. This resetting of the target performance may require a review of the overall structure of the building 100 in order to increase the maximum response story drift angle, the maximum response absolute acceleration, and the like.

[Details of optimum tuning formula, additional stiffness ratio, additional weight ratio and optimum damping formula in FIG. 9]
Next, the process of deriving the optimum tuning formula, added stiffness ratio, added weight ratio and optimum damping formula shown in FIG. M. The reason why the specifications of the tuning system can be set will be explained.

- Derivation of the "optimum tuning formula" in Fig. 9 (1) First D. M. For tuning system 1:
For the multi-mass point system numerical analysis model of the building 100 shown in FIG.

becomes.

And the complex eigenvalue solution of the above equation is

becomes.

where ω ₁ and h ₁ are the natural circular frequency and viscous damping constant of the first mode. Also, ω ₂ and h ₂ are determined by D.I. M. D.D., a new vibration mode that occurs when the tuned system is coupled with building 100; M. is the natural circular frequency of the mode and the viscous damping constant.

Next, from the relationship between the solution of the quartic equation and the coefficients,

becomes.

By substituting [Formula 2] into [Formula 3],

becomes.

Assuming that the optimum tuning condition is h ₁ =h ₂ , substituting (A) in [Eq. M. The "optimal tuning formula" of tuning system 1 is

becomes.

D. T ₁ and T _1,DM are the natural period of the 1st order mode and tuned to the 1st order mode. M. ₁ ω _∞ and ₁ T _∞ denote the natural circular frequency and natural period of the first-order mode when _{1 cd} =∞ and ₁ m _d =∞. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Moreover, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. The theory that this is approximately established will be described later.

(2) the second D.I. M. For tuning system 2:
For the numerical analysis model of the multi-mass point system of the building 100 shown in FIG.

becomes.

The complex eigenvalue solution of the above equation is

becomes.

where ω ₁ and h ₁ are the natural circular frequency and viscous damping constant of the first mode, and ω ₂ and h ₂ are the D.O. M. D.D., a new vibration mode that occurs when the tuned system is coupled with building 100; M. is the natural circular frequency of the mode and the viscous damping constant.

Next, from the relationship between the solution of the quartic equation and the coefficients,

becomes.

Substituting [Equation 8] into [Equation 9] gives

becomes.

Assuming that the optimum tuning condition is h ₁ =h ₂ , substituting (A) in Equation 10 for (C) and arranging, the second D.D. M. The "optimal tuning formula" of tuning system 2 is

becomes.

D. T ₁ and T _1,DM are the natural period of the 1st order mode and tuned to the 1st order mode. M. ₁ ω ₀ and 1 T 0 mean the natural period of the mode, and 1 ω 0 and ₁ T ₀ mean the natural circular frequency and natural period of the first-order mode when ₁ C _d =0, ₁ m _d =0, and ₁ k _d =0. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. The theory that this is approximately established will be described later.

(3) Third D. M. For tuning system 3:
For the numerical analysis model of the multi-mass point system of the building 100 shown in FIG. 7, if the order n of the mass point and the order j of the controlled object mode are each set to 1 and λ is the solution of the complex eigenvalue, then the eigenequation is:

becomes.

The complex eigenvalue solution of the above equation is

becomes.

where ω ₁ and h ₁ are the natural circular frequency and viscous damping constant of the first mode. Also, ω ₂ and h ₂ are determined by D.I. M. D.D., a new vibration mode that occurs when the tuned system is coupled with building 100; M. is the natural circular frequency of the mode and the viscous damping constant.

Next, from the relationship between the solution of the quartic equation and the coefficients,

becomes.

Substituting [Equation 14] into [Equation 15],

becomes.

Assuming that the optimum tuning condition is h ₁ =h ₂ , substituting (A) in [Formula 16] for (C) and arranging, the third D. M. The "optimal tuning formula" of tuning system 3 is

becomes.

D. T ₁ and T _1,DM are the natural period of the 1st order mode and tuned to the 1st order mode. M. ₁ ω _∞ and ₁ T _∞ denote the natural circular frequency and natural period of the first-order mode when _{1 cd} =∞ and ₁ k _d =∞. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. 7 is j, the "optimal tuning formula" can be extended as in the following [Equation 18]. The theory that this is approximately established will be described later.

(4) Fourth D. M. For tuning system 4:
For the numerical analysis model of the multi-mass point system of the building 100 shown in FIG.

becomes.

The complex eigenvalue solution of the above equation is

becomes.

where ω ₁ and h ₁ are the natural circular frequency and viscous damping constant of the first mode. Also, ω ₂ and h ₂ are determined by D.I. M. D.D., a new vibration mode that occurs when the tuned system is coupled with building 100; M. is the natural circular frequency of the mode and the viscous damping constant.

Next, from the relationship between the solution of the quartic equation and the coefficients,

becomes.

Substituting [Formula 20] into [Formula 21] gives

becomes.

Assuming that the optimum tuning condition is h ₁ =h ₂ , substituting (A) in Equation 22 into (C) and arranging, the fourth D. M. The "optimal tuning formula" of the tuning system 4 is

becomes.

D. T ₁ and T _1,DM are the natural period of the 1st order mode and tuned to the 1st order mode. M. ₁ ω ₀ and ₁ T ₀ denote the natural circular frequency and natural period of the first mode when _{1 cmd} =0, _{1 ckd} =0, ₁ m _d =0, ₁ k _d =0. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. The theory that this is approximately established will be described later.

・Derivation of the additional stiffness ratio in FIG. 9 (1) First D. M. For tuning system 1:
For the multi _- mass point system _numerical analysis model of _the building 100 _shown in _{FIG .}

becomes.

Since ₁ ω ₀ ² = k ₁ /m ₁ , expanding the above equation yields

becomes.

Here, assuming that the additional stiffness ratio of the first-order mode is ₁ κ _k = ₁ k _d /k ₁ ,

becomes.
In other words, it can be seen that the "additional stiffness ratio" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. The theory that this is approximately established will be described later.

(2) the second D.I. M. For tuning system 2:
For the multi-mass point system numerical analysis model of the building 100 shown in FIG. 6, if the order of n is 1 and _{1 cd} = ∞ and ₁ m _d = ∞, then the square of the natural circular frequency ₁ ω _{mc ∞} of the primary mode is

becomes.

Since ₁ ω ₀ ² = k ₁ /m ₁ , expanding the above equation yields

becomes.

Here, assuming that the additional stiffness ratio of the first-order mode is ₁ κ _k = ₁ k _d /k ₁ ,

becomes.
In other words, it can be seen that the "additional stiffness ratio" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. The theory that this is approximately established will be described later.

(3) Fourth D. M. For tuning system 4:
For the multi-mass point system numerical analysis model of the building 100 shown in FIG. 8, if the order of n is 1 and _{1 cmd} = ∞, ₁ m _d = ∞, and _{1 ckd} = 0, the square of the natural circular frequency ₁ ω _{mc ∞} of the primary mode is

becomes.

Since ₁ ω ₀ ² = k ₁ /m ₁ , expanding the above equation yields

becomes.

Here, assuming that the additional stiffness ratio of the first-order mode is ₁ κ _k = ₁ k _d /k ₁ ,

becomes.
In other words, it can be seen that the "additional stiffness ratio" is a relational expression of the natural period.

If the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. 8 is j, the "additional stiffness ratio" can be expanded as in the following [Equation 36].

- Derivation of the added weight ratio in Fig. 9 (1) Second D. M. For tuning system 2:
For the multi-mass point system numerical analysis model of the building 100 shown in FIG. 6, if the order of n is 1 and _{1 cd} = ∞ and ₁ k _d = ∞, then the square of the natural circular frequency ₁ ω _{ck ∞} of the primary mode is

becomes.

Since ₁ ω ₀ ² = k ₁ /m ₁ , expanding the above equation yields

becomes.

Here, if the added weight ratio of the first mode is ₁ κ _m = ₁ m _d /m ₁ ,

becomes.
In other words, it can be seen that the "additional weight ratio" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. The theory that this is approximately established will be described later.

(2) the third D. M. For tuning system 3:
For the multi-mass system numerical analysis model of the building 100 shown in FIG. 7, if the order of n is 1 and _{1 cd} = ∞ and ₁ k _d = ∞, then the square of the natural circular frequency ₁ ω _∞ of the primary mode is

becomes.

Since ₁ ω ₀ ² = k ₁ /m ₁ , expanding the above equation yields

becomes.

Here, if the added weight ratio of the first mode is ₁ κ _m = ₁ m _d /m ₁ ,

becomes.
In other words, it can be seen that the "additional weight ratio" is a relational expression of the natural period.

Further, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. 7 is jth, the "additional weight ratio" can be expanded as in the following [Equation 44]. The theory that this is approximately established will be described later.

(3) Fourth D. M. For tuning system 4:
For the multi-mass point system numerical analysis model of the building 100 shown in FIG. 8, if the order of n is 1 and _{1 ckd} =∞, _{1 kd} = ∞, and _{1 cmd} = 0, then the square of the natural circular frequency ₁ ω _ck∞ of the primary mode is

becomes.

Since ₁ ω ₀ ² = k ₁ /m ₁ , expanding the above equation yields

becomes.

Here, if the added weight ratio of the first mode is ₁ κ _m = ₁ m _d /m ₁ ,

becomes.
In other words, it can be seen that the "additional weight ratio" is a relational expression of the natural period.

Moreover, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. The theory that this is approximately established will be described later.

- Derivation of the "optimum attenuation formula" in Fig. 9 (1) First D. M. For tuning system 1:
Assuming that the optimum tuning condition is h ₁ =h ₂ , substituting (D) in Equation 4 into (B) and arranging, the first D. M. The viscous damping constant at the time of optimum tuning of the tuning system 1 is

becomes.

D. T ₁ and T _1,DM are the natural period of the 1st order mode and tuned to the 1st order mode. M. denotes the natural period of the mode, h ₁ and h _1,DM are the viscous damping constants of the first mode and D.M. M. ₁ ω _∞ and 1 T ∞ mean the viscous damping constant of the mode, and 1 ω ∞ and ₁ T _∞ mean the natural circular frequency and natural period of the first-order mode when _{1 cd} =∞ and ₁ m _d =∞. In other words, it can be seen that the viscous damping constant at the time of optimum tuning is a relational expression of the natural period.

Further, if T ₁ =T _{1, DM} = ₁ T _∞ , the maximum viscous damping constant h _max at the time of optimum tuning is

becomes.

Further, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. 5 is j, the maximum viscous damping constant hj _,max of the j-th mode can be extended as shown in the following [Equation 51] at the time of optimum tuning. The theory that this is approximately established will be described later.

Furthermore, the damping adjustment coefficient α is added to the above formula to set the "optimum damping formula" as follows.

_hj and hj _,DM are the viscous damping constants of the jth mode and D. M. The viscous damping constant of the mode, and the damping adjustment coefficient α is 1 or more. In other words, it can be seen that the viscous damping constants of both are in the range of 0 to hj _,max depending on the damping adjustment coefficient α.

(2) the second D.I. M. For tuning system 2:
Assuming that the optimum tuning condition is h ₁ =h ₂ , substituting (D) in Equation 10 into (B) and arranging, the second D. M. The viscous damping constant at the time of optimum tuning of the tuning system 2 is

becomes.

D. T ₁ and T _1,DM are the natural period of the 1st order mode and tuned to the 1st order mode. M. is the natural period of the mode, h ₁ and h _1,DM are the viscous damping constants of the first mode and the D.D. M. ₁ ω _{mc ∞} and ₁ T _{mc ∞} mean the natural circular frequency and natural period of the first mode when _{1 cd} = ∞ _{and 1} m _d = ∞, and ₁ ω _{ck ∞} and ₁ T _{ck ∞} mean the natural circular frequency _and natural period of the first mode when 1 cd = ∞ and ₁ k _d = ∞. In other words, it can be seen that the viscous damping constant at the time of optimum tuning is a relational expression of the natural period.

Also, if T ₁ =T _{1 and DM} = ₁ T ₀ , the maximum viscous damping constant h _max at the time of optimum tuning is

becomes.

Also, if the number of vibration modes in the numerical analysis model of the building 100 shown in FIG. 6 is j, the maximum viscous damping constant hj _,max of the j-th mode can be expanded as shown in [Equation 55] below at the time of optimum tuning. The theory that this is approximately established will be described later.

Furthermore, the damping adjustment coefficient α is added to the above formula to set the "optimum damping formula" as follows.

_hj and hj _,DM are the viscous damping constants of the jth mode and D. M. The viscous damping constant of the mode, and the damping adjustment coefficient α is 1 or more. That is, it can be seen that the viscous damping constants of both are in the range of 0 to hj _,max by the damping adjustment coefficient α.

(3) Third D. M. For tuning system 3:
Assuming that the optimum tuning condition is h ₁ =h ₂ , substituting (D) of Equation 16 into (B) and arranging, the third D. M. The viscous damping constant at the time of optimum tuning of the tuning system 3 is

becomes.

D. T ₁ and T _1,DM are the natural period of the 1st order mode and tuned to the 1st order mode. M. denotes the natural period of the mode, h ₁ and h _1,DM are the viscous damping constants of the first mode and D.M. M. ₁ ω _∞ and 1 T ∞ mean the viscous damping constant of the mode, and 1 ω ∞ and ₁ T _∞ mean the natural circular frequency and natural period of the first-order mode when ₁ c _d =∞ and ₁ k _d =∞. In other words, it can be seen that the viscous damping constant at the time of optimum tuning is a relational expression of the natural period.

Further, if T ₁ =T _{1, DM} = ₁ T _∞ , the maximum viscous damping constant h _max at the time of optimum tuning is

becomes.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. 7 is j, the maximum viscous damping constant hj _,max of the j-th mode can be expanded as shown in [Equation 59] below at the time of optimum tuning. The theory that this is approximately established will be described later.

Furthermore, the damping adjustment coefficient α is added to the above formula to set the "optimum damping formula" as follows.

_hj and hj _,DM are the viscous damping constants of the jth mode and the D.D. M. The viscous damping constant of the mode, and the damping adjustment coefficient α is 1 or more. In other words, it can be seen that the viscous damping constants of both are in the range of 0 to hj _,max depending on the damping adjustment coefficient α.

(4) Fourth D. M. For tuning system 4:
Assuming that the optimum tuning condition is h ₁ =h ₂ , substituting (D) in Equation 22 into (B) and arranging as _{1 cmd} = _{1 ckd} = _{1 cd} yields the fourth D. M. The viscous damping constant at the time of optimum tuning of the tuning system 4 is

becomes.

D. T ₁ and T _1,DM are the natural period of the 1st order mode and tuned to the 1st order mode. M. is the natural period of the mode, h ₁ and h _1,DM are the viscous damping constants of the first mode and the D.D. M. ₁ ω _{mc ∞} and ₁ T _{mc ∞} mean the natural circular frequency and natural period of the first mode when _{1 cd} = ∞ _{and 1} m _d = ∞, and ₁ ω _{ck ∞} and ₁ T _{ck ∞} mean the natural circular frequency _and natural period of the first mode when 1 cd = ∞ and ₁ k _d = ∞. For the sake of simplification, ignoring the item ( _{1 cd} /( ₁ ω ₀ ·m ₁ )) ² in the above equation, it can be seen that the viscous damping constant at the time of optimum tuning is a relational expression of the natural period.

Also, if T ₁ =T _{1 and DM} = ₁ T ₀ , the maximum viscous damping constant h _max at the time of optimum tuning is

becomes.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. 8 is jth, the maximum viscous damping constant hj _,max of the jth mode can be extended as shown in the following [Equation 63] at the time of optimum tuning. The theory that this is approximately established will be described later.

Furthermore, the damping adjustment coefficient α is added to the above formula to set the "optimum damping formula" as follows.

_hj and hj _,DM are the viscous damping constants of the jth mode and D. M. The damping adjustment coefficient α, which is the viscous damping constant of the mode, is a numerical value of 1 or more. In other words, it can be seen that the viscous damping constants of both are in the range of 0 to hj _,max depending on the damping adjustment coefficient α.

Optimum tuning or optimum damping by using the optimum tuning formula, added stiffness ratio, added weight ratio and optimum damping formula Using the optimum tuning formula, added stiffness ratio, added weight ratio and optimum damping formula, D. M. The reason why it is possible to optimally tune or optimally attenuate the control object mode (arbitrary vibration mode) of the building by setting the specifications of the tuning system will be explained.

Note that, as described above, the optimum tuning includes the controlled object mode (arbitrary vibration mode) of the building and the D.O. M. The viscous damping constant of the tuning system should match or substantially match that of the new vibration mode "DM mode" generated by coupling with the building. Also, the term "optimal damping" refers to effectively reducing vibration in the controlled object mode by giving the controlled object mode an optimum viscous damping constant.

<Explanation about the theory that holds approximately>
Next, the above-mentioned approximately established theory will be explained.

The eigenvalue analysis calculation of a multi-mass point system can obtain the eigenvalue result of each order mode by utilizing the property that the eigenvectors of each order mode are mutually linearly independent (orthogonality). By using the property of orthogonality, it is possible to separate the reference coordinates (response) of each order mode to the disturbance, which is the basic principle of modal analysis. The optimum tuning formula, added stiffness ratio, added weight ratio, and optimum damping formula described above are derived from the numerical analysis model of a single mass point system, but since they are all relational expressions of the natural period, they can also be applied to the controlled object mode (arbitrary vibration mode) of the numerical analysis model of the multi-mass point system and the three-dimensional system by utilizing the orthogonal property of the eigenvector. Note that the present embodiment is based on complex eigenvalue analysis.

[Design method example]
Next, an example of a design method using the optimum tuning formula, added stiffness ratio, added weight ratio and optimum damping formula shown in FIG. 9 will be described. This setting method is an example of setting the specifications so as to satisfy both the optimum tuning and the optimum attenuation.

(bilayer)
First, a design example in which the building 100 (see FIG. 73) is two stories and the controlled object mode is the primary mode will be shown. Also, since the building 100 has two stories, a two-story shear model is used.

FIG. 14A is a table summarizing analysis conditions for the building 100 (double-layer shear model). Note that FL in the table indicates a floor, FL1 is the first floor (bottom floor) of the building, and FL2 is the second floor from the bottom of the building. And the first D.C. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. The tuning system 4 is installed in the first layer (bottom layer).

Also, in this design, the specification setting procedure from step 53 to step 58 in the flow chart of FIG. 13 will be explained.

・First D. M. tuning system 1
First, the first D.I. M. A design example of the tuning system 1 will be described.

As described in step 53 of the flow chart, the rigid element 30 of the first D.M. tuning system 1 is set up as shown in FIG. 15(A). In the case of this example, as described above, the first layer (bottom layer) contains the first D.I. M. A tuning system 1 is installed.

Complex eigenvalue analysis is then performed to determine the natural period of the first mode of the building 100, as described in step 54 of the flow chart. The obtained natural periods ₁ T ₀ and ₁ T _∞ are shown in the table of FIG. 14(B) and the table of FIG. 15(B), respectively. Further, as shown in the table of FIG. 14B, the setting of the attenuation (internal structural attenuation) of the building 100 is Rayleigh type and the primary and secondary are set to 2% each. ₁ T ₀ is 2.81 seconds as shown in FIG. 14(B). ₁ T _∞ is 2.50 seconds as shown in FIG. 15(B).

Next, as described in step 55 of the flowchart, the natural period (FIGS. 14B and 15B) and the addition stiffness ratio formula (see FIG. 9) are used to calculate the addition stiffness ratio ₁ κ _k of the first mode. In this example, the additional stiffness ratio ₁ κ _k of the primary mode is 0.26.

Next, as described in step 56 of the flow chart, the optimum viscous damping constant h ₁ for the first order mode is calculated using the added stiffness ratio ₁ κ _k (0.26) and the optimum damping equation (see FIG. 9). In this example, the optimum viscous damping constant _h1 is 0.18 (18%).

Next, as described in step 57 of the flow chart, a complex eigenvalue analysis is performed to satisfy the first-order mode optimal tuning equation (see FIG. 9). M. Find element 10 ( ₁ m _d ). The results are shown in the table of FIG. 16(A) and the table of FIG. 16(B).

Finally, as described in step 58 of the flow chart, a complex eigenvalue analysis is performed to satisfy the first order mode optimal viscous damping constant h ₁ (18%). That is, the damping element 20 ( _{1 cd} ) is obtained so that the viscous damping constants of the first order mode and the first order (D.M) mode are both 18%.

The final specifications and eigenvalue results are shown in the tables of FIGS. 17(A) and 17(B).

The “primary mode” is the primary mode of the building 100, and the primary (D.M.) mode is the primary mode of the “D.M. mode”, which is a new vibration mode coupled with the building 100 described above.

・Second D. M. Tuning system 2
Next, the second D.I. M. A design example of the tuning system 2 will be described.

As described in step 53 of the flow chart, the rigid element 30 of the second D.M. tuning system 2 is set as shown in FIG. 18(A). Also in the case of this example, it is installed in the first layer (bottom layer) as described above.

Complex eigenvalue analysis is then performed to determine the natural period of the first mode of the building 100, as described in step 54 of the flow chart. The determined natural periods ₁ T ₀ and ₁ T _∞ are shown in the table of FIG. 14(B) and the table of FIG. 18(B), respectively. ₁ T _∞ is 2.50 seconds as shown in FIG. 18(B).

Next, as described in step 55 of the flowchart, the natural period (FIGS. 14B and 18B) and the addition stiffness ratio formula (see FIG. 9) are used to calculate the addition stiffness ratio ₁ κ _k of the first mode. In this example, the additional stiffness ratio ₁ κ _k of the primary mode is 0.26.

Next, as described in step 56 of the flow chart, the optimum viscous damping constant h ₁ for the first order mode is calculated using the added stiffness ratio ₁ κ _k (0.26) and the optimum damping equation (see FIG. 9). In this example, the optimum viscous damping constant _h1 is 0.18 (18%).

Then, as described in step 57 of the flow chart, complex eigenvalue analysis is performed to satisfy the first-order mode optimal tuning equation (see FIG. 9). M. Find element 10 ( ₁ m _d ). The results are shown in the table of FIG. 19(A) and the table of FIG. 19(B).

Finally, as described in step 58 of the flow chart, a complex eigenvalue analysis is performed to satisfy the first order mode optimal viscous damping constant h ₁ (18%). That is, the damping element 20 ( _{1 cd} ) is obtained so that the viscous damping constants of the first order mode and the first order (D.M) mode are both 18%.

The final specifications and eigenvalue results are shown in the tables of FIGS. 20(A) and 20(B).

The “primary mode” is the primary mode of the building 100, and the primary (D.M.) mode is the primary mode of the “D.M. mode”, which is a new vibration mode coupled with the building 100 described above.

Here, in this example, as shown in FIG. 18A, since the rigidity of the rigid element 30 is set in advance, the additional rigidity ratio is used in step 56 of the flow chart. However, first D. M. When the specifications of the element 10 are set, the additional weight ratio is used in step 56 of the flow chart. Then, in step 57 of the flow chart, a complex eigenvalue analysis is performed to obtain the stiffness so as to satisfy the first-order mode optimum tuning equation (see FIG. 9).

・Third D. M. tuning system 3
Next, the third D.I. M. A design example of the tuning system 3 will be described.

The D.M. tuning system 3 of the third D.M. M. Element 10 is set as shown in FIG. Also in the case of this example, it is installed in the first layer (bottom layer) as described above.

Complex eigenvalue analysis is then performed to determine the natural period of the first mode of the building 100, as described in step 54 of the flow chart. The obtained natural periods ₁ T ₀ and ₁ T _∞ are shown in the table of FIG. 14(B) and the table of FIG. 21(B), respectively. Also, ₁ T _∞ is 3.16 seconds as shown in FIG. 21(B).

Next, as described in step 55 of the flowchart, the natural period (FIGS. 14B and 21B) and the addition weight ratio formula (see FIG. 9) are used to calculate the addition weight ratio ₁ κ _m of the first mode. In this example, the added weight ratio ₁ κ _m of the primary mode is 0.26.

Next, as described in step 56 of the flow chart, the optimum viscous damping constant h ₁ for the primary mode is calculated using the added weight ratio ₁ κ _m (0.26) and the optimum damping formula (see FIG. 9). In this example, the optimum viscous damping constant _h1 is 0.18 (18%).

Next, as described in step 57 of the flow chart, a complex eigenvalue analysis is performed to obtain the stiffness element 10 ( ₁ k _d ) so as to satisfy the first-order mode optimum tuning equation (see FIG. 9). The results are shown in the table of FIG. 22(A) and the table of FIG. 22(B).

Finally, as described in step 58 of the flow chart, a complex eigenvalue analysis is performed to satisfy the first order mode optimum viscous damping constant h1 (18%). That is, the damping element 20 ( _{1 cd} ) is obtained so that the viscous damping constants of the first order mode and the first order (D.M) mode are both 18%.

The final specifications and eigenvalue results are shown in the tables of FIGS. 23(A) and 23(B).

The “primary mode” is the primary mode of the building 100, and the primary (D.M.) mode is the primary mode of the “D.M. mode”, which is a new vibration mode coupled with the building 100 described above.

Also, the third D.I. M. When the tuning system 3 is used, an attenuation constant of 0.10 can be obtained not only in the primary mode but also in the secondary mode, as shown in FIG. 23(B).

・Fourth D. M. tuning system 4
Next, the fourth D.I. M. A design example of the tuning system 4 will be described.

As described in step 53 of the flow chart, the fourth D.E. M. The rigid element 30 of the tuning system 4 is set as shown in FIG. 24(A). Also in the case of this example, it is installed in the first layer (bottom layer) as described above.

Complex eigenvalue analysis is then performed to determine the natural period of the first mode of the building 100, as described in step 54 of the flow chart. The determined natural periods ₁ T ₀ and ₁ T _∞ are shown in the table of FIG. 14(B) and the table of FIG. 24(B), respectively. Also, ₁ T _∞ is 2.50 seconds as shown in FIG. 24(B).

Next, as described in step 55 of the flowchart, the natural period (FIGS. 14B and 24B) and the addition stiffness ratio formula (see FIG. 9) are used to calculate the addition stiffness ratio ₁ κ _k of the first mode. In this example, the additional stiffness ratio ₁ κ _k of the primary mode is 0.26.

Next, as described in step 56 of the flow chart, the optimum viscous damping constant h ₁ for the first order mode is calculated using the added stiffness ratio ₁ κ _k (0.26) and the optimum damping equation (see FIG. 9). In this example, the optimum viscous damping constant _h1 is 0.18 (18%).

Next, as described in step 57 of the flow chart, a complex eigenvalue analysis is performed to satisfy the first-order mode optimal tuning equation (see FIG. 9). M. Find element 10 ( ₁ m _d ). The results are shown in the table of FIG. 25(A) and the table of FIG. 25(B).

Finally, as described in step 58 of the flow chart, a complex eigenvalue analysis is performed to satisfy the first order mode optimal viscous damping constant h ₁ (18%). That is, the first damping element 22 ( _{j cmd} ) and the second damping element 24 ( _{j ckd} ) are obtained so that the viscous damping constants of the first order mode and the first order (D.M) mode are both 18%.

The final specifications and eigenvalue results are shown in the tables of FIGS. 26(A) and 26(B).

The “primary mode” is the primary mode of the building 100, and the primary (D.M.) mode is the primary mode of the “D.M. mode”, which is a new vibration mode coupled with the building 100 described above. Also, the fourth D.I. M. When the tuning system 4 is used, a damping constant of 0.12 can be obtained not only in the primary mode but also in the secondary mode, as shown in FIG. 26(B).

Here, in this example, as shown in FIG. 24A, since the rigidity of the rigid element 30 is set in advance, the additional rigidity ratio is used in step 56 of the flow chart. However, first D. M. When the specifications of the element 10 are set, the additional weight ratio is used in step 56 of the flow chart. Then, in step 57 of the flow chart, a complex eigenvalue analysis is performed to obtain the stiffness so as to satisfy the first-order mode optimum tuning equation (see FIG. 9).

[Four D.C. M. Attenuation reduction effect of tuning system and its comparison]
The first D. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. The attenuation reduction effect and the difference thereof when the tuning system 4 is installed will be described.

(1) Case of double-layer shear model First, the case of the above-mentioned double-layer shear model will be described. The internal structure damping is Rayleigh type and 2% is given to each of the primary and secondary.

- When the controlled object mode is the primary mode First, the case where the controlled object mode is the primary mode will be described.

As in the previous design example, the optimum viscous damping constant for the primary mode is set to 18%.

The first D. M. The specifications of the tuning system 1 are shown in FIG. 27(A), and the eigenvalues are shown in FIG. 27(B). The second D. M. The specifications of the tuning system 2 are shown in FIG. 28(A), and the eigenvalues are shown in FIG. 28(B). The third D. M. The specifications of the tuning system 3 are shown in FIG. 29(A), and the eigenvalues are shown in FIG. 29(B). Fourth D. M. The specifications of the tuning system 4 are shown in FIG. 30(A), and the eigenvalues are shown in FIG. 30(B). 27 to 30 are the same as FIGS. 17, 20, 23 and 26 described above.

27(B), 28(B), 29(C), and 30(B) showing the eigenvalues, it can be seen that the viscous damping constant is added to the second-order mode even if the mode to be controlled is the first-order mode.

And the viscous damping constant of the second mode is

The first D. M. 0.02 for tuning system 1
The second D. M. 0.05 for tuning system 2
The third D. M. 0.10 for tuning system 3
Fourth D. M. 0.12 for tuning system 4

and, in particular, the fourth D. M. The tuning system 4 and the third D.I. M. It can be seen that the viscous damping constant for the second mode of the tuned system 3 is large.

35 and 36 show resonance curves (including internal structural damping) when the primary mode is controlled. Note that the horizontal axis in FIG. 35 is an equidistant scale, and the horizontal axis in FIG. 36 is a logarithmic scale.

As can be seen from this resonance curve, even if the mode to be controlled is the primary mode, the response to the secondary mode is also reduced. And from these, in particular, the third D.C. M. Tuning system 3 and the fourth D.I. M. It can be seen that the second-order mode response reduction effect of the tuning system 4 is large.

Therefore, when expecting the second-order mode response reduction effect, the third D.C. M. Tuning system 3 and the fourth D.I. M. Using a tuning system 4 is useful.

- When the controlled object mode is the secondary mode Next, the case where the controlled object mode is the secondary mode will be described.

The optimum viscous damping constant for the second order mode is set to be 18%.

The first D. M. The specifications of the tuning system 1 are shown in FIG. 31(A), and the eigenvalues are shown in FIG. 31(B). The second D. M. The specifications of the tuning system 2 are shown in FIG. 32(A), and the eigenvalues are shown in FIG. 32(B). The third D. M. The specifications of the tuning system 3 are shown in FIG. 33(A), and the eigenvalues are shown in FIG. 33(B). Fourth D. M. The specifications of the tuning system 4 are shown in FIG. 34(A), and the eigenvalues are shown in FIG. 34(B).

31(B), 32(B), 33(B), and 34(B) showing the eigenvalues, it can be seen that the viscous damping constant is added to each of the primary modes even if the controlled object mode is the secondary mode.

And the viscous damping constant of the first mode is

The first D. M. 0.10 for tuning system 1
The second D. M. 0.04 for tuning system 2
The third D. M. 0.02 for tuning system 3
Fourth D. M. 0.10 for tuning system 4

and, in particular, the first D.C. M. Tuning system 1 and the fourth D.I. M. It can be seen that the viscous damping constant for the first mode of the tuned system 4 is large.

37 and 38 show resonance curves (including internal structural damping) when the secondary mode is controlled. Note that the horizontal axis in FIG. 37 is an equidistant scale, and the horizontal axis in FIG. 38 is a logarithmic scale.

As can be seen from this resonance curve, even if the mode to be controlled is the secondary mode, the response to the primary mode is also reduced. And from these, in particular, the first D.C. M. Tuning system 1 and the fourth D.I. M. It can be seen that the primary mode response reduction effect of the tuning system 4 is large.

Therefore, when the first-order mode response reduction effect is expected, the first D.C. M. Tuning system 1 and the fourth D.I. M. Using a tuning system 4 is useful.

(2) Five-layer shear model The specifications of the five-layer shear model are shown in Fig. 39(A), and the eigenvalue results are shown in Fig. 39(B). As can be seen from FIG. 39B, the primary natural period is 1.96 seconds and the secondary natural period is 0.70 seconds. The internal structural damping is Rayleigh type and 2% is given to each of the primary and secondary.

Each D.I. M. The tuning system was installed on the second and fifth floors, and the D.I. M. The control coping mode of the tuning system is the primary mode, and the D.I. M. The control handling mode of the tuned system is the secondary mode. Also, each mode is designed so that the optimum viscous damping constant is about 10%.

The first D. M. The specifications of the tuning system 1 are shown in FIG. 40(A), and the eigenvalues are shown in FIG. 40(B). The second D. M. The specifications of the tuning system 2 are shown in FIG. 41(A), and the eigenvalues are shown in FIG. 41(B). The third D. M. The specifications of the tuning system 3 are shown in FIG. 42(A), and the eigenvalues are shown in FIG. 42(B). Fourth D. M. The specifications of the tuning system 4 are shown in FIG. 43(A), and the eigenvalues are shown in FIG. 43(B).

44 and 45 show resonance curves (including internal structure damping). Note that the horizontal axis in FIG. 44 is an equidistant scale, and the horizontal axis in FIG. 45 is a logarithmic scale.

As can be seen from this resonance curve, any D.I. It can be seen that the response of the M-tuned system is reduced to approximately the same extent.

Note that the third D.E.D. M. Tuning system 3 and the fourth D.I. M. It can be seen that the response reduction effect of the tuning system 4 is great. Therefore, when emphasizing the response reduction effect of higher modes after the third mode, the third D. M. Tuning system 3 and the fourth D.I. M. Using a tuning system 4 is useful.

(3) Eight-layer shear model The specifications of the eight-layer shear model are shown in Fig. 46(A), and the eigenvalue results are shown in Fig. 46(B). As can be seen from the figure, the mass of each layer was set to 1.0 tons, and the layer stiffness was set to have a trapezoidal distribution in which the stiffness ratio was 0.5 at the top layer and 1.0 at the bottom layer, and the natural period of the primary mode was 2.10 seconds. Further, the internal structure damping is Rayleigh type and 2% is given to each of the primary mode and the secondary mode.

Each D.I. M. Tuning systems are installed on the first and second floors. Both the first layer and the second layer were D.I. M. The control coping mode of the tuned system is the first order mode and was designed so that the optimum viscous damping constant is around 20%.

The first D. M. The specifications of the tuning system 1 are shown in FIG. 47(A), and the eigenvalues are shown in FIG. 47(B). The second D. M. The specifications of the tuning system 2 are shown in FIG. 48(A), and the eigenvalues are shown in FIG. 48(B). The third D. M. The specifications of the tuning system 3 are shown in FIG. 49(A), and the eigenvalues are shown in FIG. 49(B). Fourth D. M. The specifications of the tuning system 4 are shown in FIG. 50(A), and the eigenvalues are shown in FIG. 50(B).

47(B), 48(B), 49(B), and 50(B) showing the eigenvalues, it can be seen that the viscous damping constant is added to each of the secondary and higher modes even if the controlled mode is the primary mode.

And its size is the third D.I. M. Tuning system 3≈fourth D. M. Tuning system 4 > second D. M. Tuning system 2 > first D. M. This is the order of the tuning system 1 .

Resonance curves (including internal structural damping) are shown in FIGS. Note that the horizontal axis of FIG. 51 is an equidistant scale, and the horizontal axis of FIG. 52 is a logarithmic scale.

As can be seen from the resonance curves, any D.I. M. It can be seen that the response of the tuning system is reduced to approximately the same extent.

Note that the third D.E.D. M. Tuning system 3 and the fourth D.I. M. It can be seen that the response reduction effect of the tuning system 4 is great.

Therefore, when emphasizing the response reduction effect of higher modes after the second mode, the third D. M. Tuning system 3 and the fourth D.I. M. Using a tuning system 4 is useful.

FIG. 53(A) shows the maximum response acceleration acting on each layer, and FIG. 53(B) shows the maximum response displacement acting on each layer.

From FIG. 53(A), the magnitude of the effect of reducing the maximum response acceleration is the third D. M. Tuning system 3≈fourth D. M. Tuning system 4 > second D. M. Tuning system 2 > first D. M. A tuning system 1 . Therefore, from the viewpoint of the effect of reducing the maximum response acceleration, the third D. M. Tuning system 3 and the fourth D.I. M. The use of tuning system 4 has been found to be effective.

Also, from FIG. 53(B), the magnitude of the effect of reducing the maximum response displacement is M. It can be seen that there is little difference in the tuning system.

• Mounting rigidity Next, as shown in FIGS. 54 and 55, the third D. M. Tuning system 3 and the fourth D.I. M. A case where the tuning system 4 includes a mounting rigid element 50 (k _d2 ) such as a brace member will be described. In addition, in this comparative example, the first D.I. M. Tuning system 1 and a second D.I. M. The rigid elements 30 (k _d ) of the tuning system 2 correspond to rigid elements such as the brace members described above.

A third D.D. considering the mounting stiffness element 50 (k _d2 ). M. The specifications of the tuning system 3 are shown in FIG. 56(A), and the eigenvalues are shown in FIG. 56(B). A fourth D.D. considering the mounting stiffness element 50 (k _d2 ). M. The specifications of the tuning system 4 are shown in FIG. 57(A), and the eigenvalues are shown in FIG. 57(B). Resonance curves (including internal structural damping) are shown in FIGS. FIG. 60(A) shows the maximum response acceleration acting on each layer, and FIG. 60(B) shows the maximum response displacement acting on each layer.

As can be seen from these figures, even if there is a mounting rigidity of 50, a viscous damping constant is also added to higher-order modes of the second and higher modes. In addition, it can be seen that the effect of reducing the response is great for the secondary mode as well. Furthermore, it also has the effect of reducing the maximum response acceleration. Therefore, even when the mounting rigidity 50 is considered, the third D.D. M. Tuning system 3 and the fourth D.I. M. It is useful to use a tuning system 4, which can be either optimally tuned or optimally damped.

(4) Steel-frame high-rise building Next, a case of application to a steel-frame high-rise building will be described.

- Shear Link Type First, as shown in FIG. 62, the case where a shear link mechanism 210 is used will be described.

A steel-framed building 200 as an example of the structure in FIG. 62 simulates a 10-layer model of the damping structure theme structure in appendix A2 of the passive damping design and construction manual 3rd edition. A shear link mechanism 210 ensures that each D.C. M. The tuning system is arranged along the horizontal direction.

As shown in FIG. 62, each D.C. M. The tuning system was provided on the first layer. As shown in FIG. 61, the input seismic motion was the random number phase of the level 2 notification wave. The third D. M. Tuning system 3 and the fourth D.I. M. In the tuning system 4, a mounting rigid element 50 (k _d2 ) is considered as a rigid element such as the brace member of the shear link mechanism 210 described above (see FIGS. 54 and 55). In addition, in this comparative example, the first D.I. M. Tuning system 1 and a second D.I. M. The rigid element 30 (k _d ) of the tuning system 2 corresponds to the rigid element such as the brace member of the shear link mechanism 210 described above (see FIG. 62).

FIG. 64 shows the first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. 1 shows set specifications and basic performance of a tuning system 4 and an oil damper as a comparative example.

A resonance curve is shown in FIG. Based on the resonance curve of FIG. 63, the first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. It can be seen that by installing the tuning system 4 the first order mode is suppressed.

The maximum response displacement is shown in FIG. 65(A), the maximum response acceleration is shown in FIG. 65(B), the maximum response story shear force is shown in FIG. 65(C), the maximum response story shear force coefficient is shown in FIG. 65(D), the maximum response story drift angle is shown in FIG. Furthermore, the respective reduction rates of maximum response displacement, maximum response acceleration, maximum response story shear force, maximum response story shear force coefficient, maximum response story drift angle, and maximum response overturning moment are shown in FIGS.

65 and 66, the shear link mechanism 210 is used for the first D.D. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. A response reduction effect by installing the tuning system 4 can be confirmed.

Toggle Type Next, as shown in FIG. 68, the case of using a toggle mechanism 310 will be described.

A steel-framed building 300 as an example of the structure in FIG. 68 is a simulation of a 20-story model of the damping structure theme structure in appendix A2 of the third edition of the passive damping design and construction manual. Each D.I. M. The tuning system was provided in the 1st to 10th layers.

Note that the toggle mechanism is a type of link mechanism composed of two links and a slider, and is a mechanism in which a member moves in the input direction upon input and the output is increased by a force boosting structure. Also, regarding the toggle mechanism, for example, Japanese Unexamined Patent Publication No. 10-169244 can be referred to.

As shown in FIG. 67, the input seismic motion was the random number phase of the level 2 notification wave. The third D. M. Tuning system 3 and the fourth D.I. M. Tuning system 4 considers mounting rigid element 50 (k _d2 ) as a rigid element such as the toggle arm member of toggle mechanism 310 described above (see FIGS. 54 and 55). In addition, in this comparative example, the first D.I. M. Tuning system 1 and a second D.I. M. Rigid element 30 (k _d ) of tuning system 2 corresponds to a rigid element such as the toggle arm member of toggle mechanism 310 described above (see FIG. 68).

FIG. 70 shows the first D.I. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. 1 shows set specifications and basic performance of a tuning system 4 and an oil damper as a comparative example.

A resonance curve is shown in FIG. The maximum response displacement is shown in FIG. 71(A), the maximum response acceleration is shown in FIG. 71(B), the maximum response story shear force is shown in FIG. 71(C), the maximum response story shear force coefficient is shown in FIG. 71(D), the maximum response story drift angle is shown in FIG. Furthermore, the respective reduction rates of maximum response displacement, maximum response acceleration, maximum response story shear force, maximum response story shear force coefficient, maximum response story drift angle, and maximum response overturning moment are shown in Figs.

71 and 72, the first D.C. using the toggle mechanism 310 is shown. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. A response reduction effect by installing the tuning system 4 can be confirmed.

<Action and effect>
Next, the operation and effects of this embodiment will be described.

As described above, the first D.I.D. is an example of a vibration damping device installed in buildings 100, 101, 200, and 300 having a plurality of vibration modes. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. By setting the specifications for optimal tuning or optimal damping of the tuning system 4 (including the case where the D.M. tuning system 3 and 4 considers the mounting rigidity element), the vibration of the buildings 100, 101, 200, 300 during an earthquake can be effectively reduced.

A first D.D. M. Tuning system 1, the second D.C. M. Tuning system 2, third D. M. Tuning system 3 and the fourth D.I. M. The tuning system 4 (including the D.M. tuning systems 3 and 4 considering mounting stiffness elements) can be optimally tuned or optimally damped for any vibration mode.

<For Fig. 74>
Next, the case of using the design formula shown in FIG. 74 will be described.

In this specification setting method, the controlled object mode (arbitrary vibration mode) of the building 100 and the D. M. The controlled object mode or D.M. By giving the M. mode the optimum viscous damping constant, the D. mode is designed to effectively reduce the vibration of the controlled mode. M. The specifications of the element 10, the damping element 20, the first damping element 22, the second damping element 24 and the rigid element 30 are set.

- Derivation of the "optimal tuning formula" in Fig. 74 (1) First D. M. For tuning system 1:
For the multi-mass system numerical analysis model of the building 100 shown in FIG. 5, the order n of the mass point and the order j of the controlled object mode are each set to 1, X is the one-story relative displacement of the model, and Y is the input displacement of the harmonic ground motion.

becomes.

Here, ₁ κ _k = ₁ k _d /k ₁ , ₁ κ _m = ₁ m _d /m ₁ , hd = _{1 cd /} (2√m _{1 ·} k ₁ ), λ = ω / ₁ ω ₀ , where ω and ₁ ω ₀ are the natural circular frequency of the harmonic ground motion and the natural circular vibration of the first mode of the analytical model when ₁ κ _k = 0 is a number.

Using the relative displacement response magnification curve, if the intersection point of the response magnification curves of _hd = 0 and _hd = ∞ is set to a fixed point P and a fixed point Q, the fixed point equation is as follows:

becomes.

Taking the solutions of the fixed-point equations as λ _P ² and λ _Q ² , we obtain the following relationship.

In the optimum tuning, since the fixed point P and the fixed point Q have the same response magnification, the following equation holds.

Substituting [Equation 67] into [Equation 68] and arranging, the first D. M. The "optimal tuning formula" of tuning system 1 is

becomes.

D. T ₁ and T _1,DM are in tune with the natural period of the first mode and the first mode when ₁ c _d =0. M. means the natural period of the mode, and ₁ T _∞ means the natural period of the first mode when _{1 cd} =∞ and ₁ m _d =∞. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG.

(2) the second D.I. M. For tuning system 2:
For the multi-mass point system numerical analysis model of the building 100 shown in FIG. 6, the order n of the mass point and the order j of the controlled object mode are each set to 1, X is the one-story relative displacement of the model, and Y is the input displacement of the harmonic ground motion.

becomes.

Here, ₁ κ _k = ₁ k _d /k ₁ , ₁ κ _m = ₁ m _d /m ₁ , hd = _{1 cd /} (2√m _{1 ·} k ₁ ), λ = ω / ₁ ω ₀ , where ω and ₁ ω ₀ are the natural circular frequency of the harmonic ground motion and the natural circular vibration of the first mode of the analytical model when ₁ κ _k = 0 is a number.

Using the relative displacement response magnification curve, if the intersection point of the response magnification curves of _hd = 0 and _hd = ∞ is set to a fixed point P and a fixed point Q, the fixed point equation is as follows:

becomes.

Taking the solutions of the fixed-point equations as λ _P ² and λ _Q ² , we obtain the following relationship.

In the optimum tuning, since the fixed point P and the fixed point Q have the same response magnification, the following equation holds.

Substituting (A) and (B) of Equation 73 into Equation 74 and arranging, the second D. M. The "optimal tuning formula" of tuning system 2 is

becomes.

D. T ₁ and T _1,DM are in tune with the natural period of the first mode and the first mode when ₁ c _d =∞. M. ₁ T ₀ means the natural period of the mode when _{1 cd} =0, ₁ m _d =0, and ₁ k _d =0. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

(3) Third D. M. For tuning system 3:
For the multi-mass point system numerical analysis model of the building 100 shown in FIG. 7, the order n of the mass point and the order j of the controlled object mode are each set to 1, X is the one-story relative displacement of the model, and Y is the input displacement of the harmonic ground motion.

becomes.

Here, ₁ κ _k = ₁ k _d /k ₁ , ₁ κ _m = ₁ m _d /m ₁ , hd = _{1 cd /} (2√m _{1 ·} k ₁ ), λ = ω / ₁ ω ₀ , where ω and ₁ ω ₀ are the natural circular frequency of the harmonic ground motion and the natural circular vibration of the first mode of the analytical model when ₁ κ _k = 0 is a number.

Using the relative displacement response magnification curve, if the intersection point of the response magnification curves of _hd = 0 and _hd = ∞ is set to a fixed point P and a fixed point Q, the fixed point equation is as follows:

becomes.

Taking the solutions of the fixed-point equations as λ _P ² and λ _Q ² , we obtain the following relationship.

In the optimum tuning, since the fixed point P and the fixed point Q have the same response magnification, the following equation holds.

Substituting (A) and (B) of [Equation 73] into [Equation 80], the third D. M. The "optimal tuning formula" of tuning system 3 is

becomes.

D. T ₁ and T _1,DM are in tune with the natural period of the first mode and the first mode when ₁ c _d =0. M. means the natural period of the mode. ₁ T ₀ means the natural period of the primary mode when _{1 cd} =0 and ₁ k _d =0, and ₁ T _∞ means the natural period of the primary mode when _{1 cd} =∞ and ₁ k _d =∞. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG.

(4) Fourth D. M. For tuning system 4:
For the multi-mass point system numerical analysis model of the building 100 shown in FIG. 8, the order n of the mass point and the order j of the controlled object mode are each set to 1, X is the one-story relative displacement of the model, and Y is the input displacement of the harmonic ground motion.

becomes.

where ₁ κ _k = ₁ k _d /k ₁ , ₁ κ _m = ₁ m _d /m ₁ , h _md = ₁ cm d /(2 √ m _{1 ·} k ₁ ), h _kd = _{1 ckd} / (2 √ m _{1 ·} k ₁ ), λ = _ω / 1 ω ₀ and _ω and ₁ ω ₀ are These are the natural circular frequency of the harmonic ground motion and the natural circular frequency of the first-order mode of the analytical model when ₁ κ _k =0.

Using the relative displacement response magnification curve, if the intersection of the response magnification curve at h _md =0, h _kd =0 and the response magnification curve at ₁ κ _k =0 is set to a fixed point P and a fixed point Q, the fixed point equation is as follows:

becomes.

Taking the solutions of the fixed-point equations as λ _P ² and λ _Q ² , we obtain the following relationship.

In the optimum tuning, since the fixed point P and the fixed point Q have the same response magnification, the following equation holds.

By substituting (A) and (B) of Equation 85 into Equation 86 and arranging, the fourth D. M. The "optimal tuning formula" of the tuning system 4 is

becomes.

D. T ₁ and T _1,DM are in tune with the natural period of the first mode and the first mode when _{1 cmd} =0, _{1 ckd} =0. M. ₁ T ₀ means the natural period of the mode when ₁ k _d =0. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

- Derivation of the "optimum attenuation formula" in Fig. 74 (1) First D. M. For tuning system 1:
The response magnification of the fixed point P and the fixed point Q can be expanded as shown in the following formula using Equation 68.

Substituting (A) and (C) of [Equation 67] into [Equation 89], we get

becomes.

Assuming that the response magnification at resonance is 1/(2h ₁ ), the first D.I. M. The "optimal damping equation" for tuning system 1 is obtained as follows.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

(2) the second D.I. M. For tuning system 2:
The response magnification of the fixed point P and the fixed point Q can be expanded as shown in the following formula using Equation 74.

Substituting (A) and (C) of [Equation 73] into [Equation 93], we get

becomes.

Assuming that the response magnification at resonance is 1/(2h ₁ ), the second D.I. M. The "optimal damping equation" for tuning system 2 is obtained as follows.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

(3) Third D. M. For tuning system 3:
The response magnification of fixed point P and fixed point Q can be expanded as shown in the following formula using [Equation 80].

Substituting (A) and (C) of [Equation 79] into [Equation 97], we get

becomes.

Assuming that the response magnification at resonance is 1/(2h ₁ ), the third D.I. M. The "optimum damping formula" for the tuning system 3 is obtained as follows.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _m is the added weight ratio, and the derivation process is the same as the design formula in FIG. 9 .

(4) Fourth D. M. For tuning system 4:
The response magnification of the fixed point P and the fixed point Q can be expanded as shown in the following formula using Equation 86.

Substituting (A) and (C) of [Formula 85] into [Formula 101], we get

becomes.

Assuming that the response magnification at resonance is 1/(2h ₁ ), the fourth D. M. The "optimal damping equation" for tuning system 4 is obtained as follows.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

<For the design formula in FIG. 75>
Next, the case of using the design formula shown in FIG. 75 described above will be described.

In this specification setting method, the controlled object mode (arbitrary vibration mode) of the building 100 and the D. M. A controlled object mode or a D.M. By giving the M. mode the optimum viscous damping constant, the D. mode is designed to effectively reduce the vibration of the controlled mode. M. The specifications of the element 10, the damping element 20, the first damping element 22, the second damping element 24 and the rigid element 30 are set.

- Derivation of the "optimum tuning formula" in Fig. 75 (1) First D. M. For tuning system 1:
For the multi-mass system numerical analysis model of the building 100 shown in FIG. 5, the order n of the mass points and the order j of the controlled object mode are each set to 1, X is the one-story relative displacement of the model, and Y is the input displacement of the harmonic ground motion.

becomes.

Here, ₁ κ _k = ₁ k _d /k ₁ , ₁ κ _m = ₁ m _d /m ₁ , hd = _{1 cd /} (2√m _{1 ·} k ₁ ), λ = ω / ₁ ω ₀ , where ω and ₁ ω ₀ are the natural circular frequency of the harmonic ground motion and the natural circular vibration of the first mode of the analytical model when ₁ κ _k = 0 is a number.

Using the absolute acceleration response magnification curve, if the intersection of the response magnification curves of _hd = 0 and _hd = ∞ is set to a fixed point P and a fixed point Q, the fixed point equation is:

becomes.

Taking the solutions of the fixed-point equations as λ _P ² and λ _Q ² , we obtain the following relationship.

In the optimum tuning, since the fixed point P and the fixed point Q have the same response magnification, the following equation holds.

Substituting [Equation 107] into [Equation 108], the first D. M. The "optimal tuning formula" of tuning system 1 is

becomes.

D. T ₁ and T _1,DM are in tune with the natural period of the first mode and the first mode when ₁ c _d =0. M. means the natural period of the mode, and ₁ T _∞ means the natural period of the first mode when _{1 cd} =∞ and ₁ m _d =∞. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

(2) the second D.I. M. For tuning system 2:
For the multi-mass system numerical analysis model of the building 100 shown in FIG. 6, the order n of the mass points and the order j of the controlled object mode are each set to 1, X is the one-story relative displacement of the model, and Y is the input displacement of the harmonic ground motion.

becomes.

Here, ₁ κ _k = ₁ k _d /k ₁ , ₁ κ _m = ₁ m _d /m ₁ , hd = _{1 cd /} (2√m _{1 ·} k ₁ ), λ = ω / ₁ ω ₀ , where ω and ₁ ω ₀ are the natural circular frequency of the harmonic ground motion and the natural circular vibration of the first mode of the analytical model when ₁ κ _k = 0 is a number.

Using the absolute acceleration response magnification curve, if the intersection of the response magnification curves of _hd = 0 and _hd = ∞ is set to a fixed point P and a fixed point Q, the fixed point equation is:

becomes.

Taking the solutions of the fixed-point equations as λ _P ² and λ _Q ² , we obtain the following relationship.

In the optimum tuning, since the fixed point P and the fixed point Q have the same response magnification, the following equation holds.

Substituting (A) and (B) of [Equation 113] into [Equation 114], the second D. M. The "optimal tuning formula" of tuning system 2 is

becomes.

D. T ₁ and T _1,DM are in tune with the natural period of the first mode and the first mode when ₁ c _d =∞. M. ₁ T ₀ means the natural period of the mode when _{1 cd} =0, ₁ m _d =0, and ₁ k _d =0. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG.

(3) Third D. M. For tuning system 3:
For the multi-mass system numerical analysis model of the building 100 shown in FIG. 7, the order n of the mass points and the order j of the controlled object mode are each set to 1, X is the one-story relative displacement of the model, and Y is the input displacement of the harmonic ground motion.

becomes.

Here, ₁ κ _k = ₁ k _d /k ₁ , ₁ κ _m = ₁ m _d /m ₁ , hd = _{1 cd /} (2√m _{1 ·} k ₁ ), λ = ω / ₁ ω ₀ , where ω and ₁ ω ₀ are the natural circular frequency of the harmonic ground motion and the natural circular vibration of the first mode of the analytical model when ₁ κ _k = 0 is a number.

Using the absolute acceleration response magnification curve, if the intersection of the response magnification curve when _hd = 0 and the response magnification curve when ₁ κ _k = 0 and the natural circular frequency ratio is (1 + ₁ κ _m )λ ² is a fixed point P and a fixed point Q, then the fixed point equation is as follows:

becomes.

Taking the solutions of the fixed-point equations as λ _P ² and λ _Q ² , we obtain the following relationship.

In the optimum tuning, since the fixed point P and the fixed point Q have the same response magnification, the following equation holds.

Substituting (A) and (B) of [Equation 119] into Equation 120, the third D. M. The "optimal tuning formula" of tuning system 3 is

becomes.

D. T ₁ and T _1,DM are in tune with the natural period of the first mode and the first mode when ₁ c _d =0. M. means the natural period of the mode, and ₁ T _∞ means the natural period of the first mode when _{1 cd} =∞ and ₁ k _d =∞. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _m is the added weight ratio, and the derivation process is the same as the design formula in FIG. 9 .

(4) Fourth D. M. For tuning system 4:
For the multi-mass system numerical analysis model of the building 100 shown in FIG. 8, the order n of the mass points and the order j of the controlled object mode are each set to 1, X is the one-story relative displacement of the model, and Y is the input displacement of the harmonic ground motion.

becomes.

where ₁ κ _k = ₁ k _d /k ₁ , ₁ κ _m = ₁ m _d /m ₁ , h _md = ₁ cm d /(2 √ m _{1 ·} k ₁ ), h _kd = _{1 ckd} / (2 √ m _{1 ·} k ₁ ), λ = _ω / 1 ω ₀ and _ω and ₁ ω ₀ are These are the natural circular frequency of the harmonic ground motion and the natural circular frequency of the first-order mode of the analytical model when ₁ κ _k =0.

Using the absolute acceleration response magnification curve, if the intersection point of the response magnification curve at h _md =0, h _kd =0 and the response magnification curve at ₁ κ _k =0 is set to a fixed point P and a fixed point Q, the fixed point equation is as follows:

becomes.

Taking the solutions of the fixed-point equations as λ _P ² and λ _Q ² , we obtain the following relationship.

In the optimum tuning, since the fixed point P and the fixed point Q have the same response magnification, the following equation holds.

Substituting (A) and (B) of [Equation 125] into [Equation 126], the fourth D. M. The "optimal tuning formula" of the tuning system 4 is

becomes.

D. T ₁ and T _1,DM are in tune with the natural period of the first mode and the first mode when _{1 cmd} =0, _{1 ckd} =0. M. ₁ T ₀ means the natural period of the mode when ₁ k _d =0. In other words, it can be seen that the "optimum tuning formula" is a relational expression of the natural period.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG.

・Derivation of the “optimal damping formula” in FIG.
(1) The first D.I. M. For tuning system 1:
The response magnification of fixed point P and fixed point Q can be expanded as shown in the following equation using [Equation 108].

Substituting (A) and (C) of [Equation 107] into [Equation 129], we get

becomes.

Assuming that the response magnification at resonance is 1/(2h ₁ ), the first D.I. M. The "optimal damping equation" for tuning system 1 is obtained as follows.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

(2) the second D.I. M. For tuning system 2:
The response magnification of fixed point P and fixed point Q can be expanded as shown in the following equation using [Equation 114].

Substituting (A) and (C) of [Equation 113] into [Equation 133], we get

becomes.

Assuming that the response magnification at resonance is 1/(2h ₁ ), the second D.I. M. The "optimal damping equation" for tuning system 2 is obtained as follows.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

(3) Third D. M. For tuning system 3:
The response magnification of fixed point P and fixed point Q can be expanded as shown in the following equation using [Equation 120].

Substituting (A) and (B) of [Equation 119] into [Equation 137], we get

becomes.

Assuming that the response magnification at resonance is 1/(2h ₁ ), the third D.I. M. The "optimum damping formula" for the tuning system 3 is obtained as follows.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _m is the added weight ratio, and the derivation process is the same as the design formula in FIG. 9 .

(4) Fourth D. M. For tuning system 4:
The response magnification of fixed point P and fixed point Q can be developed as shown in the following equation using [Equation 126].

Substituting (A) and (B) of [Equation 125] into [Equation 141], we get

becomes.

Assuming that the response magnification at resonance is 1/(2h ₁ ), the fourth D. M. The "optimal damping equation" for tuning system 4 is obtained as follows.

Also, if the number of vibration modes of the numerical analysis model of the building 100 shown in FIG. Note that _j κ _k is the additional stiffness ratio, and the derivation process is the same as the design formula in FIG. 9 .

<Others>
The present invention is not limited to the above embodiments.

For example, in the example of the setting method described using the [flowchart of setting method of specifications] in FIG. 13, the specifications are set so as to satisfy both the optimum tuning and the optimum attenuation, but the present invention is not limited to this. The specifications may be set so as to satisfy at least one of optimum tuning and optimum attenuation. In other words, the specifications of the damping device may be set so that the damping target mode is optimally tuned or damped.

In the case of only the optimum tuning, the specifications of the inertia mass element and the rigidity element constituting the vibration damping device may be set using the optimum tuning formula and one of the added stiffness ratio and the added weight ratio in the setting method described above.

In the case of only the optimum damping, the specifications of the damping elements constituting the vibration damping device may be set using the optimum damping formula, one of the additional stiffness ratio or the additional weight ratio, and the optimum viscous damping constant in the setting method described above.

In addition, in the example of the setting method explained using the [flowchart of setting method of specifications] of FIG. M. The mode may be given an optimum viscous damping constant. It should be noted that D.I. M. Even when giving the optimum viscous damping constant to the mode, the setting method is the same as described above.

For example, in the above embodiment, the viscous damping constant was calculated with the damping adjustment coefficient α of the optimum damping formula shown in FIG. 9 set to 2 (α=2), but the present invention is not limited to this. It is sufficient that the attenuation adjustment coefficient α is 1 or more.

Further, for example, in the above embodiment, the present invention is applied to a single-story building 101 or a multi-story high-rise building such as buildings 100, 200, and 300 as an example of a structure, but the present invention is not limited to these. For example, the present invention can be applied to base-isolated buildings such as basic base-isolated structures and intermediate base-isolated structures, architectural structures such as steel towers, civil engineering structures such as bridges, base-isolated structures in buildings such as base-isolated floors and ceilings, and vibration control devices installed in rack warehouses. In short, the present invention can be applied to a vibration damping device installed in a structure having multiple vibration modes.

Also, the present invention can be similarly applied to a vibration damping device provided in a structure having only one vibration mode to achieve optimum tuning or optimum damping. In this case, the "controlled object mode" is one vibration mode of the building.

Also, the damping device installed in the structure is the first D.I. M. tuning system, the second D.I. M. tuning system, the third D.C. M. A tuning system and a fourth D.I. M. A tuning system (including the D.M. tuning system 3 and 4 considering the mounting stiffness factor) may be used in combination.

Also, the vibration of the structure is not limited to that caused by an earthquake. For example, it may be vibration of a structure caused by wind. Alternatively, it may be due to a vibration source inside or outside the structure. Note that the vibration sources include facility equipment such as motors, running automobiles, heavy machinery, jumping of spectators in concert halls, and the like.

Furthermore, various aspects can be implemented without departing from the gist of the present invention.

1 First D. M. Tuning system (an example of a damping device)
2 Second D. M. Tuning system (an example of a damping device)
3 Third D. M. Tuning system (an example of a damping device)
4 Fourth D. M. Tuning system (an example of a damping device)
10 inertial mass element 11 first element 12 second element 20 damping element 22 first damping element 24 second damping element 30 rigid element 50 mounting rigid element 100 building 101 building 200 building 300 building

Claims (20)

A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
any one of the one vibration mode or the plurality of vibration modesFind the natural period using complex eigenvalue analysis for the structural vibration mode of ,
The structure vibration mode for which the natural period is obtained is The damping device occurs in conjunction with the structureCouplingSetting the specifications of the inertial mass element, the damping element, and the rigid element that constitute the vibration damping device using an optimum tuning formula, an optimum damping formula, one of an additional stiffness ratio or an additional weight ratio, and an optimum viscous damping constant so as to achieve optimum tuning and optimum damping depending on the vibration mode;
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
any one of the one vibration mode or the plurality of vibration modesFind the natural period using complex eigenvalue analysis for the structural vibration mode of ,
The structure vibration mode for which the natural period is obtained is The damping device occurs in conjunction with the structureCouplingsetting the specifications of the inertial mass element and the rigid element that make up the vibration damping device using an optimum tuning formula and one of the added stiffness ratio and the added weight ratio so as to achieve optimal tuning according to the vibration mode;
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
any one of the one vibration mode or the plurality of vibration modesFind the natural period using complex eigenvalue analysis for the structural vibration mode of ,
The structure vibration mode for which the natural period is obtained is The damping device occurs in conjunction with the structureCouplingSetting the specifications of the damping element that constitutes the vibration damping device using an optimum damping formula, one of the additional stiffness ratio or the additional weight ratio, and the optimum viscous damping constant so as to achieve optimum damping depending on the vibration mode,
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the inertia mass element, the damping element, and the rigid element that constitute the vibration damping device, using an optimum tuning formula, an optimum damping formula, one of an added stiffness ratio or an added weight ratio, and an optimum viscous damping constant, so that the one vibration mode or any one of the plurality of vibration modes is optimally tuned and damped by a new vibration mode generated by the vibration damping device coupled with the structure;
setting a target performance of the magnitude of shaking of the structure at the time of disturbance;
a step of setting installation positions and installation numbers of the vibration damping devices so as to satisfy the target performance;
having
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the inertial mass element and the rigid element constituting the vibration damping device, using an optimum tuning formula and one of the added stiffness ratio and the added weight ratio, so that the one vibration mode or any one of the plurality of vibration modes is optimally tuned by a new vibration mode generated by the vibration damping device coupled with the structure;
setting a target performance of the magnitude of shaking of the structure at the time of disturbance;
a step of setting installation positions and installation numbers of the vibration damping devices so as to satisfy the target performance;
having
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the damping element constituting the vibration damping device using an optimum damping formula, one of the added stiffness ratio or the added weight ratio, and the optimum viscous damping constant so that the one vibration mode or any one of the plurality of vibration modes is optimally damped by a new vibration mode generated by the vibration damping device coupled with the structure;
setting a target performance of the magnitude of shaking of the structure at the time of disturbance;
a step of setting installation positions and installation numbers of the vibration damping devices so as to satisfy the target performance;
having
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the inertia mass element, the damping element, and the rigid element that constitute the vibration damping device, using an optimum tuning formula, an optimum damping formula, one of an added stiffness ratio or an added weight ratio, and an optimum viscous damping constant, so that the one vibration mode or any one of the plurality of vibration modes is optimally tuned and damped by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
wherein the inertial mass element, the damping element and the rigid element are arranged in series;
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the inertial mass element and the rigid element constituting the vibration damping device, using an optimum tuning formula and one of the added stiffness ratio and the added weight ratio, so that the one vibration mode or any one of the plurality of vibration modes is optimally tuned by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
wherein the inertial mass element, the damping element and the rigid element are arranged in series;
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the damping element constituting the vibration damping device using an optimum damping formula, one of the added stiffness ratio or the added weight ratio, and the optimum viscous damping constant so that the one vibration mode or any one of the plurality of vibration modes is optimally damped by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
wherein the inertial mass element, the damping element and the rigid element are arranged in series;
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the inertia mass element, the damping element, and the rigid element that constitute the vibration damping device, using an optimum tuning formula, an optimum damping formula, one of an added stiffness ratio or an added weight ratio, and an optimum viscous damping constant, so that the one vibration mode or any one of the plurality of vibration modes is optimally tuned and damped by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
the damping element and the rigid element are arranged in parallel,
the inertial mass element is arranged in series with the damping element and the rigid element arranged in parallel;
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the inertial mass element and the rigid element constituting the vibration damping device, using an optimum tuning formula and one of the added stiffness ratio and the added weight ratio, so that the one vibration mode or any one of the plurality of vibration modes is optimally tuned by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
the damping element and the rigid element are arranged in parallel,
the inertial mass element is arranged in series with the damping element and the rigid element arranged in parallel;
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the damping element constituting the vibration damping device using an optimum damping formula, one of the added stiffness ratio or the added weight ratio, and the optimum viscous damping constant so that the one vibration mode or any one of the plurality of vibration modes is optimally damped by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
the damping element and the rigid element are arranged in parallel,
the inertial mass element is arranged in series with the damping element and the rigid element arranged in parallel;
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the inertia mass element, the damping element, and the rigid element that constitute the vibration damping device, using an optimum tuning formula, an optimum damping formula, one of an added stiffness ratio or an added weight ratio, and an optimum viscous damping constant, so that the one vibration mode or any one of the plurality of vibration modes is optimally tuned and damped by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
The damping element has a first damping element and a second damping element,
A first element in which the inertial mass element and the first damping element are arranged in parallel, and a second element in which the rigid element and the second damping element are arranged in parallel are arranged in series.
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the inertial mass element and the rigid element constituting the vibration damping device, using an optimum tuning formula and one of the added stiffness ratio and the added weight ratio, so that the one vibration mode or any one of the plurality of vibration modes is optimally tuned by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
The damping element has a first damping element and a second damping element,
A first element in which the inertial mass element and the first damping element are arranged in parallel, and a second element in which the rigid element and the second damping element are arranged in parallel are arranged in series.
How to set the specifications of the damping device. A method for setting specifications of a vibration damping device provided in a structure having one or more vibration modes and having an inertial mass element, a damping element and a rigid element, comprising:
setting the specifications of the damping element constituting the vibration damping device using an optimum damping formula, one of the added stiffness ratio or the added weight ratio, and the optimum viscous damping constant so that the one vibration mode or any one of the plurality of vibration modes is optimally damped by a new vibration mode generated by the vibration damping device coupled with the structure;
The damping device is
In the numerical analysis model,
The damping element has a first damping element and a second damping element,
A first element in which the inertial mass element and the first damping element are arranged in parallel, and a second element in which the rigid element and the second damping element are arranged in parallel are arranged in series.
How to set the specifications of the damping device. setting a target performance of the magnitude of shaking of the structure at the time of disturbance;
a step of setting installation positions and installation numbers of the vibration damping devices so as to satisfy the target performance;
having
A method for setting specifications of a vibration damping device according to any one of claims 1 to 3 and claims 7 to 15. The damping device is
In the numerical analysis model,
the inertial mass element and the damping element are arranged in parallel,
the rigid element is arranged in series with the inertial mass element and the damping element arranged in parallel;
A method for setting specifications of a vibration damping device according to any one of claims 1 to 16. The damping device is
In the numerical analysis model,
wherein the inertial mass element, the damping element and the rigid element are arranged in series;
A method for setting specifications of a vibration damping device according to any one of claims 1 to 6 and claims 10 to 16. The damping device is
In the numerical analysis model,
the damping element and the rigid element are arranged in parallel,
the inertial mass element is arranged in series with the damping element and the rigid element arranged in parallel;
A method for setting specifications of a vibration damping device according to any one of claims 1 to 9 and claims 12 to 16. The damping device is
In the numerical analysis model,
The damping element has a first damping element and a second damping element,
A first element in which the inertial mass element and the first damping element are arranged in parallel, and a second element in which the rigid element and the second damping element are arranged in parallel are arranged in series.
A method for setting specifications of a vibration damping device according to any one of claims 1 to 12 and 16.