JP7529459B2 - Anti-vibration mechanism - Google Patents

Description

The present invention relates to an anti-vibration mechanism.

Machines that generate large vertical vibrations at a certain frequency, such as rotary presses and presses, will cause significant vibration damage to the surrounding area if they are installed directly on a foundation, so they are often installed on a floating foundation with spring elements such as air springs between them. Also, in facilities such as live music studios, large numbers of people exercise to the music, causing vibration damage to surrounding buildings, so there is concern that this will cause vibration damage to surrounding buildings, and vibration prevention measures are required.
A common vibration isolation measure to avoid such vibration disorders is to use a vibration isolation mechanism that supports the structure via soft spring elements as a floating floor or floating foundation, rather than integrating the floor or foundation on which the vibration sources, people and equipment, rest, into the structure.

As a specific example, the floating foundation of the shaking table is shown in FIG. 6, and a vibration model when a floating foundation or a floating floor is applied is shown in FIG.
As shown in Fig. 6, a floating foundation 12 is installed on an outer foundation 11 via an air spring 13. When an excitation force F is input to the floating foundation 12, a reaction force R acts on the outer foundation 11.
7, in the vibration isolation mechanism 100A, a support spring element 13 with spring stiffness K and a damping mechanism 14 with a damping coefficient C are provided between a structural floor 11 and a floating floor 12 with a mass M that is provided so as to be displaceable relative to the structural floor 11. The support spring element 13 and the damping mechanism 14 are arranged in parallel.

Figure 8 shows the reaction force response magnification to the excitation force as a frequency transfer function for the vibration model shown in Figure 7. As shown in Figure 8, the reaction force magnification at a frequency of 1 Hz (at resonance) decreases as the damping constant increases, but in the high frequency range, the smaller the damping constant, the smaller the reaction force (improving vibration isolation performance), so the damping constant h is often set to about 0.05. In general, the resonance frequency is set to 1/2 or less of the frequency of the object to be isolated, so there is little possibility of a large excitation input occurring at the resonance frequency. However, in the unlikely event of excitation at 1 Hz, a reaction force 10 times the excitation force is generated, which could damage the spring elements or cause the floating floor to collide with the structural floor.

Therefore, the following Patent Document 1 proposes a system that improves vibration isolation performance while simultaneously reducing response in the resonance range.

FIG. 9 shows a vibration model disclosed in the first embodiment of Patent Document 1.
As shown in Fig. 9, in the vibration isolation mechanism 100B, a support spring element 13 with spring stiffness K, a series spring element 21 with spring stiffness _kd , an inertial mass damper 22 with inertial mass ψ _d , and a damping mechanism 23 with a damping coefficient c _d1 or a damping mechanism 24 with a damping coefficient c _d2 are provided between a structural floor 11 and a floating floor 12 with mass M that is provided so as to be displaceable relative to the structural floor 11. The series spring element 21 and the inertial mass damper 22 are arranged in series and in parallel with the support spring element 13. The series spring element 21 and the damping mechanism 23 are arranged in parallel. The inertial mass damper 22 and the damping mechanism 24 are arranged in parallel. The damping coefficient c _d1 of the damping mechanism 23 is set to 0.

Figure 10 shows the graph of Figure 8 with the addition of anti-vibration mechanism 100B. As shown in Figure 10, in the high frequency range of 3 Hz or more, the anti-vibration characteristics are superior to those obtained when the damping constant is increased to make the peak in the resonance range the same (for example, damping constant h = 0.2), but there is an issue that the results are inferior at frequencies below 3 Hz.

The reaction force generated by vibrations at a certain frequency (rhythm), such as from mechanical vibrations or live music performances, often causes vibration disorders and becomes a problem. While these facilities have been seen as nuisances and have been relocated to the suburbs, there is also a demand to locate them in urban areas with good transport links, and so a solution that can deal with this effectively without causing resonance problems is needed.

On the other hand, a method has been proposed in which an inertial mass damper is placed in parallel with a spring element to significantly reduce reaction force in a specific frequency range (see Patent Document 2 below).

However, the vibration isolation mechanism described in Patent Document 2 also has the same characteristic of a large response magnification at the resonance frequency. In this case, if the damping constant is increased, the response magnification at resonance will decrease, but the reaction force reduction effect in a specific frequency range will decrease. Figure 11 shows a vibration model in which an inertial mass ψ = 0.17M is added in parallel to the spring element in the vibration model shown in Figure 7, and Figure 12 shows the transfer function (reaction force response magnification). In the vibration isolation mechanism 100C shown in Figure 11, a support spring element 13 with spring stiffness K, a damping mechanism 14 with damping coefficient C, and an inertial mass damper 15 with inertial mass ψ are provided between the structure floor 11 and the floating floor 12 with mass M that is provided so as to be displaceable relative to the structure floor 11. The support spring element 13, the damping mechanism 14, and the inertial mass damper 15 are arranged in parallel. The vibration isolation mechanism 100C is a mechanism that significantly attenuates the reaction force against the excitation force to about 1/10 at 2 to 4 Hz. In this vibration model, as the damping constant increases, the response magnification at resonance decreases, but the effect of significantly reducing the reaction force response magnification near 2.5 Hz becomes smaller. This means that there is a trade-off between reducing the response magnification at resonance and significantly reducing the reaction force response magnification near 2.5 Hz, and that conventional vibration isolation technology cannot achieve both.

For this reason, the applicant has devised a vibration isolation system that can reduce damping to ensure vibration isolation performance in the high frequency range while suppressing excessive response at the resonance point (Patent Document 1 below).

FIG. 13 shows a vibration model disclosed in the second embodiment of Patent Document 1.
As shown in Fig. 13, in the vibration isolation mechanism 100D, a support spring element 13 with spring stiffness K, an inertial mass damper 25 with inertial mass ψ, a series spring element 21 with spring stiffness kd, an inertial mass damper 22 with inertial mass _ψd , and a damping mechanism 23 with a damping coefficient c _d1 or a damping mechanism ₂₄ with a damping coefficient c _d2 are provided between a structural floor 11 and a floating floor 12 with a mass M that is provided so as to be displaceable relative to the structural floor 11. The series spring element 21 and the inertial mass damper 22 are arranged in series and in parallel with the support spring element 13. The inertial mass damper 25 is arranged in parallel with the support spring element 13. The series spring element 21 and the damping mechanism 23 are arranged in parallel. The inertial mass damper 22 and the damping mechanism 24 are arranged in parallel. The damping coefficient c _d1 of the damping mechanism 23 is set to 0.

In addition, Patent Document 2 and Non-Patent Document 1 disclose that a vibration control system similar to a TMD (dynamic vibration absorber) can be constructed using an inertial mass and a series spring. The optimal values of vibration parameters according to fixed point theory are described in Non-Patent Document 2 below.

JP 2019-199904 A JP 2008-82541 A

Kazuhiko Isoda, Tetsuya Hanzawa, Kazuo Tamura, "Study on response characteristics of one-mass system vibration model using response reduction mechanism combined with rotational inertia mass damper", Journal of Structural Engineering, Architectural Institute of Japan, Vol. 74, No. 642, August 2009, pp. 1469-1476 Kazuhiko Isoda, Tetsuya Hanzawa, Kazuo Tamura, "Basic Study on Low-Rise Concentrated Vibration Control Using Inertial Mass Damper", Journal of Structural Engineering, Architectural Institute of Japan, Vol. 78, No. 686, April 2013, pp. 713-722

Figure 14 shows the graph of Figure 12 with the addition of anti-vibration mechanism 100D. As shown in Figure 14, vibration is blocked near 2.5 Hz, and excellent anti-vibration properties are exhibited, but below 2.1 Hz, the results are inferior to the case where a damping constant is added (line h = 0.1) that produces a larger peak in the resonance range, and there is concern that sufficient performance cannot be exhibited in the range of 2.0 to 3.5 Hz (near 2 Hz) where vertical vibration is a problem. In addition, the excitation vibration frequency during walking is about 1.6 Hz, and at this frequency, the peak in the resonance range is inferior to the conventional technology (line h = 0.2) that uses only the same spring and damping, resulting in a characteristic that shows almost no anti-vibration effect, so it is desired to improve the anti-vibration properties in the range higher than the resonance frequency (up to about twice the frequency).

Furthermore, the vibration isolation mechanisms described in Patent Document 2 and Non-Patent Document 1 above are not capable of preventing resonance of the floating floor while exhibiting greater vibration isolation effects than conventional mechanisms across the entire frequency range above the resonant frequency.

Therefore, the present invention has been made in consideration of the above circumstances, and provides an anti-vibration mechanism that exhibits excellent vibration frequency characteristics (obtaining a large vibration-damping effect over the entire frequency range above the resonant frequency while preventing resonance of the vibrating body).

In order to achieve the above object, the present invention employs the following means.
In other words, the vibration-damping mechanism of the present invention is a vibration-damping mechanism for reducing the reaction force acting on a structure when a vibrating body installed on a structure via a support spring element is vibrated, and is characterized in that it comprises a series spring and a first inertial mass damper which are installed in parallel with the support spring element between the structure and the vibrating body and are arranged in series with each other so as to be in tune with the natural frequency of the vibrating body, and which is provided with a first damping mechanism arranged in parallel with the series spring, and no second damping mechanism arranged in parallel with the first inertial mass damper.

In the vibration isolation mechanism configured in this manner, the first inertial mass damper and series spring, which are arranged in series with each other between the structure and the vibrating body, are arranged in parallel with the support spring element, and the first damping mechanism is arranged in parallel with the series spring. The second damping mechanism is not arranged in parallel with the first inertial mass damper. This shifts the reaction force response magnification (transfer function) to the low frequency side near the resonant frequency, preventing resonance of the vibrating body while providing a large vibration isolation effect over the entire frequency range above the resonant frequency.

The vibration isolation mechanism according to the present invention preferably further comprises a second inertial mass damper disposed between the structure and the vibrating body in parallel with the support spring element.

In a vibration isolation mechanism configured in this way, by further installing a second inertial mass damper in parallel between the structure and the vibrating body, the reaction force response magnification (transfer function) is significantly reduced in a specific frequency range, making it possible to significantly reduce the reaction force.

The vibration isolation mechanism of the present invention can exhibit excellent vibration frequency characteristics (obtaining a large vibration isolation effect over the entire frequency range above the resonant frequency while preventing resonance of the floating floor).

5A and 5B are diagrams illustrating an example of a vibration model of the vibration isolation mechanism according to the first embodiment of the present invention. 1 is a graph showing the transfer function of reaction forces (the sum of reaction forces acting on a structural floor) for a vibration model of the vibration isolation mechanism according to the first embodiment of the present invention and a conventional vibration model. FIG. 11 is a diagram showing an example of a vibration model of an anti-vibration mechanism according to a second embodiment of the present invention. 13 is a graph showing the transfer function of reaction forces (the sum of reaction forces acting on the structural floor) for a vibration model of the vibration isolation mechanism according to the second embodiment of the present invention and a conventional vibration model. FIG. 2 is a cross-sectional view showing a schematic diagram of an inertial mass damper of an anti-vibration mechanism according to a first embodiment and a second embodiment of the present invention. FIG. 1 is a diagram showing a conventional floating foundation structure. FIG. 7 is a diagram showing a vibration model of the vibration isolation mechanism of FIG. 6 . 8 is a graph showing a relationship between an excitation frequency and a reaction force response magnification with respect to an excitation force in the vibration model shown in FIG. 7 . FIG. 1 is a diagram showing a vibration model of a conventional vibration isolation mechanism. The vibration isolation mechanism shown in FIG. 7 is added to the graph in FIG. FIG. 1 is a diagram showing a vibration model of a conventional vibration isolation mechanism. 12 is a graph showing the relationship between the excitation frequency and the reaction force response magnification with respect to the excitation force in the vibration model shown in FIG. 11 . FIG. 1 is a diagram showing a vibration model of a conventional vibration isolation mechanism. The graph in FIG. 12 is supplemented with the vibration isolation mechanism shown in FIG.

First Embodiment
An anti-vibration mechanism according to a first embodiment of the present invention will be described with reference to FIGS. 1 and 2. FIG.
FIG. 1 is a diagram showing an example of a vibration model of an anti-vibration mechanism according to a first embodiment of the present invention.
1, an anti-vibration mechanism 1A according to this embodiment is provided between a structural floor (structural body) 11 and a floating floor (vibrating body) 12 of mass M that is provided so as to be displaceable relative to the structural floor 11. The anti-vibration mechanism 1A includes a support spring element 13 with spring stiffness K, a series spring element (series spring) 21 with spring stiffness _kd , an inertial mass damper (first inertial mass damper) 22 with an inertial mass _ψd , and a first damping mechanism 23 with a damping coefficient _cd1 .

The series spring element 21 and the inertial mass damper 22 are arranged in series. The series spring element 21 and the inertial mass damper 22 are arranged in parallel with the support spring element 13 between the structural floor 11 and the floating floor 12. The series spring element 21 and the first damping mechanism 23 are arranged in parallel. The second damping mechanism 24 is not provided in parallel with the inertial mass damper 22. In the drawing, the damping coefficient c _d2 of the second damping mechanism 24 is shown as 0, which indicates that the second damping mechanism 24 is not provided in parallel with the inertial mass damper 22.
Incidentally, the conventional damping mechanism 14 shown in FIG. 7 does not necessarily have to be provided.

Generally, damping is provided in parallel with the inertial mass damper, but in this embodiment, only the first damping mechanism 23 is provided.

When the floating floor 12 and the supporting spring elements 13 are set in a known vibration system with a series spring element 21 having a spring stiffness _kd , it is complicated to formulate the optimal values of the inertial mass _ψd of the inertial mass damper 22 and the damping coefficient _cd1 of the first damping mechanism 23 that minimize the reaction force response magnification in the resonance range. Therefore, the transfer function (reaction force response magnification) of the following equation (1) is found by iterative calculation using a spreadsheet software.

A design example will be described below.
The floating floor 12 has a mass M of 1000 tons, and the supporting spring element 13 has a spring stiffness K of 39.5 kN/mm. The natural frequency of this system is f ₁ =1 Hz.
When the spring stiffness k _d of the series spring element 21 is set to 0.2K=7.9 kN/mm, the optimum specifications are obtained by the above method, as given by the following formula (2).

On the other hand, when a large damping constant of h = 0.2 (damping coefficient C = 2513 kN·s/m = 25.13 kN/kine for the design example specifications) is applied to the conventional vibration model shown in Figure 7, the transfer function is as shown in Figure 8, and the maximum response magnification at the resonance point is 2.73.

Fig. 2 is a graph showing the transfer function of reaction forces (the sum of reaction forces acting on the structural floor) for the vibration model of the vibration isolation mechanism according to the first embodiment of the present invention and the conventional vibration model shown in Fig. 7. The horizontal axis shows the excitation frequency f (Hz), and the vertical axis shows the reaction force response magnification R/F, which is shown on a logarithmic scale.
As shown in Fig. 2, it can be seen that with the anti-vibration mechanism 1A (the present invention), the response magnification in the resonance region can be reduced (similar to a vibration system with a large damping constant) without increasing the response magnification in the high frequency region above the resonance frequency (while maintaining it similar to a vibration system with a small damping constant in the prior art). The anti-vibration mechanism 1A (the present invention) has a characteristic that the maximum response magnification in the resonance region is 2.58 times, which is similar to the high damping structure with h = 0.2, and has a characteristic that it has almost no resonance, but in the high frequency range of 1.5 Hz or more, it is smaller than h = 0.2, and the resonance characteristics can be improved while maintaining the vibration isolation performance.

In the vibration isolation mechanism 1A configured in this manner, the inertial mass damper 22 and the series spring element 21, which are arranged in series with each other, are arranged in parallel with the support spring element 13 between the structural floor 11 and the floating floor 12, and the first damping mechanism 23 is arranged in parallel with the series spring element 21. The second damping mechanism 24 is not arranged in parallel with the inertial mass damper 22. This shifts the reaction force response magnification (transfer function) to the low frequency side near the resonance frequency, and a large vibration isolation effect can be obtained over the entire frequency range above the resonance frequency while preventing the vibration body from resonating. The conventional technology in Patent Document 1 has the vibration characteristics (reaction force response magnification) shown in Figure 10, while the present application, which has the vibration characteristics shown in Figure 2, has a vibration isolation effect superior to that of the conventional technology in the frequency range above the resonance frequency.

In addition, because the ratio of the minimum vibration frequency that exerts the vibration-damping effect to the resonant vibration frequency can be made smaller than before, it is possible to set a higher natural frequency (resonant vibration frequency) of the floating floor 12 for the vibration frequency to be damped, suppressing sinking of the floating floor 12 and preventing it from becoming a soft spring.

In addition, the tuned vibration control mechanism, which consists of an inertial mass damper 22 and a series spring element 21 arranged in series, is a linear element, so it can provide stable damping characteristics regardless of the magnitude of the excitation force. Therefore, it can function effectively even during a large earthquake.

Second Embodiment
Next, a vibration isolation mechanism according to a second embodiment of the present invention will be described with reference mainly to FIGS.
In the following embodiments, the same members as those used in the above-described embodiments are designated by the same reference numerals, and the description thereof will be omitted.
FIG. 3 is a diagram showing an example of a vibration model of the vibration isolation mechanism according to the second embodiment of the present invention.
As shown in FIG. 3, in an anti-vibration mechanism 1B according to this embodiment, an inertial mass damper (second inertial mass damper) 25 having an inertial mass ψ is further provided in parallel to the anti-vibration mechanism 1A according to the first embodiment.

Specifically, the vibration isolation mechanism 1B is provided between a structural floor (structural body) 11 and a floating floor (vibrating body) 12 of mass M that is provided so as to be displaceable relative to the structural floor 11. The vibration isolation mechanism 1B includes a support spring element 13 with spring stiffness K, an inertial mass damper (second inertial mass damper) 25 with inertial mass ψ, a series spring element (series spring) 21 with spring stiffness _kd , an inertial mass damper (first inertial mass damper) 22 with inertial mass _ψd , and a first damping mechanism 23 with a damping coefficient _cd1 .

The series spring element 21 and the inertial mass damper 22 are arranged in series. Between the structural floor 11 and the floating floor 12, the series spring element 21 and the inertial mass damper 22, the support spring element 13, and the inertial mass damper 25 are arranged in parallel. The series spring element 21 and the first damping mechanism 23 are arranged in parallel. The second damping mechanism 24 is not provided in parallel with the inertial mass damper 22. In the drawing, the damping coefficient c _d2 of the second damping mechanism 24 is shown as 0, which indicates that the second damping mechanism 24 is not provided in parallel with the inertial mass damper 22.
Incidentally, the conventional damping mechanism 14 shown in FIG. 7 does not necessarily have to be provided.

Generally, damping is provided in parallel with the inertial mass damper, but in this embodiment, only the first damping mechanism 23 is provided.

When the floating floor 12 and the supporting spring elements 13 are set in a known vibration system with a series spring element 21 having a spring stiffness _kd , it is complicated to formulate the optimal values of the inertial mass _ψd of the inertial mass damper 22 and the damping coefficient _cd1 of the first damping mechanism 23 that minimize the reaction force response magnification in the resonance range. Therefore, the transfer function (reaction force response magnification) of the following equation (3) is found by iterative calculation using a spreadsheet software.

A design example will be described below.
The floating floor 12 has a mass M of 1000 tons, and the supporting spring element 13 has a spring stiffness K of 39.5 kN/mm. The natural frequency of this system is f ₁ =1 Hz.
The inertial mass ψ of the inertial mass damper 25 installed in parallel to the support spring element 13 is set to 0.2 M so that the reaction force against the excitation force is significantly attenuated to about 1/10 at 2 to 4 Hz. In addition, in a tuned vibration control mechanism in which the series spring element 21 and the inertial mass damper 22 are arranged in series, the spring stiffness of the series spring element 21 is set to k _d = 0.14 K = 5.53 kN/mm, and the optimal values for the inertial mass ψ _d of the inertial mass damper 22 and the damping coefficient c _d1 of the first damping mechanism 23 are found by iterative calculation, yielding the following equation (4).

On the other hand, when a large damping constant of h = 0.2 (damping coefficient C = 2513 kN·s/m = 25.13 kN/kine for the design example specifications) is applied to the conventional vibration model shown in Figure 11, the transfer function is as shown by the dashed line in Figure 12, and the maximum response magnification at the resonance point is 2.56.

Fig. 4 is a graph showing the transfer function of the reaction force (the sum of the reaction forces acting on the structural floor) for the vibration model of the vibration isolation mechanism according to the first embodiment of the present invention and the conventional vibration model shown in Fig. 11. The horizontal axis shows the excitation frequency f (Hz), and the vertical axis shows the reaction force response magnification R/F on a logarithmic scale.
As shown in Fig. 4, it can be seen that with the anti-vibration mechanism 1B (present invention), the response magnification in the resonance region can be reduced (similar to a vibration system with large damping, with a damping constant of h = 0.2) without increasing the response magnification in the high frequency region (while maintaining it similar to a vibration system with low damping, with a damping constant of h = 0.1 to 0.05). The anti-vibration mechanism 1B (present invention) has a characteristic that the maximum response magnification in the resonance region is 2.73 times, similar to a highly damped structure with h = 0.2, and has a characteristic that it has almost no resonance, but in the high frequency range of 1.2 Hz or more, it is reduced to about h = 0.05 to 0.1, and the resonance characteristics can be improved while maintaining the anti-vibration performance.

FIG. 5 is a cross-sectional view showing a schematic example of the inertial mass damper 22 of the vibration isolation mechanism 1A and the inertial mass dampers 22, 25 of the vibration isolation mechanism 1A.
As shown in Fig. 5, at one end side (the left side of the paper in Fig. 5) of a ball screw 101 extending in the direction of axis O, a ball nut 102 is constrained from displacement in the direction of axis O while being rotatable. At one end (the end on the right side of the paper in Fig. 5) of the ball screw 101, the ball nut 102 is constrained from rotation while being displaceable in the direction of axis O. A flywheel (rotating weight) 103 is integrated with the ball nut 102. A steel ball 104 is provided in the ball screw 101. Note that the frictional resistance between the ball screw 101 and the ball nut 102 and the rotational moment of inertia of the ball nut 102 are ignored here.

Let F be the axial force (reaction force) when the flywheel 103, which has a rotational moment of inertia _Iθ , is rotated by an angle θ due to a displacement x of the ball screw 101 in the axial O direction. If the lead (thread pitch) of the ball screw 101 _{is Ld} , the flywheel 103 is disk-shaped with a diameter D and a mass m, then x = _θLd /(2π), and thus the following equation (5) is established.

From the above formula (5), the reaction force F is proportional to the relative acceleration between the ball screw 101 and the ball nut 102, and the inertial mass ψ is the inertial mass in the direction of the axis O. Although it depends on the shape and dimensions of the flywheel 103 and the lead _Ld of the ball screw 101, the inertial mass ψ is several hundred to several thousand times the mass m of the flywheel 103, and a device can be provided in which a huge inertial mass ψ can be realized with a small mass of the flywheel 103.

In the vibration isolation mechanism 1B configured in this manner, the inertial mass damper 22 and the series spring element 21, which are arranged in series with each other, are arranged in parallel with the support spring element 13 between the structural floor 11 and the floating floor 12, and the first damping mechanism 23 is arranged in parallel with the series spring element 21. The second damping mechanism 24 is not arranged in parallel with the inertial mass damper 22. This shifts the reaction force response magnification (transfer function) to the low frequency side near the resonance frequency, and a large vibration isolation effect can be obtained over the entire frequency range above the resonance frequency while preventing the vibration body from resonating. The conventional technology in Patent Document 1 has the vibration characteristics (reaction force response magnification) shown in Figure 10, while the present application, which has the vibration characteristics shown in Figure 2, has a vibration isolation effect superior to that of the conventional technology in the frequency range above the resonance frequency.

In addition, by further installing an inertial mass damper 25 in parallel between the structural floor 11 and the floating floor 12, the reaction force response magnification (transfer function) is significantly reduced in a specific frequency range, making it possible to significantly reduce the reaction force.

The shapes and combinations of the components shown in the above-mentioned embodiment are merely examples, and may be modified in various ways based on design requirements, etc., without departing from the spirit of the present invention.

1A, 1B... Vibration isolation mechanism 11... Structure floor (structure)
12...Floating floor (vibration body)
13... Support spring element 21... Series spring element (series spring)
22...Inertial mass damper (first inertial mass damper)
23... First damping mechanism 24... Second damping mechanism 25... Inertial mass damper (second inertial mass damper)

Claims (2)

A vibration isolation mechanism for reducing a reaction force acting on a structure when a vibrator installed on the structure via a support spring element is vibrated, comprising:
a series spring and a first inertial mass damper, the series spring and the first inertial mass damper being disposed in parallel with the support spring element between the structure and the vibrating body and being arranged in series with each other so as to be in tune with a natural frequency of the vibrating body;
a first damping mechanism disposed in parallel with the series spring;
There is no second damping mechanism arranged in parallel with the first inertial mass damper;
A vibration-damping mechanism characterized by satisfying the following formula ₍₁₎ , where M is the mass of the vibrating body, K is the spring stiffness of the support spring element, kd is the spring stiffness of the series spring, _{ψd is} the inertial mass of the first inertial mass damper , and cd1 is the damping coefficient of the first damping mechanism.

A vibration isolation mechanism for reducing a reaction force acting on a structure when a vibrator installed on the structure via a support spring element is vibrated, comprising:
a series spring and a first inertial mass damper, the series spring and the first inertial mass damper being disposed in parallel with the support spring element between the structure and the vibrating body and being arranged in series with each other so as to be in tune with a natural frequency of the vibrating body;
a first damping mechanism disposed in parallel with the series spring;
a second damping mechanism is not provided in parallel with the first inertial mass damper;
a second inertial mass damper disposed in parallel with the support spring element between the structure and the vibrating body;
A vibration -damping mechanism characterized by satisfying the following formula (3), where M is the mass of the vibrating body, K is the spring stiffness of the support spring element, kd is the spring stiffness of the series spring, _{ψd is} the inertial mass of the first inertial mass damper, ψ is the inertial mass of the second inertial mass damper , and _cd1 is the damping coefficient of the first damping mechanism.

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