JP6108370B2 - Optimal design method of low-rise centralized control system using viscous damping - Google Patents

Description

  The present invention relates to a design method for a building having a damping structure, and more particularly to an optimum design method for a low-rise concentrated damping system using viscous damping.

As a means for reducing the response of multi-layer structures to earthquakes and wind loads, a low-rise centralized control system has been proposed in which damping devices are centrally arranged in the low-layer part of the structure. According to this kind of system, since there is no need to install a vibration control device in the high-rise part in the case of a new construction, the degree of freedom in building planning is increased, and in the case of existing renovation, the renovation point is limited to the lower floors. Work can be done while staying and the construction period can be shortened.
In this type of system, the lower the floor rigidity of the lower floor, the more effective, but it is difficult to reduce the rigidity of the lower floor to reduce the response in an actual structure (the building shape depends on the architectural plan, application, design, etc. Is currently determined, and structural advantages cannot be pursued). For this reason, there are many cases where a large response reduction effect cannot be expected even if a large number of seismic dampers such as oil dampers are used.

In recent years, inertia mass dampers have been developed and put into practical use that exhibit inertia mass effects that are orders of magnitude greater than the actual mass by rotating the weight using the moment of inertia of the weight.
As a low-rise centralized control system that installs such an inertial mass damper in the low-rise part of the structure and exhibits a large vibration suppression effect even for a structure with low damping, the present applicant has first described a tower-like structure. (See Patent Document 1).
This is because inertia mass dampers are arranged between the lower layers without interposing a series spring, compared with a conventional tuned vibration control mechanism using an inertia mass damper and a series spring. The inertial mass increases, but the response reduction effect is significant (response magnification can be reduced).

JP 2011-17140 A

However, in the low-rise centralized vibration control system provided in Patent Document 1, when the rigidity of the low-rise part is sufficiently larger than the rigidity of the high-rise part (the rigidity of the low-rise part is k ₁ and the rigidity of the high-rise part is K ₂ ) At this time, it is limited to k ₁ >> k ₂ ) and is not widely applicable to general structures.
Therefore, it is necessary to develop an effective and appropriate low-rise concentrated control system that can be extended to more general structures and widely applied.

  In view of the above circumstances, the present invention uses a fixed point theory to determine the optimum specifications for viscous damping so that a low-rise centralized control system as shown in Patent Document 1 can be applied to general structures. It is something that is set.

That is, the invention according to claim 1 is a low-layer concentration that suppresses the response of the multi-layer structure by adding a viscous damping element in parallel with the layer rigidity between the low-layer portions of the multi-layer structure having a small attenuation. This is an optimal design method applied to a seismic control system. The low and high layers are modeled as a single mass system, and the low layer mass m ₁ , low layer stiffness k ₁ , high layer mass m ₂ , high layer When the stiffness k _{2 of the} part is set, the damping coefficient c ₁ by the viscous damping element is set based on the following equation so as to be optimized by acceleration.

The invention according to claim 2 is a low-rise centralized vibration control for suppressing a response of the multi-layer structure by adding a viscous damping element in parallel with the layer rigidity between the layers of the low-layer portion of the multi-layer structure having a small attenuation. This is an optimal design method applied to the system, where the lower layer and the upper layer are modeled as one mass system, and the mass m _{1 of} the lower layer, the stiffness k _{1 of the} lower layer, the mass m _{2 of} the higher layer, When the stiffness is k ₂ , the damping coefficient c ₁ due to the viscous damping element is set based on the following equation in order to optimize the displacement.

  By applying the present invention to the design of a low-rise centralized control system in which a viscous damping element is installed in the lower part of a multi-layered structure with low damping, it is possible to apply viscosity under conditions that are almost valid for general structures. The optimum specifications of the damping element and the maximum response magnification as its effect can be formulated. Therefore, the low-rise concentrated control system can be widely applied to a wide range of structures, and the design work for it can be done easily and efficiently. Can be implemented.

The reference example of the low-rise concentrated control system which is an application object of the present invention is shown, (a) is a schematic diagram of a structural form, and (b) is a vibration model diagram. It is a figure which shows the example of a setting of the optimal specification by this invention with a reference example and a comparative example. It is a figure which shows the effect by the example of setting of the optimal specification by this invention with a reference example and a comparative example, (a) is a figure which shows the acceleration response magnification at the time of optimizing with acceleration, (b) is optimized with displacement It is a figure which shows the displacement response magnification in the case.

FIG. 1 shows a reference example of a low-rise concentrated vibration control system to which the optimum design method of the present invention is to be applied. FIG. 1A is a schematic diagram of a structural form, and FIG. 1B is a vibration model diagram.
This low-rise centralized vibration control system is intended for a multi-layer structure composed of a low-rise part 1 and a high-rise part 2 in the same manner as the tower-like structure damping structure disclosed in Patent Document 1. By adding an inertia mass damper 3 and an oil damper 4 as a viscous damping element in parallel between the layers, the response of the entire structure can be effectively suppressed.

In this optimum design method, the low-mass portion 1 and the high-rise portion 2 are each modeled as a one-mass system, and the inertia mass ψ ₁ by the inertia mass damper 3 is minimized so as to minimize the acceleration and displacement response magnification peaks of the high-rise portion 2. And the damping coefficient c ₁ by the oil damper 4 (viscous damping element) are optimally set based on the following methods, respectively.

In particular, in this optimum design method, the mass m _{1 of} the lower layer portion 1, the rigidity k ₁ of the lower layer portion 1, the mass m _{2 of} the higher layer portion 2, and the rigidity k ₂ of the higher layer portion 2 of the target multilayer structure On condition that the relationship of the following equation is satisfied, the inertial mass ψ ₁ by the inertial mass damper 3 and the damping coefficient c ₁ by the oil damper 4 are optimized.
In a general multilayer structure, the rigidity k ₁ of the lower layer part 1 is larger than the rigidity k _{2 of the} higher layer part 2, and the mass m ₂ of the higher layer part 2 is larger than the mass m ₁ of the lower layer part 1. The condition of the formula is satisfied in most cases, and therefore the present invention is widely applicable to general multilayer structures.

In this optimum design method, the inertial mass damper 3 and the oil damper 4 are optimized for both acceleration and displacement using a fixed point theory, and the optimization is performed for the upper mass point (mass m _{1 of the} high-rise part 2). The maximum value of the response magnification over the entire frequency range is minimized (this is the same method as a general TMD (dynamic vibration absorption mechanism)).
In this reference example, since the fixed point theory is applied, the attenuation of the main body structure (main system) is ignored.

Hereinafter, reference examples of the present invention will be specifically described.
In the optimum design method of this reference example, the inertial mass ψ ₁ and the damping coefficient c ₁ are optimized by acceleration or optimized by displacement.

In the case of optimization with acceleration, the following equation is used for the maximum response magnification for the inertial mass ψ ₁ and the damping coefficient c ₁ and the acceleration optimized thereby.

When optimizing with displacement, the inertial mass ψ ₁ and the damping coefficient c ₁ , and the maximum response magnification for the displacement optimized thereby are as follows:

If the lower layer is a multilayer, it is modeled as a one-mass system by the following.
Assuming that the lower layer mass matrix M ₁ , stiffness matrix K ₁ , natural primary mode vector φ ₁ obtained by eigenvalue analysis, and primary mode stimulation coefficient β ₁ , mass m ₁ and stiffness k ₁ are respectively expressed by the following equations.
Here, φ ₁ ^T represents a transposed vector of the mode vector φ ₁ .

It is also optimal if the damping matrix C ₁ by the oil damper arranged in each layer, the inertia mass matrix Ψ ₁ by the inertia mass damper, the damping coefficient c _{0 of} the unit oil damper, and the inertia mass ψ ₀ of the unit inertia mass damper are optimal. The damping coefficient c ₁ and the inertial mass ψ _{1 obtained} by conversion are respectively expressed by the following equations.

Incidentally, element matrix C ₀ by a damping coefficient c ₀ of the unit, the element matrix [psi ₀ by inertial mass [psi ₀ units, the unit amount in the main diagonal as with stiffness matrix as the following equation, the sign on its side The reverse unit amount. These matrices are matrixes for the unit dampers, and this number becomes n times in the n layers.

The unit inertia mass Ψ ₀ and the damping coefficient c ₀ required for the inertia mass damper and the oil damper are obtained by the following equations, respectively.

Furthermore, when the high layer part is a multilayer, a one-mass system model is made as follows.
Assuming that the mass matrix M ₂ , the stiffness matrix K _{2 of} the upper layer part, the eigen primary mode vector φ ₂ obtained by eigenvalue analysis, and the primary mode stimulation coefficient β ₂ , the mass m ₂ and the stiffness k ₂ are respectively expressed by the following equations.

  Hereinafter, with reference to the model shown in FIG. 1, an optimization method in the case where the inertial mass damper is arranged in parallel with the layer rigidity of the lower layer portion will be described in detail.

As described above, the structural damping is ignored, the mass m _{i of} each mass point, the absolute displacement x _i , the stiffness k _{i of} each layer, the inertia mass ψ ₁ provided in the lower layer, the additional damping c ₁ (the oil damper as the viscous damping element) (Attenuation coefficient), the following equation holds.

If the Fourier transform of the displacement x _i is X _i , the excitation angular frequency ω is as follows:

Here, the displacement of the mass point m ₂ is expressed by the following equation.

Under the above conditions (conditions for applying the present invention to the low-rise concentrated vibration control system), the acceleration (absolute displacement) of the mass m ₂ with respect to the fixed end and the transfer function of the displacement are as follows.

Therefore, the acceleration and absolute displacement response magnification | X ₂ / X ₀ | and the relative displacement response magnification | (X ₂ −X ₀ ) / X ₀ |

(1) Optimization regarding acceleration and absolute displacement Optimization is performed from the response magnification of acceleration and absolute displacement. Using the fixed-point theory for the square of the response magnification (| X ₂ / X ₀ | ² ) with the square of the vibration frequency ratio (ξ ² ) as a variable, the inertial mass ratio μ and the damping constant h _Find the optimal value of ₀₁ . When the above equation (6) does not change regardless of the attenuation constant h ₀₁ , it is expressed by the following equation.

If this is rearranged, it becomes a quadratic equation about ξ ² .

Here, if the vibration frequencies at the fixed points (P, Q) are ξ _P and ξ _Q , the following equation is obtained from the root formula of the quadratic equation.

When h ₀₁ → ∞, assuming that the value of equation (6) is the same at the two fixed points, the following equation is obtained.

From the condition of ξ _P ≠ ξ _Q , the following equation is obtained.

  From the equations (10) and (12), the inertial mass ratio μ is obtained by the following equation.

  The vibration frequency at the fixed point (P, Q) is obtained by substituting into the equation (9).

  The response magnification at the fixed point (P, Q) is represented by the following equation from the equations (11) and (14).

The optimum value of attenuation is calculated from the following equation by setting ν = ξ ² so that equation (6) becomes a maximum at a fixed point.

  The following formula is obtained by rearranging formula (17) using the relationship between formula (9) and formula (15).

Here, when h ₀₁ ² is solved, the following equation is obtained.

  The following equation is obtained as an average value of attenuation.

Therefore, the optimum values of inertia mass and damping Ψ _opt and c _opt are as follows.

In a general structure, the mass ratio m ₁ / m ₂ <1 and the stiffness ratio k ₁ / k ₂ > 1, so the influence of m ₁ is small, and Ψ _opt and c _opt increase as the stiffness ratio k ₁ / k ₂ increases. Will increase. Further, when the lower layer rigidity k ₁ is reduced, the rigidity ratio k ₁ / k ₂ is reduced, and therefore the optimum value may be reduced.
On the other hand, the maximum response values obtained in the following equation (15), is determined only by the rigidity ratio k ₁ / k ₂ regardless of the weight, it can be seen to decrease as the stiffness ratio k ₁ / k ₂ becomes small.

  Further, the frequency of the fixed point (P, Q) that causes this is obtained by the following equation from the equation (14).

A simple example for the above case is shown. In the case of m ₁ = 2000 tons, m ₂ = 10000 tons, k ₁ = 1000 kN / mm, k ₂ = 200 kN / mm, the optimum specifications are as follows from the equations (27), (28), and (15).

(2) Optimization related to displacement Optimize from the displacement response magnification. Using the equation (7) as an object, the optimum values of the inertial mass ratio μ and the damping constant h ₀₁ are obtained using fixed point theory. The condition in which the equation (7) does not change regardless of h ₀₁ is expressed by the following equation.

If this is rearranged, it becomes a quadratic equation about ξ ² .

Here, if the vibration frequencies at the fixed points (P, Q) are ξ _P and ξ _Q , the following equation is obtained from the root formula of the quadratic equation.

When h ₀₁ → ∞, assuming that the value of equation (7) is the same at the two fixed points, the following equation is obtained.

From the condition of ξ _P ≠ ξ _Q , the following equation is obtained.

  From the equations (30), (31), and (33), the inertial mass ratio μ is obtained by the following equation.

  The vibration frequency at the fixed point (P, Q) is obtained by substituting the equation (34) into the equation (29).

  The response magnification at the fixed point (P, Q) is represented by the following equation from the equations (32) and (35).

The optimum value of attenuation is calculated from the following equation with ν = ξ ² so that equation (7) becomes a maximum at a fixed point.

Solving for h ₀₁ ² using the relationship between equations (29) and (36), the following equation is obtained.

  The following equation is obtained as an average value of attenuation.

Therefore, the optimum values of inertia mass and damping Ψ _opt and c _opt are as follows.

  The maximum response value and the frequency of the fixed point (P, Q) can be obtained by the following equation from the equations (36) and (35).

  In the case of the same example as above, the values for the equations (25) to (27) are as follows, which is almost the same as the case of optimization by acceleration.

(3) Example = Optimization without adding inertial mass When the inertial mass is not added in the model shown in FIG. 1 (when ψ ₁ = 0), optimization is performed from the response magnification of acceleration and displacement, Compared with the response reduction effect when inertial mass is added.

Here, h ₀₁ ′ = c ₁ / 2m ₂ ω ₀₂ is set, and the optimum value of the damping constant h ₀₁ ′ is obtained from μh ₀₁ = h ₀₁ ′ and ξ ² μ = 0 in Equations (6) and (7).

  First, the acceleration and absolute displacement are optimized. The following formula is obtained from the formula (6).

The optimum attenuation can be obtained by the following equation as ν = ξ ² and η = h ₀₁ ′ ² from the condition for obtaining the attenuation that minimizes the maximum value in the above equation.

  It is sufficient to solve this together. The second expression becomes a quadratic expression related to ν and becomes the following expression.

Therefore, ν ₁ taking the maximum value of the first-order mode is expressed by the following equation.

  When the first equation of the equation (50) is arranged with respect to η using the equation (51), the following equation is obtained.

  Moreover, the maximum response magnification is calculated | required by following Formula from (49) Formula and (51) Formula.

Calculate using the same example as above. In the case of m ₁ = 2000 tons, m ₂ = 10000 tons, k ₁ = 1000 kN / mm, k ₂ = 200 kN / mm, the optimum specifications are as follows from the equations (52) and (55).

  Compared with the results of the equations (26) and (27), it can be seen that the addition of the inertial mass can reduce the maximum response magnification to 1/3 even if the attenuation is half.

In the case of m ₁ = 0, since the mass ratio m ₁ / m ₂ = 0, the value of ν ₁ obtained from the equation (51) is substituted into the equations (58) and (59) as follows. .

Comparing this result with the results of equations (58) and (59), it can be seen that the difference due to the presence or absence of m ₁ is small.

  Next, the displacement is optimized. From the equation (7), the following equation is obtained.

  The optimum attenuation is obtained by the following equation as in the case of acceleration.

  It is sufficient to solve this together. The second expression becomes a quadratic expression related to ν and becomes the following expression.

Therefore, ν ₁ taking the maximum value of the first-order mode is expressed by the following equation.

  The following equation is obtained by rearranging the first equation of the equation (50) with respect to η using the equation (51).

  Further, the maximum response magnification is given by the following equation from equations (61) and (63).

  When the trial calculation is performed with the same example as above, the optimum specifications are as follows from the equations (64) and (67).

  Compared with the results of the equations (46) and (47), it can be seen that the addition of the inertial mass can reduce the maximum response magnification to 1/3 even if the attenuation is half.

In the case of m ₁ = 0, since the mass ratio m ₁ / m ₂ = 0, when the value of ν ₁ obtained from the equation (51) is substituted into the equations (58) and (59), the following is obtained: It can be seen that the difference with or without m ₁ is small.

The optimum specifications for each case described above are collectively shown in FIG. When m ₁ = 0, it can be seen that the maximum response magnification can be reduced to the square root without adding the inertial mass.

  The response magnification obtained in the above example is shown together in FIG. (A) shows the acceleration response magnification when optimized by acceleration, and (b) shows the displacement response magnification when optimized by displacement. In either case, it can be seen that the response magnification is improved by adding attenuation. When only damping is added, the maximum value of the response magnification is about 11. However, by adding inertial mass, the maximum value is reduced to about 3.3 times, and the resonance characteristic is greatly improved. You can see that

The effects of applying the optimum design method of the present invention in designing a low-rise centralized control system are listed below.
(1) In the low-rise centralized vibration control system to which the present invention is applied, the damping performance of the entire structure is greatly improved even if a damping damper (viscous damping element) is arranged only in the low-rise section, and the response to earthquakes and winds is improved. Can be reduced. In addition, it is possible to greatly suppress the post-swing after the disturbance has subsided.

(2) With the optimum design method of the present invention, the optimum specifications of the damping damper (viscous damping element) and the maximum response magnification as its effect can be formulated under conditions that are generally valid for general structures. Widely applicable to a wide range of structures.

(3) The maximum response magnification can be reduced for both acceleration and displacement, compared to the conventional design method of a vibration control mechanism using viscous damping by oil dampers. The maximum response magnification is the maximum value over the entire frequency range of the ratio of the response (the acceleration of mass m ₂ of the mass of the upper layer and the relative displacement with respect to the ground surface) to the input (earthquake acceleration and excitation displacement on the ground surface). In the vicinity of the resonance frequency, the value is 1.0 or more. In a structure in which resonance is a problem, the maximum response magnification is several tens of times, which can be reduced, which is a very significant effect.

(4) The required damping coefficient is small compared to the conventional vibration control mechanism using viscous damping with oil dampers.

(5) It is a passive vibration control structure that is effective from minute amplitude to large amplitude, and does not require external energy. Electricity and computer control are not required, and the simple mechanism is highly reliable and low cost.

(6) The vibration control mechanism targeted by the present invention is not necessarily limited to a new structure. Damping elements may be added by retrofitting the lower layer of existing structures. By doing in this way, it is possible to reinforce the vibration control while using the existing structure.

(7) The conventional TMD mechanism is installed near the top of the structure, and its weight is a load on the tower. However, in the present invention, the damping element is only installed in the lower part of the structure. It is not necessary to install a large weight at a high place, and the load on the entire structure is small.

1 Low-rise part 2 High-rise part 3 Inertial mass damper 4 Oil damper (viscous damping element)

Claims (2)

It is an optimal design method applied to a low-rise centralized control system that suppresses the response of the multi-layer structure by adding a viscous damping element in parallel with the layer rigidity for the multi-layer structure with low damping. There,
The low-rise part and the high-rise part are modeled as a single mass system, and the acceleration is optimized when the mass m _{1 of} the low part, the stiffness k _{1 of the} low part, the mass m _{2 of} the high part, and the stiffness k _{2 of the} high part An optimum design method for a low-rise centralized control system using viscous damping, characterized in that the damping coefficient c ₁ by the viscous damping element is set based on the following equation.

It is an optimal design method applied to a low-rise centralized control system that suppresses the response of the multi-layer structure by adding a viscous damping element in parallel with the layer rigidity for the multi-layer structure with low damping. There,
The low-rise part and the high-rise part are each modeled as a one-mass system, and are optimized by displacement when the mass m _{1 of} the low part, the stiffness k _{1 of the} low part, the mass m _{2 of} the high part, and the rigidity k _{2 of the} high part An optimum design method for a low-rise centralized control system using viscous damping, characterized in that the damping coefficient c ₁ by the viscous damping element is set based on the following equation.

Images