JP3724817B2 - Damping structure of flexible structure - Google Patents

Description

【００１７】
式（１），（２）を２自由度系について考えると、
【００１８】
【数２】

【００１９】
となり、一般に式（１），（２）は成立しない。これらの式を成立させ、２自由度系物理モデルに変換するためには、以下の拘束条件を満たす必要がある。
【００２０】
【数３】

【００２１】
ここで、式（５），（６）は質量集中の条件を満たすための拘束条件であり、式（６）は、剛性マトリックスの整合性を取るための条件である。仮にこの拘束条件がなければ、ばねを２個とする仮定を満たさなくなる。これらの式を物理モデルへの変換誤差と考え、以下の誤差関数ε₁ ，ε₂ を定義する。
【００２２】
【数４】

【００２３】
これらの誤差関数ε₁ ，ε₂ を零にするように正規モードを修正するため、正規モードに対する誤差関数の感度行列を次のように置く。
【００２４】
【数５】

【００２５】
ここで、Φ₁₁，Φ₂₁，Φ₁₂，Φ₂₂について、修正量を各々δΦ₁₁，δΦ₂₁，δΦ₁₂，δΦ₂₂とする。誤差関数ε₁ ，ε₂ を各々零とする正規モードの修正量を求めるには、以下の式を解けばよい。
【００２６】
【数６】

【００２７】
これを、最小ノルム解を使った一般化逆行列により、次の修正量を求める。
【００２８】
【数７】

【００２９】
この修正量により誤差を零に収束させ、式（５），（６）を満足する正規モードを得る。
【００３０】
ところで、ｉ次モードのｊ点における等価質量をＭeqijとすると、等価質量と正規モードとの関係は、以下の通りとなる。
【００３１】
【数８】

【００３２】
実験モード解析より求めた固有モードと、１自由度等価質量同定法により求めた各モードにおける最大振幅点での等価質量より、正規モード行列Φを求めることができる。ただし、正負については、伝達関数における位相を参照することにより決定した。
【００３３】
以上述べてきた手法により、並列構造物の低次元化物理モデルを作成する。正規モード行列修正前の質量行列Ｍと剛性行列Ｋを次に示す。なお、添字の１，２は、各々構造物１、２を意味する。
【００３４】
【数９】

【００３５】
式（１１）による収束計算の結果得られた質量行列Ｍと剛性行列Ｋを次に示す。
【００３６】
【数１０】

【００３７】
ちなみに、これらの質量、剛性をもとに作成した低次元化物理モデルと分布定数モデルの周波数応答はほとんど一致し、この時、式（７），（８）の誤算関数は零に収束している。以上の手法により、 2-2自由度系並列構造物の低次元化モデルを作成した。固有モード行列修正後の物理パラメータを次に示す。
【００３８】
Ｍ₁₁＝15.4 Kg ，Ｍ₁₂＝8.8 Kg，Ｍ₂₁＝8.9 Kg，Ｍ₂₂＝8.4 Kg
Ｋ₁₁＝21800 N/m ，Ｋ₁₂＝37500 N/m ，Ｋ₂₁＝61600 N/m ，Ｋ₂₂＝70000 N/m
次に、集中定数系モデルに基づき状態方程式を導入する。
【００３９】
図１０に、２個の 2自由度系集中定数モデルと２個のアクチュエータとからなる４自由度系力学モデルを示す。この力学モデルの運動方程式は次のようになる。
【００４０】
【数１１】

【００４１】
ここで、状態変数ベクトルを次のように定義し、
【００４２】
【数１２】

【００４３】
式（１３）〜（１４）を状態方程式で表すと次のようになる。
【００４４】
【数１３】

【００４５】
ただし、
【００４６】
【数１４】

【００４７】
Ｋｃ1 ，Ｋｃ2 ：各々のアクチュエータの制御力と制御量ｕの関係を示す力変換計数
制御系の設計には、ＬＱ制御理論を用いる。この場合設計パラメータは、次に示す２次形式評価関数Ｊに与える重み係数行列ＱとＲである。
【００４８】
【数１５】

【００４９】
ただし、
【００５０】
【数１６】

【００５１】
ここで、ｑ₁ 〜ｑ₄ は、質量Ｍ₁₁，Ｍ₁₂，Ｍ₂₁，Ｍ₂₂の各々の速度に、ｑ₅ 〜ｑ₈ は、各々の変位に掛かる重み係数であり、ｒ₁ ，ｒ₂ は、各々の制御量に掛かる重み係数である。
【００５２】
ＬＱ制御理論に基づけば、この評価関数を最小にする制御量ｕは次のように定式化されており、リカッチ方程式を解くことによりフィードバックゲインを決定することができる。
【００５３】
【数１７】

【００５４】
ただし、
【００５５】
【数１８】

【００５６】
ここに、Ｐは次のリカッチ方程式の解である。
【００５７】
【数１９】

【００５８】
図１１に、変位項に重みを掛けた場合のインパルス応答を示す。破線が制御を掛けない場合、実線が制御を掛けた場合であり、 2-2自由度系並列構造の各点において速やかに振動が収束していることが分かる。
【００５９】
図１２に、速度項に重みを掛けた場合のインパルス応答を示す。破線が制御を掛けない場合、実線が制御を掛けた場合であり、変位項に重みを掛けた場合と同様、速やかに各点の振動が収束しているのが分る。
【００６０】
図１３に、変位項に重みを掛けた場合と速度項に重みを掛けた場合の周波数応答を示す。同図（ａ）が構造物１、同図（Ｂ）が構造物２の応答であり、破線が制御を掛けない場合、細い実線が変位項に重みを掛けて制御した場合、太い実線が速度項に重みを掛けた場合である。
【００６１】
次に、振動制御実験における 2-2自由度系並列構造の諸元を図１４に示す。各構造物は、それぞれ２個の質量Ｍ₁₁，Ｍ₁₂及びＭ₂₁，Ｍ₂₂を２枚の鉄板で挾み込んだ平行板ばね構造となっている。
【００６２】
実験装置の構成を図１５に示す。各質量Ｍ₁₁とＭ₂₁間及びＭ₁₂とＭ₂₂間は、それぞれ電磁力タイプのアクチュエータＡ₁ 及びＡ₂ で連結され、制御を掛けない場合、各構造物１，２は独立の関係にある。制御系のコントローラーにはパーソナルコンピュータを使用している。並列構造物の４個の絶対変位は、Ａ／Ｄ変換器を介してコンピュータに入力され、差分により速度信号を得る。これら８個の状態量と、重み係数に応じて算出したゲインにより、並列構造物の振動制御を行なっている。本実験装置の等価質量と等価剛性は、先に正規モードを修正して求めた値を使用している。
【００６３】
図１６，図１７に、変位項に重みを掛けた場合と速度項に重みを掛けた場合のインパルス応答の実験結果を示す。破線が制御を掛けていない場合、実線が制御を掛けた場合である。両図より、制御を掛けると、各点の振動が速やかに収束していることが分る。
【００６４】
図１８に、重み係数行列による周波数応答の実験結果を示す。同図（ａ）が構造物１、同図（ｂ）が構造物２の周波数応答であり、それぞれ、破線が制御を受けていない場合、細い実線が変位項に重みを掛け、Ｒ＝diag［1 ×10^-6 1 ×10^-6］で制御した場合、太い実線が速度項に重みを掛け、Ｒ＝diag［1 ×10^-3 1 ×10^-3］で制御した場合である。
【００６５】
このように、低次元化モデル作成法による振動制御の手法は、多自由度系並列構造物の振動制御にも有効であることが、実験により確認された。
【００６６】
【実施例】
以下に、本発明の好適実施例について詳細に説明する。
【００６７】
図１は、制振構造を適用してなる超高層ビルの前提技術を示す図であり、２つの超高層ビル３，４を互いに隣接させて並列に配置すると共に、両超高層ビル３，４間を２つのアクチュエータ５によって連結した構造になっている。アクチュエータ５は、図２に示すようにコ字形の部材７の内側に永久磁石８を固定すると共に、これに挿入される凸状部材９の周囲に電磁石コイル６を巻き付けて概略構成されており、一方の部材７が超高層ビル４の壁面に固定され、他方の部材９が他方の超高層ビル３の壁面に固定されることで、２つの超高層ビル３，４を連結している。この場合、一方のアクチュエータ５は超高層ビル３，４の最上層階部に、他方のアクチュエータ５は中層階部に設けられており、これら２つのアクチュエータ５を制御することによって各超高層ビル３，４の１次と２次の振動モ−ドを同時に制御できるようになっている。すなわち、各アクチュエータ５の設置場所並びに制御量は、前述の 2-2自由度系並列構造物の低次元化モデル作成法による振動制御の手法に基いて設定されており、ビル内の制御室に設けられた図示しないコンピュータによって各アクチュエータ５が連係をとりながら駆動されることにより、アクチュエータ５を介して超高層ビル３と４同志が力を及ぼし合い、互いに相手の超高層ビルの振動を抑制するようになっている。このように構成された各超高層ビル３，４によれば、地震や強風によるビル揺れを確実に防止でき、安全かつ快適な居住環境が実現できる。
【００６８】
さて、図３は、本発明の制振構造を適用してなる超高層ビルの一実施例を示す図である。この場合、一方の超高層ビル１０は断面形状がコ字形に形成され、その内側に他方の超高層ビル１１を収容した構造になっている。これら２つの超高層ビル１０，１１間は、計２５個のアクチュエータ５によって高さ方向、周方向共に５箇所ずつ連結されている。各アクチュエータ５の設置場所並びに制御量は、前述の 2-2自由度系並列構造物の低次元化モデル作成法を拡張した多自由度系の振動制御の手法により設定されており、図示しないコンピュータによって各アクチュエータ５が連係をとりながら駆動されることにより、アクチュエータ５を介して超高層ビル１０と１１同志が力を及ぼし合い、互いに相手の超高層ビルの振動を抑制するようになっている。この場合、アクチュエータ５が周方向に配設されているので、超高層ビル１０，１１のすべての方向の振動を防止することができる。
【００６９】
また、図４に示すように一方の超高層ビル１２を凸形、他方の超高層ビル１３を凹型とし、両者を嵌合状態に隣接させると共に複数のアクチュエータ５で連結した構造や、図５に示すように一方の超高層ビル１４を角筒状に形成すると共にその内部に他方の超高層ビル１５を設け、内外の超高層ビル１４，１５間を複数のアクチュエータ５で連結した構造も制振構造として極めて有効である。図５に示す構造の場合、内側の超高層ビル１５はエレベ−タ空間等に使用すればよい。
【００７０】
次に、本発明の制振構造の極めて斬新な応用例として、超高層無重力落下施設について、図６を用いて説明する。超高層無重力落下施設は、最近我国でも運転が開始された地下無重力落下施設を地上高く拡張することで、無重力時間の長時間化を図るものである。地下無重力落下施設では、炭鉱の縦坑などを利用して，直径1800mm、長さ8400mmのカプセルを490m自由落下させ、その内部に設けた１立法メートルの内カプセルの中で10秒間、10^-5Ｇ程度の世界最大規模の超微小重力環境を得ている。この種の施設では、無重力環境を実現するために、落下中のカプセルが落下筒の内壁と接触しないように電磁力ガイド方式を導入し、隙間制御が行われる。この隙間制御は、落下筒が揺れないことが必須条件であり、地上側のタワーが揺れると制御不可能になる。
【００７１】
そこで、図６に示す超高層無重力落下施設にあっては、３本の円柱状タワー１６，１７，１８を互いに隣接させて並列に配置すると共に、隣接するタワー１６，１７，１８同志を所々アクチュエータ５で連結した構造になっている。この場合、３本のタワー１６，１７，１８は大、中、小とそれぞれに径が異なっており、剛性の違いなどを考慮した多自由度系並列構造物のモデルに基づき、タワー１６，１７，１８とアクチュエータ５とから成る振動制御系を最適化するようにｎ（高さ方向の設置数）×３個のアクチュエータが配設されている。そして、図示しないコンピュータによって各アクチュエータ５が連係をとりながら駆動されることにより、タワー１６と１７同志、タワー１７と１８同志、タワー１８と１６同志がそれぞれ力を及ぼし合い、互いに相手のタワーの振動を抑制して、各々のタワーの１次からｎ次モ−ドの振動を制御するようになっている。
【００７２】
この場合、最も大径のタワー１６が無重力落下施設を構成しており、その内部に同軸に設けられたカプセル２１落下用の落下筒（内筒）１９は、地下施設の落下筒２０に接続されている。タワー１６を地上高く建てることにより、これを落下層として使用し、地下施設を制動層として使用できる。その場合、タワーの高さを地上2000m とすれば約20秒の無重力時間が得られ、また、地上4500m とすれば約30秒の無重力時間が得られる。この程度長時間の無重力状態が実現されれば、様々な新素材や医薬品の製造が期待できる。なお、中、小径のタワー１７，１８の内部は、エレベ−タ空間や制御施設や監視施設などに使用できる。
【００７３】
図７は前提技術の応用例を示したものである。図７に示す超高層無重力落下施設は、無重力落下による新材料の製造工場と居住施設などとの一体化を図ったもので、２つの超高層ビル２２，２３を隣接させて並列に配置すると共に、超高層ビル２２，３３同志の左右２箇所を所々アクチュエータ５で連結した構造になっている。この場合、２つの超高層ビル２２，２３は共に横幅に対して厚みが小さい構造のビルであり、その構造を考慮した多自由度系並列構造物のモデルに基づき、各々の超高層ビル２２，２３の１次からｎ次の振動モ−ドをｎ段のアクチュエータにより同時に制御する構造になっている。この構成によれば、アクチュエータ５が左右に設けられているので、超高層ビル２２，２３の捩じれ方向の振動も防止できる。
【００７４】
この場合、カプセル２１を落下させるための落下筒２４は、同図（ｂ）に示すように両方の超高層ビル２２，２３の内部にそれぞれ設けられており、地下でＵ字管状に連結されている。このように２本の落下筒２４同志を下部で連結しておけば、一方の超高層ビル２２の落下筒２４を使用して落下させたカプセル２１を、その勢いで他方のビル２３の落下筒２４の途中まで上昇させることができるので、カプセル２１の引上げ時間を短縮できると共に、連続運転が可能になる。超高層ビル２２，２３の落下筒２４以外の領域は任意の目的に使用できるが、超高層階は居住空間に適さないので、環境試験などの実験施設とし、中・下層階に新材料製造のための調製施設、準備室、居住区などが設けられる。また、屋上部はスカイダイビングの発進基地などに利用できる。かくして、新材料の製造工場、環境試験施設、レジャ−施設及び居住施設が同居するインテリジェントビルが実現される。
【００７５】
以上、本発明の制振構造を地上構造物に適用した場合について説明したが、本発明はこれに限定されるものではなく、宇宙ステーションや人工衛星など、３次元柔軟骨組構造の宇宙構造物にも適用することができる。
【００７６】
【発明の効果】
以上要するに本発明の柔軟構造物の制振構造は、隣接した柔軟構造物同士の相互作用によって各構造物に制振力を作用させるので、固有周期の長い構造物に対しても確実に制振力を作用させてその振動を抑えることができる。しかも、隣接する構造物同士を複数の振動モード形を求め、各振動モード形の最大振幅点をアクチュエータの設置場所として定め、その場所でアクチュエータにより柔軟構造物同士を連結することにより、柔軟構造物の振動を数次のモードに亘って同時に抑えることができる。
【図面の簡単な説明】
【図１】 制振構造を適用してなる超高層ビルの前提技術を示す図であり、（ａ）は斜視図、（ｂ）は平面図である。
【図２】 図１の要部を示す図であり、（ａ）は平面図、（ｂ）は側面図である。
【図３】 本発明の制振構造を適用してなる超高層ビルの一実施例を示す図であり、（ａ）は斜視図、（ｂ）は（ａ）のＢ−Ｂ断面図である。
【図４】 本発明の制振構造を適用してなる超高層ビルの他の実施例を示す要部断面図で
ある。
【図５】 本発明の制振構造を適用してなる超高層ビルの他の実施例を示す要部断面図である。
【図６】 本発明の制振構造を適用してなる超高層無重力落下施設の一実施例を示す図であり、（ａ）は斜視図、（ｂ）は（ａ）のＢ−Ｂ断面図、（ｃ）は（ａ）の平面図である。
【図７】 制振構造を適用してなる超高層無重力落下施設の前提技術の応用例を示す図であり、（ａ）は斜視図、（ｂ）は（ａ）のＢ−Ｂ断面図、（ｃ）は（ａ）のＡ−Ａ断面図である。
【図８】 2-2自由度系並列構造物をモデル化した概略斜視図である。
【図９】 図８の各構造物とその振動モ−ド形とを示す図である。
【図１０】 ４自由系力学モデルを示す図である。
【図１１】 変位項に重みを掛けた場合の各質点のインパルス応答（数値解析結果）を示す図である。
【図１２】 速度項に重みを掛けた場合の各質点のインパルス応答（数値解析結果）を示す図である。
【図１３】 重み係数による周波数応答の比較（数値解析結果）を示す図である。
【図１４】 2-2自由度系並列構造物の構造と諸元を示す図である。
【図１５】 2-2自由度系並列構造物の振動を制御するための実験装置の構成図である。
【図１６】 変位項に重みを掛けた場合の各質点のインパルス応答（実験結果）を示す図である。
【図１７】 速度項に重みを掛けた場合の各質点のインパルス応答（実験結果）を示す図である。
【図１８】 重み計数による周波数応答の比較（実験結果）を示す図である。
【符号の説明】
１ 柔軟構造物
２ 柔軟構造物
３ 超高層ビル（柔軟構造物）
４ 超高層ビル（柔軟構造物）
５ アクチュエータ
１０ 超高層ビル（柔軟構造物）
１１ 超高層ビル（柔軟構造物）
１２ 超高層ビル（柔軟構造物）
１３ 超高層ビル（柔軟構造物）
１４ 超高層ビル（柔軟構造物）
１５ 超高層ビル（柔軟構造物）
１６ 超高層タワー（柔軟構造物）
１７ 超高層タワー（柔軟構造物）
１８ 超高層タワー（柔軟構造物）
２２ 超高層ビル（柔軟構造物）
２３ 超高層ビル（柔軟構造物）[0001]
[Industrial application fields]
The present invention relates to a vibration control technique for preventing vibration of a structure designed to have a flexible structure, such as a skyscraper, a skyscraper, or a space structure, and more particularly, a structural system that is a vibration control target. The present invention relates to a vibration damping structure for a flexible structure based on a design method in which a vibration control system is integrated.
[0002]
[Prior art]
Most of today's skyscrapers and towers are designed to have a flexible structure based on the idea of absorbing the energy of earthquakes and winds and protecting the buildings from their destruction by shaking to some extent during earthquakes and strong winds. Has been.
[0003]
On the other hand, in addition to such a concept of earthquake-resistant structure, the concept of a vibration control structure that actively suppresses the vibration in response to a shake caused by an earthquake has been established in recent years. This is a 300m-class skyscraper recently built in Japan with a pendulum type weight on the top floor of the building and using the inertial reaction force of the weight to actively suppress shaking. In addition, an active vibration damping device using the inertial reaction force of the weight is installed on the top floor, and its vibration damping effect is recognized. As a result, 800m, 1000m and 2000m skyscraper plans have been announced, and the era of such skyscrapers is about to arrive.
[0004]
[Problems to be solved by the invention]
However, in order to realize the above-mentioned high-rise building plan, it is indispensable to establish a technology for preventing building shaking due to strong winds. Since the natural period of the vibration becomes longer as the building becomes taller, a high-rise building exceeding 800 m cannot be dealt with only by the vibration damping device using the inertial reaction force of the weight as described above. In other words, such a super high-rise building has a slow shaking displacement speed and a large amplitude. Therefore, the damping device provided on the top floor of the building is displaced with the shaking of the building, and the inertial reaction force of the weight Can no longer act on the building.
[0005]
Further, in such a high-rise building, vibrations of a high-order mode such as a secondary mode and a tertiary mode are generated in a low frequency range in addition to the primary vibration mode. (See FIG. 9). Therefore, it is necessary to be able to suppress such vibrations of a plurality of modes simultaneously.
[0006]
The present invention was devised in view of the above circumstances, and its purpose is to flexibly suppress vibrations of a flexible structure such as a skyscraper, a skyscraper, or a space structure simultaneously in several modes. The object is to provide a structure damping structure.
[0007]
[Means for Solving the Problems]
In order to achieve the above object, the vibration damping structure for a flexible structure according to the present invention is a flexible structure damping structure for damping a flexible structure such as two high-rise buildings. The other flexible structure is formed in a prismatic shape so as to be accommodated inside the one flexible structure, and a plurality of vibration mode shapes to be controlled by these flexible structures are formed. In addition, an actuator for connecting the flexible structures to each other at intervals in the height direction is provided on the three sides facing the outside of the prismatic flexible structure and the inside of the U-shaped flexible structure, The installation positions in the height direction of the actuators are set at the maximum amplitude points of the plural vibration mode shapes, and the structures exert a force through the actuators to vibrate each other's structures. To restrain each other And creating a lumped parameter model at each actuator installation location, and setting an optimum control amount for each actuator so as to optimize a vibration control system composed of the structure and the actuator. . Further, in the vibration damping structure of a flexible structure for damping a flexible structure such as two high-rise buildings, the one flexible structure is formed in a convex shape in cross section, and the other flexible structure is The cross-sectional shape is formed in a concave shape so as to be fitted to one flexible structure, and a plurality of vibration mode shapes to be controlled of the flexible structure are obtained, and both the front end sides of the convex flexible structure and the concave flexible shape are obtained. The flexible structures are spaced apart from each other in the height direction so as to connect both ends of the concave flexible structure and both sides of the convex flexible structure facing each other so as to connect the inside of the structure. The actuators are connected to each other, and the installation positions in the height direction of the actuators are set to the maximum amplitude points of the plural-order vibration mode shapes so that the structures exert forces through the actuators. The structure of each other The lumped parameter system model is created at the installation location of each actuator, and the optimal control amount of each actuator is optimized to optimize the vibration control system composed of the structure and the actuator. Is set. Further, in a vibration damping structure for a flexible structure for damping a flexible structure such as two high-rise buildings, one flexible structure is formed in a rectangular tube shape, and the other flexible structure is Formed into a prismatic shape to be accommodated inside the flexible structure, and obtained a plurality of vibration mode shapes to be controlled of the flexible structure, and the outside of the prismatic flexible structure and the prismatic flexible structure. Actuators for connecting the flexible structures to each other on the four sides facing each other at an interval in the height direction, and the installation positions in the height direction of the actuators are set in the respective vibration modes. The maximum amplitude point is set so that the structures exert forces through the actuators to suppress the vibration of the other structure, and a lumped parameter model is installed at each actuator installation location. Product And, and sets the optimum control amount of each actuator so as to optimize the vibration control system consisting of the above structure and the actuator. Further, in a vibration damping structure for a flexible structure for damping a flexible structure such as three high-rise buildings, the flexible structure is arranged adjacent to each other in a triangular shape, and a plurality of flexible structures to be controlled are to be controlled. In order to obtain the next vibration mode shape, actuators for connecting adjacent flexible structures are provided at intervals in the height direction, and the installation positions in the height direction of the actuators are set to the above-mentioned plural vibration modes. The maximum amplitude point of the shape is set so that the structures exert a force through the actuator and suppress the vibration of the other structure, and the lumped constant at the installation location of each actuator A system model is created, and an optimum control amount for each actuator is set so as to optimize a vibration control system composed of the structure and the actuator.
[0009]
[Action]
According to the vibration damping structure of the present invention as described above, a damping force acts on each structure due to the interaction between adjacent structures, and thus no weight is required to apply an inertial reaction force to each structure. Therefore, it can cope with vibration control of a structure having a long natural period. In addition, the vibrations of the structure can be simultaneously suppressed over several modes by connecting adjacent structures to each other by an actuator in correspondence with the vibration mode.
[0010]
In the following, a method for simultaneously controlling the primary and secondary vibration modes of each structure with two actuators for a parallel structure composed of two flexible structures will be studied. Thus, the effectiveness of the vibration damping structure of the present invention is confirmed.
[0011]
First, the method for creating a reduced-order model for parallel structures and the design of the control system are described. In this method, after obtaining the natural vibration and vibration mode of a structure, a lumped parameter system model is created at an arbitrary point, and the natural mode and sensitivity analysis methods are used. Based on the idea of correcting.
[0012]
Fig. 8 shows the 2-2 degree-of-freedom parallel structure to be controlled. The left side is the structure 1 and the right side is the structure 2. Each of the structures 1 and 2 has a parallel leaf spring structure in which two masses are sandwiched by two plates, and these are connected by two actuators A1 and A2. Therefore, a parallel structure is considered as two independent two-degree-of-freedom structures, and a two-degree-of-freedom system reduction model is created.
[0013]
FIG. 9 shows the vibration modes of the structures 1 and 2 obtained by the experimental mode analysis.
[0014]
From this mode shape, the structure 1 and the structure 2 are reduced to a two-degree-of-freedom system at points 10 and 19 respectively.
[0015]
First, assuming a physical model having two masses and a spring, a physical model having two degrees of freedom is created from the primary and secondary modal parameters. Here, the eigenmode normalized so that the mode matrix is a unit matrix is called a normal mode. The relationship between the mass matrix M and the stiffness matrix K when the normal mode matrix is Φ and the frequency matrix is Ω is as follows.
[0016]
[Expression 1]

[0017]
Considering equations (1) and (2) for a two-degree-of-freedom system,
[0018]
[Expression 2]

[0019]
Therefore, in general, formulas (1) and (2) do not hold. In order to establish these equations and convert them into a two-degree-of-freedom physical model, the following constraint conditions must be satisfied.
[0020]
[Equation 3]

[0021]
Here, Expressions (5) and (6) are constraint conditions for satisfying the condition of mass concentration, and Expression (6) is a condition for obtaining consistency of the stiffness matrix. If this constraint is not present, the assumption of two springs will not be satisfied. These equations are considered as errors in conversion to a physical model, and the following error functions ε ₁ and ε ₂ are defined.
[0022]
[Expression 4]

[0023]
In order to correct the normal mode so that these error functions ε ₁ and ε ₂ are zero, the sensitivity matrix of the error function for the normal mode is set as follows.
[0024]
[Equation 5]

[0025]
Here, with respect to Φ ₁₁ , Φ ₂₁ , Φ ₁₂ , and Φ ₂₂ , the correction amounts are δΦ ₁₁ , δΦ ₂₁ , δΦ ₁₂ , and δΦ ₂₂ , respectively. In order to obtain the correction amount of the normal mode in which the error functions ε ₁ and ε ₂ are each zero, the following equation may be solved.
[0026]
[Formula 6]

[0027]
The following correction amount is obtained from the generalized inverse matrix using the minimum norm solution.
[0028]
[Expression 7]

[0029]
With this correction amount, the error is converged to zero, and a normal mode satisfying equations (5) and (6) is obtained.
[0030]
By the way, when the equivalent mass at the j point of the i-th mode is Meqij, the relationship between the equivalent mass and the normal mode is as follows.
[0031]
[Equation 8]

[0032]
The normal mode matrix Φ can be obtained from the eigenmode obtained by the experimental mode analysis and the equivalent mass at the maximum amplitude point in each mode obtained by the one-degree-of-freedom equivalent mass identification method. However, positive / negative was determined by referring to the phase in the transfer function.
[0033]
A low-dimensional physical model of a parallel structure is created by the method described above. The mass matrix M and the stiffness matrix K before the normal mode matrix correction are shown below. The subscripts 1 and 2 mean the structures 1 and 2, respectively.
[0034]
[Equation 9]

[0035]
A mass matrix M and a stiffness matrix K obtained as a result of the convergence calculation by Expression (11) are shown below.
[0036]
[Expression 10]

[0037]
By the way, the frequency response of the reduced-order physical model created based on these masses and stiffness is almost the same as the distributed constant model. At this time, the miscalculation functions of equations (7) and (8) converge to zero. Yes. Using the above method, a reduced-order model of a 2-2 degree-of-freedom parallel structure was created. The physical parameters after eigenmode matrix correction are shown below.
[0038]
M ₁₁ = 15.4 Kg, M ₁₂ = 8.8 Kg, M ₂₁ = 8.9 Kg, M ₂₂ = 8.4 Kg
K ₁₁ = 21800 N / m, K ₁₂ = 37500 N / m, K ₂₁ = 61600 N / m, K ₂₂ = 70000 N / m
Next, an equation of state is introduced based on the lumped parameter system model.
[0039]
FIG. 10 shows a four-degree-of-freedom dynamic model composed of two two-degree-of-freedom lumped constant models and two actuators. The equation of motion of this dynamic model is as follows.
[0040]
[Expression 11]

[0041]
Where the state variable vector is defined as
[0042]
[Expression 12]

[0043]
Expressions (13) to (14) are represented by the following equation of state.
[0044]
[Formula 13]

[0045]
However,
[0046]
[Expression 14]

[0047]
Kc1, Kc2: LQ control theory is used to design a force conversion counting control system showing the relationship between the control force of each actuator and the control amount u. In this case, the design parameters are weight coefficient matrices Q and R given to the secondary form evaluation function J shown below.
[0048]
[Expression 15]

[0049]
However,
[0050]
[Expression 16]

[0051]
Here, q _{1 to} q ₄ are the velocities of the masses M ₁₁ , M ₁₂ , M ₂₁ , and M ₂₂ , and q _{5 to} q ₈ are the weighting coefficients applied to the respective displacements, r ₁ , r ₂ Is a weighting coefficient applied to each control amount.
[0052]
Based on the LQ control theory, the control amount u that minimizes this evaluation function is formulated as follows, and the feedback gain can be determined by solving the Riccati equation.
[0053]
[Expression 17]

[0054]
However,
[0055]
[Expression 18]

[0056]
Here, P is a solution of the following Riccati equation.
[0057]
[Equation 19]

[0058]
FIG. 11 shows an impulse response when a weight is applied to the displacement term. When the broken line is not controlled, the solid line is controlled, and it can be seen that the vibration converges quickly at each point of the 2-2 degree-of-freedom parallel structure.
[0059]
FIG. 12 shows the impulse response when the speed term is weighted. The case where the broken line is not controlled is the case where the solid line is controlled, and as in the case where the weight is applied to the displacement term, it can be seen that the vibrations at the respective points converge quickly.
[0060]
FIG. 13 shows the frequency response when the displacement term is weighted and when the speed term is weighted. (A) is the response of the structure 1 and (B) is the response of the structure 2. When the broken line is not controlled, the thin solid line is weighted and controlled, the thick solid line is the speed. This is a case where a term is multiplied by a weight.
[0061]
Next, the specifications of the 2-2 degree-of-freedom parallel structure in the vibration control experiment are shown in FIG. Each structure has a parallel leaf spring structure in which two masses M ₁₁ , M ₁₂ and M ₂₁ , M ₂₂ are sandwiched between two iron plates.
[0062]
The configuration of the experimental apparatus is shown in FIG. The masses M ₁₁ and M ₂₁ and M ₁₂ and M ₂₂ are connected by electromagnetic force type actuators A ₁ and A ₂ , respectively, and the structures 1 and 2 are in an independent relationship when not controlled. . A personal computer is used as the controller for the control system. The four absolute displacements of the parallel structure are input to the computer via the A / D converter, and the velocity signal is obtained by the difference. The vibration control of the parallel structure is performed using these eight state quantities and the gain calculated according to the weighting factor. For the equivalent mass and equivalent rigidity of the experimental apparatus, values obtained by correcting the normal mode first are used.
[0063]
FIG. 16 and FIG. 17 show the experimental results of the impulse response when the displacement term is weighted and when the velocity term is weighted. The case where the broken line is not controlled is the case where the solid line is controlled. From both figures, it can be seen that when control is applied, the vibrations at each point converge quickly.
[0064]
FIG. 18 shows the experimental results of the frequency response using the weighting coefficient matrix. (A) is the frequency response of the structure 1 and (b) is the frequency response of the structure 2. When the broken line is not controlled, the thin solid line weights the displacement term and R = diag [ 1 × 10 ⁻⁶ 1 × 10 ⁻⁶ ], the thick solid line weights the speed term, and R = diag [1 × 10 ⁻³ 1 × 10 ⁻³ ].
[0065]
As described above, it has been experimentally confirmed that the vibration control method based on the low-dimensional model creation method is also effective for vibration control of a multi-degree-of-freedom parallel structure.
[0066]
【Example】
In the following, preferred embodiments of the present invention will be described in detail.
[0067]
FIG. 1 is a diagram showing a premise technology of a high-rise building to which a vibration control structure is applied. Two high-rise buildings 3 and 4 are arranged adjacent to each other in parallel, and both high-rise buildings 3 and 4 are arranged. A structure in which the two are connected by two actuators 5 is provided. The actuator 5 is schematically configured by fixing a permanent magnet 8 inside a U-shaped member 7 as shown in FIG. 2 and winding an electromagnet coil 6 around a convex member 9 inserted therein, One member 7 is fixed to the wall surface of the skyscraper 4 and the other member 9 is fixed to the wall surface of the other skyscraper 3, thereby connecting the two skyscrapers 3 and 4. In this case, one actuator 5 is provided on the uppermost floor of the high-rise buildings 3 and 4, and the other actuator 5 is provided on the middle-level floor. By controlling these two actuators 5, each of the high-rise buildings 3 is provided. , 4 can control the primary and secondary vibration modes at the same time. In other words, the installation location and the control amount of each actuator 5 are set based on the vibration control method based on the 2-2 degree-of-freedom parallel structure low-dimensional model creation method described above. Each actuator 5 is driven while being coordinated by a provided computer (not shown), so that the high-rise buildings 3 and 4 exert forces through the actuator 5 and suppress vibrations of the other high-rise buildings. It is like that. According to each of the skyscrapers 3 and 4 thus configured, it is possible to reliably prevent the building from being shaken by an earthquake or a strong wind, and to realize a safe and comfortable living environment.
[0068]
Now, FIG. 3 is a diagram showing an embodiment of a skyscraper made by applying the damping structure of the present invention. In this case, one skyscraper 10 has a U-shaped cross-sectional shape, and has a structure in which the other skyscraper 11 is accommodated inside. The two super high-rise buildings 10 and 11 are connected to each other at five points in the height direction and the circumferential direction by a total of 25 actuators 5. The installation location and control amount of each actuator 5 are set by a multi-degree-of-freedom vibration control method that is an extension of the above-described 2-2 degree-of-freedom parallel structure structure reduction model creation method. By driving the actuators 5 in association with each other, the high-rise buildings 10 and 11 are exerting forces through the actuators 5 to suppress vibrations of the other high-rise buildings. In this case, since the actuator 5 is disposed in the circumferential direction, vibrations in all directions of the high-rise buildings 10 and 11 can be prevented.
[0069]
Also, as shown in FIG. 4, one skyscraper 12 has a convex shape, and the other skyscraper 13 has a concave shape. As shown in the figure, a structure in which one skyscraper 14 is formed in a rectangular tube shape and the other skyscraper 15 is provided in the inside, and the inner and outer skyscrapers 14, 15 are connected by a plurality of actuators 5 is also controlled It is extremely effective as a structure. In the case of the structure shown in FIG. 5, the inner skyscraper 15 may be used in an elevator space or the like.
[0070]
Next, as a very novel application example of the vibration damping structure of the present invention, a super-high-rise weightless dropping facility will be described with reference to FIG . The super high-rise gravity-free fall facility aims to extend the weight-free time by extending the underground zero-gravity fall facility, which has recently started operation in Japan, to a higher level above the ground. In the underground gravity drop facility, a pit of 1800mm in diameter and 8400mm in length is dropped 490m freely using a vertical shaft of a coal mine, and 10 ^-5 for 10 seconds in an inner capsule of 1 cubic meter provided inside. The world's largest microgravity environment of about G is obtained. In this type of facility, in order to realize a zero-gravity environment, an electromagnetic force guide system is introduced so that the capsule being dropped does not come into contact with the inner wall of the dropping cylinder, and the gap control is performed. This gap control is an essential condition that the dropping cylinder does not shake, and cannot be controlled when the tower on the ground side shakes.
[0071]
Therefore, in the super-high-rise weightless fall facility shown in FIG. 6, the three columnar towers 16, 17, and 18 are arranged adjacent to each other in parallel, and the adjacent towers 16, 17, and 18 are arranged in places as actuators. 5 is connected. In this case, the three towers 16, 17, and 18 have different diameters, large, medium, and small, and the towers 16, 17 are based on a multi-degree-of-freedom parallel structure model that takes into account differences in rigidity. , 18 and the actuator 5 are arranged so that n (the number of installations in the height direction) × 3 actuators are arranged. Each actuator 5 is driven by a computer (not shown) in cooperation with each other, so that the towers 16 and 17, the towers 17 and 18, and the towers 18 and 16 each exert forces, and the other towers vibrate. And the vibration of the first to nth modes of each tower is controlled.
[0072]
In this case, the tower 16 having the largest diameter constitutes a weightless dropping facility, and a dropping cylinder (inner cylinder) 19 for dropping the capsule 21 provided coaxially therein is connected to the dropping cylinder 20 of the underground facility. ing. By building the tower 16 high above the ground, it can be used as a falling layer and the underground facility can be used as a braking layer. In that case, if the height of the tower is 2000 m above the ground, a weightless time of about 20 seconds can be obtained, and if the height of the tower is 4500 m, a weightless time of about 30 seconds can be obtained. If such a long period of weightlessness is realized, production of various new materials and pharmaceuticals can be expected. The insides of the medium and small diameter towers 17 and 18 can be used for elevator spaces, control facilities, monitoring facilities, and the like.
[0073]
FIG. 7 shows an application example of the base technology. The super high-rise weightless fall facility shown in FIG. 7 is an integration of a new material manufacturing factory and living facilities by weightless fall, and the two high-rise buildings 22 and 23 are arranged adjacent to each other in parallel. The structure is such that the left and right locations of the high-rise buildings 22 and 33 are connected by the actuator 5 in some places. In this case, the two skyscrapers 22 and 23 are both buildings having a structure with a small thickness with respect to the width, and based on the model of the multi-degree-of-freedom parallel structure considering the structure, The first to n-th vibration modes 23 are simultaneously controlled by n-stage actuators. According to this configuration, since the actuator 5 is provided on the left and right, vibrations in the torsional direction of the high-rise buildings 22 and 23 can be prevented.
[0074]
In this case, the drop cylinder 24 for dropping the capsule 21 is provided inside each of the skyscrapers 22 and 23 as shown in FIG. Yes. If the two dropping cylinders 24 are connected at the lower portion in this way, the capsule 21 dropped using the dropping cylinder 24 of one skyscraper 22 is dropped by the momentum of the other building 23. Since it can raise to the middle of 24, the pulling time of the capsule 21 can be shortened, and continuous operation becomes possible. The areas other than the falling cylinder 24 of the high-rise buildings 22 and 23 can be used for any purpose, but the high-rise floors are not suitable for living spaces. Preparation facilities, preparation rooms, residential areas, etc. are provided. The rooftop can also be used as a starting point for skydiving. Thus, an intelligent building in which a new material manufacturing factory, an environmental test facility, a leisure facility, and a residential facility coexist is realized.
[0075]
As described above, the case where the vibration damping structure of the present invention is applied to a ground structure has been described. However, the present invention is not limited to this, and it is applicable to a space structure having a three-dimensional flexible frame structure such as a space station or an artificial satellite. Can also be applied.
[0076]
【The invention's effect】
In short, the damping structure for a flexible structure according to the present invention exerts a damping force on each structure by the interaction between adjacent flexible structures, so that even a structure with a long natural period can be reliably controlled. The vibration can be suppressed by applying force. In addition, by obtaining a plurality of vibration mode shapes between adjacent structures , determining the maximum amplitude point of each vibration mode shape as the installation location of the actuator, and connecting the flexible structures with the actuator at that location, the flexible structure Can be suppressed simultaneously over several modes.
[Brief description of the drawings]
BRIEF DESCRIPTION OF DRAWINGS FIG. 1 is a view showing a premise technique of a skyscraper to which a vibration damping structure is applied, wherein (a) is a perspective view and (b) is a plan view.
2A and 2B are diagrams illustrating a main part of FIG. 1, in which FIG. 2A is a plan view and FIG. 2B is a side view.
FIGS. 3A and 3B are diagrams showing an embodiment of a skyscraper to which the vibration damping structure of the present invention is applied, in which FIG. 3A is a perspective view, and FIG. 3B is a cross-sectional view along line BB in FIG. .
FIG. 4 is a cross-sectional view of an essential part showing another embodiment of a skyscraper to which the vibration damping structure of the present invention is applied.
FIG. 5 is a cross-sectional view of an essential part showing another embodiment of a skyscraper to which the vibration damping structure of the present invention is applied.
6A and 6B are diagrams showing an embodiment of a super high-rise weightless dropping facility to which the vibration damping structure of the present invention is applied, wherein FIG. 6A is a perspective view, and FIG. 6B is a cross-sectional view along line BB in FIG. (C) is a top view of (a).
FIG. 7 is a diagram showing an application example of the premise technology of a super-high-rise weightless fall facility to which a damping structure is applied, (a) is a perspective view, (b) is a cross-sectional view along BB in (a), (C) is AA sectional drawing of (a).
FIG. 8 is a schematic perspective view modeling a 2-2 degree-of-freedom parallel structure.
9 is a view showing each structure of FIG. 8 and its vibration mode shape. FIG.
FIG. 10 is a diagram showing a four-free system dynamic model.
FIG. 11 is a diagram showing an impulse response (result of numerical analysis) of each mass point when a weight is applied to the displacement term.
FIG. 12 is a diagram illustrating an impulse response (numerical analysis result) of each mass point when a speed term is weighted.
FIG. 13 is a diagram showing a comparison (frequency analysis result) of frequency responses by weighting factors.
FIG. 14 is a diagram showing the structure and specifications of a 2-2 degree-of-freedom parallel structure.
FIG. 15 is a configuration diagram of an experimental apparatus for controlling vibration of a 2-2 degree-of-freedom parallel structure.
FIG. 16 is a diagram showing an impulse response (experimental result) of each mass point when a weight is applied to the displacement term.
FIG. 17 is a diagram showing an impulse response (experimental result) of each mass point when a speed term is weighted.
FIG. 18 is a diagram showing a comparison (experimental result) of frequency responses by weight counting.
[Explanation of symbols]
1 Flexible structure 2 Flexible structure 3 High-rise building (flexible structure)
4 High-rise buildings (flexible structures)
5 Actuator 10 High-rise building (flexible structure)
11 Skyscraper (flexible structure)
12 High-rise buildings (flexible structures)
13 High-rise buildings (flexible structures)
14 Skyscraper (flexible structure)
15 Skyscraper (flexible structure)
16 High-rise tower (flexible structure)
17 High-rise tower (flexible structure)
18 Super high-rise tower (flexible structure)
22 Skyscraper (flexible structure)
23 High-rise buildings (flexible structures)

Claims (4)

In the damping structure of a flexible structure for damping a flexible structure such as two high-rise buildings,
One flexible structure is formed in a U-shaped cross section, and the other flexible structure is formed in a prismatic shape so as to be accommodated inside the one flexible structure. The flexible structure is connected to the three sides facing the outside of the prismatic flexible structure and the inside of the U-shaped flexible structure at intervals in the height direction. In addition to providing actuators, the installation positions in the height direction of the actuators are set at the maximum amplitude points of the plurality of vibration mode shapes, and the structures exert forces through the actuators so It is configured to suppress vibrations of the structure, and a lumped parameter model is created at each actuator installation location to optimize the vibration control system consisting of the structure and the actuator. Damping of flexible structures and sets the optimum control amount of each actuator. In the damping structure of a flexible structure for damping a flexible structure such as two high-rise buildings,
One flexible structure is formed in a convex shape in cross section, and the other flexible structure is formed in a concave shape so as to be fitted to the one flexible structure. In order to obtain the next vibration mode shape, both ends of the concave flexible structure and the convex flexible structure facing the both ends of the concave flexible structure are connected to connect both ends of the convex flexible structure and the inside of the concave flexible structure. Actuators for connecting the flexible structures are provided at an interval in the height direction so as to connect both sides of the object, and the installation positions in the height direction of the actuators are set in the above-described plurality of vibration mode shapes. is set to the maximum amplitude point, through the actuator as well as configured to mutually suppress the vibration of the structure each other's mutual exert the structure each other force, lumped system at the site of each actuator model Damping structure for flexible structures created Le, and sets the optimum control amount of each actuator so as to optimize the vibration control system consisting of the above structure and the actuator. In the damping structure of a flexible structure for damping a flexible structure such as two high-rise buildings,
One flexible structure is formed into a rectangular tube shape, and the other flexible structure is formed into a prismatic shape so as to be accommodated inside the one flexible structure. An actuator for obtaining a mode shape and connecting the flexible structures to each other at intervals in the height direction on the four sides facing the outside of the prismatic flexible structure and the inside of the prismatic flexible structure. At the same time, the installation positions in the height direction of the actuators are set at the maximum amplitude points of the plurality of vibration mode types, and the structures exert forces through the actuators, and the structures of each other Each actuator is configured to optimize the vibration control system consisting of the structure and the actuator by creating a lumped parameter system model at each actuator installation location. Damping of flexible structures and sets the optimum control amount of eta. In the damping structure of a flexible structure for damping a flexible structure such as three skyscrapers,
The flexible structures are arranged adjacent to each other in a triangular shape, and a plurality of vibration mode shapes to be controlled of the flexible structures are obtained, and an actuator for connecting adjacent flexible structures is spaced in the height direction. Each of the actuators is provided at a distance, and the installation positions in the height direction of the actuators are set to the maximum amplitude points of the plural vibration mode shapes so that the structures exert forces through the actuators and are opposed to each other. In order to suppress the vibration of each structure, a lumped parameter system model is created at the installation location of each actuator, and the vibration control system composed of the structure and the actuator is optimized. A damping structure for a flexible structure characterized by setting an optimal control amount.

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