JP2004318608A - Device for computing meshing stiffness of pair of helical gears - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a device for computing the meshing stiffness of a pair of helical gears, which accurately determines the meshing stiffness value of pairs of helical gears through simple computing processes for a wide range of gear items. <P>SOLUTION: A computing part 6 computes first and second branch points where the behavior of a meshing stiffness curve varies, and sets at different functions the approximate expression of a meshing stiffness curve for a section from the first branch point to the second branch point and the approximate expression of a meshing stiffness curve for sections from a meshing start point to the first branch point and from the second branch point to a meshing end point. To set the approximate expression of each section, the computing part 6 uses the different functions according to the result of comparison between the front meshing rate and the overlapping meshing rate of the pair of helical gears. This sets the approximate expressions that very closely match the actual meshing stiffness curve. Use of the approximate expressions set in this way accurately determines the meshing stiffness value of the pair of helical gears through simple computing processes. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

【００２９】
また、歯の曲げ撓み影響関数であるＫ_ｂは、狩野ら（狩野・斎木、「歯車用ラックの新しい曲げ撓み影響関数」、日本機械学会２００２年度年次大会講演論文集（Ｖ）２３１４号、Ｐ２７ 参照）によって提案されている歯幅方向と歯丈方向の違った撓み特性を考慮した高精度な式を使用した。
【００３０】

【００３１】
なお、式（１）〜（６）中の変数は以下の通りである。
【００３２】
（ｘ，ｙ）：撓み観測点の座標値
（ξ，η）：単位集中荷重点の座標値
Ｐ（ξ）：噛合接触線上の荷重分布
Ｅ：ヤング率［２．０６８×１０^１１Ｎｍ^２］
γ：ポアソン比［０．３］
ΔＢ：各噛合接触線上の計算分割幅
Ｘ：接触歯幅中央を０とするはすば歯対の等価作用線（図４参照）上の座標値
Ｕ：原点集中荷重時の原点での撓みの絶対値
λ：撓み楕円状分布の同心円分布への座標変換係数
ｒ：歯先を原点とする撓み同心円分布の半径
ν（ｒ）：等価同心円分布の撓み特性関数
Ｇ（η）：歯丈方向の集中荷重点直下の撓み特性関数
Ｆ（ξ）：歯幅方向の集中荷重点直下の撓み特性関数
【００３３】
以上の計算式を用い、様々な歯車諸元によるはすば歯対の噛合開始から噛合終了までの各噛合剛性の理論値を演算し解析した結果、本出願人らは、はすば歯対の噛合開始点から噛合中心点までの間に噛合剛性曲線の挙動が変化する分岐点（以下、第１の分岐点と称す）が存在するとともに、噛合中心点から噛合終了点までの間に同様の分岐点（以下、第２の分岐点と称す）が存在することを知見した。そして、この分岐点についてさらに詳細に調べた結果、その位置は正面噛合率ε_αと重なり噛合率ε_βの大小関係で決まる上、例えば図５の各例に示すような３つのケース（ケース１〜ケース３）に分類できることを知見した。なお、図５において、ケース１に示す例は、ＢＨ〔歯幅と歯丈の比（歯幅／歯丈）〕＝２，Ｈ_ｋ（歯丈比）＝２．３５，α_ｎ（圧力角）＝２０．５°，β_０（ピッチ円筒上のねじれ角）＝４°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐ（Ｋ（０））で規格化した噛合剛性曲線であり、ケース２に示す例は、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝３６°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線であり、ケース３に示す例は、ＢＨ＝６，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝３６°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線である。
【００３４】
ケース１は、正面噛合率ε_αが重なり噛合率ε_βよりも大きいケースである。この場合、図６に示すように、作用平面１００上を噛合開始点Ｓから噛合終了点Ｅへと平行移動する噛合接触線ＣＣの長さは、当該噛合接触線ＣＣが作用平面１００上を歯幅方向に横切った状態で最長となる。そして、このケースでは、噛合接触線ＣＣの長さが最長となった瞬間に第１の分岐点が存在するとともに、その長さが最長でなくなる瞬間に第２の分岐点が存在する。
【００３５】
また、ケース２は、正面噛合率ε_αが重なり噛合率ε_βと等しいケースである。この場合、図７に示すように、作用平面１００上を噛合開始点Ｓから噛合終了点Ｅへと平行移動する噛合接触線ＣＣの長さは、当該噛合接触線ＣＣが作用平面１００上を歯幅方向及び歯丈方向に横切った瞬間に最長となる。そして、このケースでは、噛合接触線ＣＣの長さが最長となった瞬間に第１，第２の分岐点が同時に存在する。
【００３６】
また、ケース３は、正面噛合率ε_αが重なり噛合率ε_βよりも小さいケースである。この場合、図８に示すように、作用平面１００上を噛合開始点Ｓから噛合終了点Ｅへと平行移動する噛合接触線ＣＣの長さは、当該噛合接触線ＣＣが作用平面１００上を歯丈方向に横切った状態で最長となる。そして、このケースでは、噛合接触線ＣＣの長さが最長となった瞬間に第１の分岐点が存在するとともに、その長さが最長でなくなる瞬間に第２の分岐点が存在する。
【００３７】
このように、各噛合剛性曲線は、分岐点位置の違いによって大きく影響される。また、各分岐点の位置は歯車諸元のみに依存して上記３つのケースの何れかとなる。ここで、噛合開始点、噛合終了点、及び第１，第２の分岐点の作用線上の座標、すなわちＸ_ｓ、Ｘ_ｅ、Ｘ_ｄ１、Ｘ_ｄ２はそれぞれ以下の通り算出される。
【００３８】
Ｘ_ｓ＝−（ε_α＋ε_β）／〔（ε_α／ｍ_ｎ）・Ｈ〕 …（７）
Ｘ_ｅ＝（ε_α＋ε_β）／〔（ε_α／ｍ_ｎ）・Ｈ〕 …（８）
Ｘ_ｄ１＝−｜（ε_α−ε_β）／〔（ε_α／ｍ_ｎ）・Ｈ〕｜ …（９）
Ｘ_ｄ２＝｜（ε_α−ε_β）／〔（ε_α／ｍ_ｎ）・Ｈ〕｜ …（１０）
【００３９】
ここで、式（７）〜（１０）中の変数は以下の通りである。
【００４０】
ｍ_ｎ：歯直角モジュール
Ｈ：歯丈
なお、式（９），（１０）からも明らかなように、第１，第２の分岐点のＸ座標は、噛合中心点Ｘ_ｐに対称であるため、これらを代表してＸ_ｄ２を単にＸ_ｄとも称す。
【００４１】
以上の点に基づき、先ず、第１，第２の分岐点間（Ｘ_ｄ１≦Ｘ≦Ｘ_ｄ２）での各噛合剛性値の近似関数について検討する。
【００４２】
図９に、ケース１に分類される各種諸元のはすば歯対の噛合剛性曲線を示す。なお、図９には、ＢＨ＝２，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝２０°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、ＢＨ＝４，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝２０°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、及び、ＢＨ＝６，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝２０°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線がそれぞれ例示されている。図９からも明らかなように、ケース１において、各噛合剛性曲線は、歯幅歯丈比ＢＨによらず、略同一の放物線上にあることがわかる。
【００４３】
また、図１０に、ケース３に分類される各種諸元のはすば歯対の噛合剛性曲線を示す。なお、図１０には、ＢＨ＝４，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝３６°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、ＢＨ＝５，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝３６°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、及び、ＢＨ＝６，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝３６°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線がそれぞれ例示されている。この場合、図１０に示すように、各はすば歯対の噛合剛性値Ｋ（Ｘ）は、歯幅Ｂの影響を受けるため、歯幅歯丈比ＢＨに応じて挙動が異なる。その一方で、各はすば歯対の噛合剛性値曲線は、図１１に示すように、各分岐点位置を一致させる座標変換を行った場合、同一の放物線上で略一致することがわかる。なお、図１１には、ＢＨ＝４及び５の第１，第２の分岐点位置を、座標変換によってＢＨ＝６の第１，第２の分岐点位置にそれぞれ一致させた例を示す。
【００４４】
以上のことからも分かるように、分岐点間の各噛合剛性曲線は、２次項と４次項とを備えた、噛合中心点を頂点とする左右対称の放物線関数で近似できる。この場合、ケース１の放物線形状は歯幅歯丈比ＢＨによらず一定であるが、ケース３の放物線形状は歯幅歯丈比ＢＨに依存して変化する。そこで、これらを考慮し、本出願人らは、分岐点間（−Ｘ_ｄ１≦Ｘ≦Ｘ_ｄ２）での近似関数を以下のように作成した。
【００４５】

【００４６】

ここで、式（１４）〜（１８）で規定される近似関数は、歯幅歯丈比ＢＨ＝６の噛合剛性曲線を基準とし、分岐点Ｘ_ｄに基づく座標変換を行うことにより各歯幅歯丈比ＢＨでの噛合剛性曲線を近似するものである。すなわち、歯幅歯丈比ＢＨ＝６のはすば歯対は、実際に使用され得る頻度の高いはすば歯対の１つであり、また、当該はすば歯対は歯幅歯丈比ＢＨが大きいため、噛合剛性値の計算数を多く設定できる。これらのことから、式（１４）〜（１８）では、実用的且つ近似精度の高い歯幅歯丈比ＢＨ＝６でのはすば歯対の噛合剛性曲線を基準として各歯幅歯丈比ＢＨでの噛合剛性曲線を近似する。この場合、分岐点のＸ座標（Ｘ_ｄ）は、次式のように歯幅歯丈比ＢＨ＝６で等価変換することができる。
【００４７】
Ｘ_{ｄ（ＢＨ＝６）}＝〔ε_β・（６／ＢＨ）−ε_α〕／〔（ε_α／ｍ_ｎ）・Ｈ〕 …（１９）
なお、歯幅歯丈比ＢＨ＝６以外の噛合剛性曲線を基準として式（１４）〜（１８）と同等の近似関数を作成してもよいことは勿論である。
【００４８】
次に、第１，第２の分岐点外（Ｘ_ｓ≦Ｘ≦Ｘ_ｄ１，Ｘ_ｄ２≦Ｘ≦Ｘ_ｅ）での各噛合剛性値の近似関数について検討する。
【００４９】
図１２に、ケース１に分類される各種諸元のはすば歯対の噛合剛性曲線のうち、分岐点が噛合開始点及び噛合終了点寄りにある場合のもの（すなわち、ねじれ角β_０が比較的小さいもの）を示す。なお、図１２には、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝４°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝８°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝１２°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝１６°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、及び、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝２０°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線がそれぞれ例示されている。
【００５０】
また、図１３に、ケース１に分類される各種諸元のはすば歯対の噛合剛性曲線のうち、分岐点が噛合中心点寄りにある場合のもの（すなわち、ねじれ角β_０が比較的大きいもの）を示す。なお、図１３には、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝２４°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝２８°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、及び、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝３２°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線がそれぞれ例示されている。
【００５１】
また、図１４に、ケース２に分類される各種諸元のはすば歯対の噛合剛性曲線を示す。なお、図１４には、ＢＨ＝３，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝３６°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、ＢＨ＝４，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝３０°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、ＢＨ＝５，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝２６°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線、及び、ＢＨ＝６，Ｈ_ｋ＝２．３５，α_ｎ＝２０．５°，β_０＝２２．５°での各噛合剛性値Ｋ（Ｘ）をその噛合中心点での噛合剛性値Ｋ_ｐで規格化した噛合剛性曲線がそれぞれ例示されている。ここで、ケース３に分類される各種諸元のはすば歯対の噛合剛性曲線も、分岐点外においては、ケース２と略同様となる。
【００５２】
図１２からも明らかなように、ねじれ角β_０が比較的小さい場合には、分岐点外での噛合剛性曲線は直線で近似することができる。また、図１３，図１４からも明らかなように、ねじれ角β_０が比較的大きい場合には、分岐点外の噛合剛性曲線は直線で近似することが困難である。これらに基づき、本出願人らは、分岐点外での各噛合剛性曲線について更なる検討を行った結果、ねじれ角β_０を歯幅歯丈比ＢＨ＝６のもの（β_{０（ＢＨ＝６）}）に等価変換した際に、β_{０（ＢＨ＝６）}＜１２°の噛合剛性曲線を直線近似することができ、β_{０（ＢＨ＝６）}≧１２°の噛合剛性曲線をＡ次の項を有する指数関数とＢ次の項を有する関数とで近似できることを知見した。そこで、これらを考慮して、本出願人らは、分岐点外（Ｘ_ｓ≦Ｘ≦Ｘ_ｄ１，Ｘ_ｄ２≦Ｘ≦Ｘ_ｅ）での近似関数を以下のように作成した。
【００５３】
なお、等価変換ねじれ角β_{０（ＢＨ＝６）}は次式で定義される。
【００５４】
β_{０（ＢＨ＝６）}＝ｔａｎ^−１〔（ＢＨ／６）・ｔａｎβ_０〕 …（２０）
【００５５】
β_{０（ＢＨ＝６）}＜１２°の場合
Ｋ（Ｘ）／Ｋ_ｐ＝ａ_０＋ａ_１・｜Ｘ｜ …（２１）
ａ_０＝（｜Ｘ_ｄ｜・Ｙ_ｓ−Ｙ_ｄ・｜Ｘ_ｓ｜）／（｜Ｘ_ｄ｜−｜Ｘ_ｓ｜） …（２２）
ａ_１＝（Ｙ_ｄ−Ｙ_ｓ）／（｜Ｘ_ｄ｜−｜Ｘ_ｓ｜） …（２３）
【００５６】

【００５７】
ここで、式（２４）〜（３０）において、各ケースでの次数Ａ，Ｂはねじれ角β_０に基づいて以下のように規定される。
【００５８】

また、式（２４）〜（３０）において、Ｙ_ｓは、噛合開始点での噛合剛性値と噛合中心点での噛合剛性値との比を示し、例えば以下のように求められる。
【００５９】

また、式（２４）〜（３０）において、Ｙ_ｄは、分岐点での噛合剛性値と噛合中心点での噛合剛性値との比〔Ｋ（Ｘ_ｄ）／Ｋ_ｐ〕を示し、式（１１）〜（１３）に基づいて設定された近似式、または、式（１４）〜（１８）に基づいて設定された近似式を用いて求められる。
【００６０】
次に、上述の各近似関数を用いてはすば歯対の噛合剛性を演算する演算装置について説明する。
【００６１】
図２において、符号１ははすば歯対の噛合剛性演算装置を示し、この噛合剛性演算装置１は、はすば歯対の諸元を入力するための入力部５と、入力部５からのの入力諸元に基づいてはすば歯対の噛合剛性値を演算する演算部６と、演算部６で実行される噛合剛性演算ルーチンを格納するとともに、入力部５からの入力諸元や演算部６での演算結果等を適宜記憶する記憶部７と、演算部６での演算結果等を出力する出力部８とを有して構成されている。
【００６２】
ここで、噛合剛性演算装置１は、例えば図３に示すコンピュータシステム１０で実現される。このコンピュータシステム１０は、例えば、コンピュータ本体１１に、キーボード１２と、ディスプレイ装置１３と、プリンタ１４とが接続ケーブル１５を介して接続されて要部が構成されている。そして、このコンピュータシステム１０において、例えば、コンピュータ本体１１に配設された各種ドライブ装置やキーボード１２等が入力部５として機能するとともに、コンピュータ本体１１に内蔵されたＣＰＵ，ＲＯＭ，ＲＡＭ等が演算部６として機能する。また、コンピュータ本体１１に内蔵されたハードディスク等が記憶部７として機能するとともに、ディスプレイ装置１３やプリンタ１４等が出力部８として機能する。
【００６３】
本実施の形態において、記憶部７に格納された噛合剛性演算ルーチンは、上述の式（７）〜（３８）を適宜用いて入力諸元に応じた噛合剛性近似式を設定するとともに、設定した噛合剛性近似式に基づいてはすば歯対の各噛合状態での噛合剛性値を演算するためのものである。
【００６４】
そして、演算部６は、記憶部７に格納された噛合剛性演算ルーチンをロードし、実行することによって、分岐点演算手段、噛合率比較手段、第１の近似式設定手段（近似式設定手段）、第２の近似式設定手段（近似式設定手段）、及び、ねじれ角比較手段としての各機能を実現する。
【００６５】
次に、上述の構成による噛合剛性演算装置１において、演算部６で実行される噛合剛性演算ルーチンについて説明する。
【００６６】
このルーチンは、はすば歯対の諸元が入力部５を通じて入力された後に実行される。ここで、本実施の形態において、噛合剛性演算装置１には、諸元として、例えば、歯直角モジュールｍ_ｎ、歯丈係数Ｋ_ｓ、頂隙係数Ｃ_ｋ、圧力角α_ｎ、歯幅Ｂ、及び、ねじれ角β_０が入力される。
【００６７】
ルーチンがスタートすると、先ず、ステップＳ１０１において、演算部６は、周知の演算方法を用いて、入力諸元に基づき、歯丈Ｈ、歯幅歯丈比ＢＨ、正面噛合率ε_α、及び、重なり噛合率ε_β等の諸元計算を行う。
【００６８】
続くステップＳ１０２において、演算部６は、式（７）及び（８）を用いて、はすば歯対の噛合開始点及び噛合終了点のＸ座標（Ｘ_ｓ、及びＸ_ｅ）を演算する。
【００６９】
続くステップＳ１０３において、演算部６は、式（９）及び（１０）を用いて、はすば歯対の第１，第２の分岐点のＸ座標（Ｘ_ｄ１、及びＸ_ｄ２）を演算する。
【００７０】
そして、ステップＳ１０４において、演算部６は、ステップＳ１０１で求めた正面噛合率ε_αと重なり噛合率ε_βとの大小関係を比較し、ε_α＞ε_βであると判定した場合には、今回の入力諸元に基づくはすば歯対はケース１に分類されるはすば歯対であると判定してステップＳ１０５に進む。
【００７１】
ステップＳ１０４からステップＳ１０５に進むと、演算部６は、式（１１）〜（１３）を用いて、第１，第２の分岐点間（Ｘ_ｄ１≦Ｘ≦Ｘ_ｄ２）の区間での噛合剛性曲線の近似式を設定する。
【００７２】
続くステップＳ１０６において、演算部６は、ステップＳ１０５で設定した近似式を用いて、分岐点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｄ〔＝Ｋ（Ｘ_ｄ）／Ｋ_ｐ〕を演算する。
【００７３】
続くステップＳ１０７において、演算部６は、式（３５）〜（３８）を用いて、噛合開始点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｓを演算する。
【００７４】
続くステップＳ１０８において、演算部６は、式（２０）を用いて、歯幅歯丈比ＢＨ＝６での等価変換ねじれ角β_{０（ＢＨ＝６）}を演算する。
【００７５】
そして、ステップＳ１０９において、演算部６は、等価変換ねじれ角β_{０（ＢＨ＝６）}と予め設定された設定ねじれ角（例えば、１２°）との大小関係を比較し、β_{０（ＢＨ＝６）}＜１２°であると判定した場合には、今回の入力諸元に基づくはすば歯対は、分岐点外（Ｘ_ｓ≦Ｘ≦Ｘ_ｄ１，Ｘ_ｄ２≦Ｘ≦Ｘ_ｅ）の区間での噛合剛性曲線を直線近似することが可能であると判定してステップＳ１１０に進む。
【００７６】
ステップＳ１０９からステップＳ１１０に進むと、演算部６は、式（２１）〜（２３）を用いて、分岐点外の区間での噛合剛性曲線の近似式を設定する。
【００７７】
そして、ステップＳ１１０からステップＳ１１７に進むと、演算部６は、ステップＳ１０５及びステップＳ１１０で設定した各区間での噛合剛性曲線の近似式を用いて、はすば歯対の噛合開始点Ｘ_ｓから噛合終了点Ｘ_ｅまで各噛合剛性値を演算し、この演算結果を出力部８を通じて出力した後、ルーチンを終了する。
【００７８】
一方、ステップＳ１０９において、β_{０（ＢＨ＝６）}≧１２°であると判定した場合には、今回の入力諸元に基づくはすば歯対は、分岐点外の区間での噛合剛性曲線を直線近似することは困難であると判定して判定してステップＳ１１１に進む。
【００７９】
ステップＳ１０９からステップＳ１１１に進むと、演算部６は、式（２４）〜（３２）を用いて、分岐点外の区間での噛合剛性曲線の近似式を設定する。
【００８０】
そして、ステップＳ１１１からステップＳ１１７に進むと、演算部６は、ステップＳ１０５及びステップＳ１１１で設定した各区間での噛合剛性曲線の近似式を用いて、はすば歯対の噛合開始点Ｘ_ｓから噛合終了点Ｘ_ｅまで各噛合剛性値を演算し、この演算結果を出力部８を通じて出力した後、ルーチンを終了する。
【００８１】
また、ε_α≦ε_βであると判定した場合には、今回の入力諸元に基づくはすば歯対はケース２またはケース３に分類されるはすば歯対であると判定してステップＳ１１２に進む。
【００８２】
ステップＳ１０４からステップＳ１１２に進むと、演算部６は、式（１４）〜（１９）を用いて、第１，第２の分岐点間（Ｘ_ｄ１≦Ｘ≦Ｘ_ｄ２）の区間での噛合剛性曲線の近似式を設定する。
【００８３】
続くステップＳ１１３において、演算部６は、ステップＳ１１２で設定した近似式を用いて、分岐点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｄ〔＝Ｋ（Ｘ_ｄ）／Ｋ_ｐ〕を演算する。
【００８４】
続くステップＳ１１４において、演算部６は、式（３５）〜（３８）を用いて、噛合開始点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｓを演算する。
【００８５】
続くステップＳ１１５において、演算部６は、式（２０）を用いて、歯幅歯丈比ＢＨ＝６での等価変換ねじれ角β_{０（ＢＨ＝６）}を演算する。
【００８６】
そして、ステップＳ１１５からステップＳ１１６に進むと、演算部６は、式（２４）〜（３０）、（３３）、（３４）を用いて、分岐点外の区間での噛合剛性曲線の近似式を設定する。
【００８７】
そして、ステップＳ１１６からステップＳ１１７に進むと、演算部６は、ステップＳ１１２及びステップＳ１１６で設定した各区間での噛合剛性曲線の近似式を用いて、はすば歯対の噛合開始点Ｘ_ｓから噛合終了点Ｘ_ｅまで各噛合剛性値を演算し、この演算結果を出力部８を通じて出力した後、ルーチンを終了する。
【００８８】
次に、上述の噛合剛性演算装置１を用いた噛合剛性値の具体的な演算例について説明する。なお、以下で出てくる各計算式とその計算結果が少し異なっているが、これは噛合剛性演算装置１上で所定の計算分割数を与えて演算を行っているためである。
【００８９】
（演算例１）
この演算例１では、はすば歯対の諸元として、噛合剛性演算装置１に、ｍ_ｎ＝１、Ｋ_ｓ＝１、Ｃ_ｋ＝０．３５、α_ｎ＝２０．５°、Ｂ＝４．７、β_０＝４°が入力された場合の例について説明する。
【００９０】
上記諸元が入力されて噛合剛性演算ルーチンがスタートすると、先ず、ステップＳ１０１において、演算部６は、入力諸元に基づき、歯丈Ｈ、歯幅歯丈比ＢＨ、正面噛合率ε_α、及び、重なり噛合率ε_β等の諸元計算を行う。
【００９１】
Ｈ＝２．３５
ＢＨ＝２
ε_α＝１．９１９
ε_β＝０．１０４
【００９２】
続くステップＳ１０２において、演算部６は、式（７）及び（８）を用いて、はすば歯対の噛合開始点及び噛合終了点のＸ座標（Ｘ_ｓ、及びＸ_ｅ）を以下の通り演算する。
【００９３】

【００９４】
続くステップＳ１０３において、演算部６は、式（９）及び（１０）を用いて、はすば歯対の第１，第２の分岐点のＸ座標（Ｘ_ｄ１、及びＸ_ｄ２）を以下の通り演算する。
【００９５】

【００９６】
そして、ε_α＝１．９１９、ε_β＝０．１０４であることから、ステップＳ１０４において、演算部６は、ε_α＞ε_βであると判定し、今回の入力諸元に基づくはすば歯対はケース１に分類されるはすば歯対であると判定してステップＳ１０５に進む。
【００９７】
ステップＳ１０４からステップＳ１０５に進むと、演算部６は、式（１１）〜（１３）を用いて、第１，第２の分岐点間（Ｘ_ｄ１≦Ｘ≦Ｘ_ｄ２）の区間での噛合剛性曲線の近似式を以下の通り設定する。
【００９８】

【００９９】
続くステップＳ１０６において、演算部６は、近似式（３９）を用いて、分岐点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｄ〔＝Ｋ（Ｘ_ｄ）／Ｋ_ｐ〕を以下の通り演算する。
【０１００】

【０１０１】
続くステップＳ１０７において、演算部６は、式（３５）〜（３８）を用いて、噛合開始点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｓを以下の通り演算する。
【０１０２】

【０１０３】
続くステップＳ１０８において、演算部６は、式（２０）を用いて、歯幅歯丈比ＢＨ＝６での等価変換ねじれ角β_{０（ＢＨ＝６）}を演算する。
【０１０４】

【０１０５】
そして、β_{０（ＢＨ＝６）}＝１．３３５°であることから、ステップＳ１０９において、演算部６は、β_{０（ＢＨ＝６）}＜１２°であると判定し、今回の入力諸元に基づくはすば歯対は、分岐点外（Ｘ_ｓ≦Ｘ≦Ｘ_ｄ１，Ｘ_ｄ２≦Ｘ≦Ｘ_ｅ）の区間での噛合剛性曲線を直線近似することが可能であると判定してステップＳ１１０に進む。
【０１０６】
ステップＳ１０９からステップＳ１１０に進むと、演算部６は、式（２１）〜（２３）を用いて、分岐点外の区間での噛合剛性曲線の近似式を以下の通り設定する。
【０１０７】

【０１０８】
そして、ステップＳ１１０からステップＳ１１７に進むと、演算部６は、ステップＳ１０５及びステップＳ１１０で設定した各区間での噛合剛性曲線の近似式（３９）、（４０）を用いて、はすば歯対の噛合開始点Ｘ_ｓから噛合終了点Ｘ_ｅまで各噛合剛性値を演算し、この演算結果を出力部８を通じて出力した後、ルーチンを終了する。
【０１０９】
この結果、図１５に示すように、式（１）〜（６）を用いて演算した噛合剛性値の理論値と極めて良く一致した噛合剛性値が得られる。
【０１１０】
（演算例２）
この演算例２では、はすば歯対の諸元として、噛合剛性演算装置１に、ｍ_ｎ＝１、Ｋ_ｓ＝１、Ｃ_ｋ＝０．３５、α_ｎ＝２０．５°、Ｂ＝７．０５、β_０＝３６°が入力された場合の例について説明する。
【０１１１】
上記諸元が入力されて噛合剛性演算ルーチンがスタートすると、先ず、ステップＳ１０１において、演算部６は、入力諸元に基づき、歯丈Ｈ、歯幅歯丈比ＢＨ、正面噛合率ε_α、及び、重なり噛合率ε_β等の諸元計算を行う。
【０１１２】
Ｈ＝２．３５
ＢＨ＝３
ε_α＝１．３４７
ε_β＝１．３１９
【０１１３】
続くステップＳ１０２において、演算部６は、式（７）及び（８）を用いて、はすば歯対の噛合開始点及び噛合終了点のＸ座標（Ｘ_ｓ、及びＸ_ｅ）を以下の通り演算する。
【０１１４】

【０１１５】
続くステップＳ１０３において、演算部６は、式（９）及び（１０）を用いて、はすば歯対の第１，第２の分岐点のＸ座標（Ｘ_ｄ１、及びＸ_ｄ２）を以下の通り演算する。
【０１１６】

なお、Ｘ_ｄ１，Ｘ_ｄ２は、実際には所定の演算分割数を与えて演算しているため、本演算例では何れも”０”となる。
【０１１７】
そして、ε_α＝１．３４７、ε_β＝１．３１９であることから、ステップＳ１０４において、演算部６は、ε_α＞ε_βであると判定し、今回の入力諸元に基づくはすば歯対はケース１に分類されるはすば歯対であると判定してステップＳ１０５に進む。
【０１１８】
ステップＳ１０４からステップＳ１０５に進むと、演算部６は、式（１１）〜（１３）を用いて、第１，第２の分岐点間（Ｘ_ｄ１≦Ｘ≦Ｘ_ｄ２）の区間での噛合剛性曲線の近似式を以下の通り設定する。
【０１１９】

【０１２０】
続くステップＳ１０６において、演算部６は、近似式（４１）を用いて、分岐点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｄ〔＝Ｋ（Ｘ_ｄ）／Ｋ_ｐ〕を以下の通り演算する。
【０１２１】

【０１２２】
続くステップＳ１０７において、演算部６は、式（３５）〜（３８）を用いて、噛合開始点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｓを以下の通り演算する。
【０１２３】

【０１２４】
続くステップＳ１０８において、演算部６は、式（２０）を用いて、歯幅歯丈比ＢＨ＝６での等価変換ねじれ角β_{０（ＢＨ＝６）}を演算する。
【０１２５】

【０１２６】
そして、β_{０（ＢＨ＝６）}＝１９．９６５°であることから、ステップＳ１０９において、演算部６は、β_{０（ＢＨ＝６）}≧１２°であると判定し、今回の入力諸元に基づくはすば歯対は、分岐点外（Ｘ_ｓ≦Ｘ≦Ｘ_ｄ１，Ｘ_ｄ２≦Ｘ≦Ｘ_ｅ）の区間での噛合剛性曲線を直線近似することが困難であると判定してステップＳ１１１に進む。
【０１２７】
ステップＳ１０９からステップＳ１１１に進むと、演算部６は、式（２４）〜（３２）を用いて、分岐点外の区間での噛合剛性曲線の近似式を以下の通り設定する。
【０１２８】

【０１２９】
そして、ステップＳ１１１からステップＳ１１７に進むと、演算部６は、ステップＳ１０５及びステップＳ１１１で設定した各区間での噛合剛性曲線の近似式（４１）、（４２）を用いて、はすば歯対の噛合開始点Ｘ_ｓから噛合終了点Ｘ_ｅまで各噛合剛性値を演算し、この演算結果を出力部８を通じて出力した後、ルーチンを終了する。
【０１３０】
この結果、図１６に示すように、式（１）〜（６）を用いて演算した噛合剛性値の理論値と極めて良く一致した噛合剛性値が得られる。
【０１３１】
（演算例３）
この演算例３では、はすば歯対の諸元として、噛合剛性演算装置１に、ｍ_ｎ＝１、Ｋ_ｓ＝１、Ｃ_ｋ＝０．３５、α_ｎ＝２０．５°、Ｂ＝１４．１、β_０＝３６°が入力された場合の例について説明する。
【０１３２】
上記諸元が入力されて噛合剛性演算ルーチンがスタートすると、先ず、ステップＳ１０１において、演算部６は、入力諸元に基づき、歯丈Ｈ、歯幅歯丈比ＢＨ、正面噛合率ε_α、及び、重なり噛合率ε_β等の諸元計算を行う。
【０１３３】
Ｈ＝２．３５
ＢＨ＝６
ε_α＝１．３４７
ε_β＝２．６３８
【０１３４】
続くステップＳ１０２において、演算部６は、式（７）及び（８）を用いて、はすば歯対の噛合開始点及び噛合終了点のＸ座標（Ｘ_ｓ、及びＸ_ｅ）を以下の通り演算する。
【０１３５】

【０１３６】
続くステップＳ１０３において、演算部６は、式（９）及び（１０）を用いて、はすば歯対の第１，第２の分岐点のＸ座標（Ｘ_ｄ１、及びＸ_ｄ２）を以下の通り演算する。
【０１３７】

【０１３８】
そして、ε_α＝１．３４７、ε_β＝２．６３８であることから、ステップＳ１０４において、演算部６は、ε_α≦ε_βであると判定し、今回の入力諸元に基づくはすば歯対はケース３に分類されるはすば歯対であると判定してステップＳ１１２に進む。
【０１３９】
ステップＳ１０４からステップＳ１１２に進むと、演算部６は、式（１４）〜（１９）を用いて、第１，第２の分岐点間（Ｘ_ｄ１≦Ｘ≦Ｘ_ｄ２）の区間での噛合剛性曲線の近似式を以下の通り設定する。
【０１４０】

【０１４１】
続くステップＳ１１３において、演算部６は、近似式（４３）を用いて、分岐点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｄ〔＝Ｋ（Ｘ_ｄ）／Ｋ_ｐ〕を以下の通り演算する。
【０１４２】

【０１４３】
続くステップＳ１１４において、演算部６は、式（３５）〜（３８）を用いて、噛合開始点での噛合剛性値と噛合中心点での噛合剛性値との比Ｙ_ｓを以下の通り演算する。
【０１４４】

【０１４５】
続くステップＳ１１５において、演算部６は、式（２０）を用いて、歯幅歯丈比ＢＨ＝６での等価変換ねじれ角β_{０（ＢＨ＝６）}を演算する。
【０１４６】

【０１４７】
そして、ステップＳ１１５からステップＳ１１６に進むと、演算部６は、式（２４）〜（３０）、（３３）、（３４）を用いて、分岐点外の区間での噛合剛性曲線の近似式を以下の通り設定する。
【０１４８】

【０１４９】
そして、ステップＳ１１６からステップＳ１１７に進むと、演算部６は、ステップＳ１１２及びステップＳ１１６で設定した各区間での噛合剛性曲線の近似式（４３）、（４４）を用いて、はすば歯対の噛合開始点Ｘ_ｓから噛合終了点Ｘ_ｅまで各噛合剛性値を演算し、この演算結果を出力部８を通じて出力した後、ルーチンを終了する。
【０１５０】
この結果、図１７に示すように、式（１）〜（６）を用いて演算した噛合剛性値の理論値と極めて良く一致した噛合剛性値が得られる。
【０１５１】
このような実施の形態によれば、噛合剛性曲線の挙動が変化する第１，第２の分岐点を演算し、第１の分岐点から第２の分岐点までの区間の噛合剛性曲線の近似式と、噛合開始点から第１の分岐点まで及び第２の分岐点から噛合終了点までの区間の噛合剛性曲線の近似式とを異なる関数に設定することにより実際の噛合剛性曲線と極めてよく一致した近似式を設定することができ、このように設定された近似式を用いることにより、簡単な演算処理で精度よくはすば歯対の噛合剛性値を把握することができる。
【０１５２】
また、はすば歯対の正面噛合率と重なり噛合率との大小関係を比較し、この比較結果に応じて噛合剛性曲線の近似式を異なる関数に設定することにより、実際の噛合剛性曲線と極めてよく一致した近似式を設定することができ、このように設定された近似式を用いることにより、簡単な演算処理で精度よくはすば歯対の噛合剛性値を把握することができる。
【０１５３】
これらの場合において、第１の分岐点から第２の分岐点までの区間の近似式を２次と４次の項を備えた関数に設定することにより、実際の噛合剛性曲線と極めて一致した近似式を設定することができる。その際、正面噛合率が重なり噛合率よりも小さい場合には、所定の歯幅歯丈比（例えば、ＢＨ＝６）の噛合剛性曲線に基づいて作成した近似関数を分岐点の座標に基づいて変形して近似式を設定することにより簡単な処理で精度よく近似式を設定することができる。
【０１５４】
また、正面噛合率が重なり噛合率よりも大きい場合には、噛合開始点から第１の分岐点まで及び第２の分岐点から噛合終了点までの区間の噛合剛性曲線の近似式を、ねじれ角と予め設定した設定ねじれ角との大小関係の比較結果に応じて異なる関数に設定することにより、近似式を精度よく設定することができる。具体的には、ねじれ角が設定ねじれ角よりも小さいと判定した際に、近似式を１次関数に設定することにより、近似式を精度よく設定することができる。その一方で、ねじれ角が設定ねじれ角よりも大きいと判定した際に、ねじれ角に基づいて規定される所定次数の指数関数であって上記ねじれ角に基づいて規定される所定次数の関数を備えた関数に設定することにより、近似式を精度よく設定することができる。その際、ねじれ角と設定ねじれ角との比較を、所定の歯幅歯丈比（例えばＢＨ＝６）で等価変換したねじれ角を用いて行うことにより、使用する関数の選択を一定の指標下で画一的に行うことができる。また、ねじれ角が設定ねじれ角よりも大きいと判定した場合の次数の規定を、所定の歯幅歯丈比（例えばＢＨ＝６）で等価変換したねじれ角を用いて行うことにより、画一的な演算で良好な次数の規定を行うことができる。
【０１５５】
また、正面噛合率が重なり噛合率よりも小さい場合には、噛合開始点から第１の分岐点まで及び第２の分岐点から噛合終了点までの区間の噛合剛性曲線の近似式を、ねじれ角に基づいて規定される所定次数の指数関数であって上記ねじれ角に基づいて規定される所定次数の関数を備えた関数に設定することにより、近似式を精度よく設定することができる。その際、次数の規定を、所定の歯幅歯丈比（例えばＢＨ＝６）で等価変換したねじれ角を用いて行うことにより、画一的な演算で良好な次数の規定を行うことができる。
【０１５６】
そして、このように精度よく設定された近似式から求めた噛合剛性値を用いて噛合固有振動数や回転方向振動等の各種解析を行った情報に基づき、はすば歯車対諸元設計等を行うことにより、静粛性や強度に優れたはすば歯車対を得ることができる。
【０１５７】
【発明の効果】
以上説明したように本発明によれば、広範囲な歯車諸元に対し、簡単な演算処理で精度よく、はすば歯対の噛合剛性値を把握することができる。
【図面の簡単な説明】
【図１】はすば歯対の噛合剛性演算ルーチンを示すフローチャート
【図２】はすば歯対の噛合剛性演算装置の概略構成図
【図３】はすば歯対の噛合剛性演算装置を実現するためのコンピュータの一例を示す概略図
【図４】はすば歯車対の作用平面を示す説明図
【図５】各ケースでのはすば歯対の噛合剛性曲線を示す図表
【図６】正面噛合率が重なり噛合率よりも大きいケース（ケース１）での噛合分岐点を示す説明図
【図７】正面噛合率が重なり噛合率と等しいケース（ケース２）での噛合分岐点を示す説明図
【図８】正面噛合率が重なり噛合率よりも小さいケース（ケース３）での噛合分岐点を示す説明図
【図９】ケース１での分岐点間の噛合剛性曲線を示す図表
【図１０】ケース３での分岐点間の噛合剛性曲線を示す図表
【図１１】図１０の各噛合剛性曲線を座標変換して示す図表
【図１２】分岐点が噛合開始点及び噛合終了点寄りにある場合におけるケース１での分岐点外の噛合剛性曲線を示す図表
【図１３】分岐点が噛合中心点寄りにある場合におけるケース１での分岐点外の噛合剛性曲線を示す図表
【図１４】ケース２での分岐点外の噛合剛性曲線を示す図表
【図１５】ケース１での噛合剛性曲線の理論値と近似値との比較結果を示す図表
【図１６】ケース２での噛合剛性曲線の理論値と近似値との比較結果を示す図表
【図１７】ケース３での噛合剛性曲線の理論値と近似値との比較結果を示す図表
【符号の説明】
１ … 噛合剛性演算装置
６ … 演算部（分岐点演算手段、噛合率比較手段、第１の近似式設定手段、第２の近似式設定手段、ねじれ角比較手段）[0001]
TECHNICAL FIELD OF THE INVENTION
BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a helical tooth pair engagement stiffness calculating device that calculates an engagement stiffness value from the start of meshing of a helical tooth pair to the end of meshing.
[0002]
[Prior art]
Conventionally, in helical gear pairs widely used in vehicles such as automobiles, trucks, railway vehicles, etc., and construction machinery, etc., in order to improve quietness and optimize strength, etc. Various analyzes such as numbers and vibrations in the rotational direction have been performed.
[0003]
In such an analysis, it is required to accurately grasp the fluctuation of the meshing rigidity from the start of meshing to the end of meshing of the helical tooth pair. This is performed by using a computer to solve an integral equation using an influence function of bending and contact deflection of the helical tooth pair.
[0004]
However, the calculation of the engagement stiffness value using the above integral equation is very difficult, and when a certain degree of calculation accuracy is required, the calculation time becomes enormous, and the input calculation conditions (load, There is a problem that the calculated value does not converge depending on the tooth surface shape or the like.
[0005]
To cope with this, for example, in Non-Patent Document 1, an involute that can display the change in spring stiffness from the start of meshing to the end of meshing in a parallel tooth with a pressure angle of 20 ° uses an approximation formula of the meshing rigidity of the helical tooth pair, A technique for calculating a variation in the meshing rigidity of a helical tooth pair is disclosed.
[0006]
[Non-patent document 1]
Umezawa, et al., "Vibration Characteristics of Helical Gears for Power Transmission (Spring Stiffness Approximate Expression)", Transactions of the Japan Society of Mechanical Engineers, Vol. 51, No. 469 (Showa 60-9), pp. 2316 to P2322
[0007]
[Problems to be solved by the invention]
However, the meshing stiffness value obtained from the approximation formula disclosed in Non-Patent Document 1 does not always sufficiently correspond to the actual meshing stiffness value. Since the gears are specialized in length, there is a possibility that practical gears having various pressure angles and tooth heights cannot be sufficiently coped with.
[0008]
The present invention has been made in view of the above circumstances, and for a wide range of gear specifications, simple arithmetic processing can be performed accurately and accurately to determine the meshing rigidity value of a helical tooth pair. It is an object to provide an engagement stiffness calculating device.
[0009]
[Means for Solving the Problems]
In order to solve the above problem, the invention according to claim 1 sets an approximation formula of a mesh stiffness curve based on specifications of a pair of helical teeth meshing with each other, and based on the approximation formula, starts a meshing start point and ends a meshing end. A helical tooth pair meshing stiffness calculating device for calculating each meshing stiffness value up to a point, wherein the first and second branch points at which the behavior of the helical tooth pair meshing stiffness curve fluctuates are determined. Branch point calculating means for calculating; first approximate expression setting means for setting an approximate expression of the meshing rigidity curve in a section from the first branching point to the second branching point; And a second approximation formula setting means for setting an approximation formula of the mesh stiffness curve in a section from the second junction to the first branch point and from the second branch point to the mesh end point.
[0010]
According to a second aspect of the present invention, in the first aspect of the present invention, the meshing rigidity calculating device for a helical tooth pair compares the magnitude relationship between the frontal meshing ratio of the helical tooth pair and the overlapping meshing ratio. The first and second approximation formula setting means include an approximation formula for the mesh stiffness curve in each section according to a comparison result between the front mesh ratio and the overlap mesh ratio. Are set to different functions.
[0011]
Further, according to a third aspect of the present invention, an approximation formula of a mesh stiffness curve is set based on the specifications of a pair of helical teeth meshing with each other, and a formula between a mesh start point and a mesh end point is set based on the approximation formula. A meshing stiffness calculating device for calculating the meshing stiffness value of the helical tooth pair, wherein the meshing ratio comparing means for comparing the magnitude relationship between the frontal meshing ratio and the overlapping meshing ratio of the helical tooth pair; Approximate expression setting means for setting an approximate expression of each of the engagement stiffness curves to a different function in accordance with a comparison result of the engagement ratio and the overlapping engagement ratio.
[0012]
Further, according to a fourth aspect of the present invention, there is provided the apparatus for calculating the meshing stiffness of a helical tooth pair according to the third aspect, wherein the behavior of the meshing stiffness curve of the helical tooth pair varies. A branch point calculating unit configured to calculate a branch point, wherein the approximation formula setting unit includes a first approximation that sets an approximation formula of the meshing rigidity curve in a section from the first branch point to the second branch point. Formula setting means, and second approximation formula setting means for setting an approximation formula of the mesh stiffness curve in a section from the mesh start point to the first branch point and from the second branch point to the mesh end point. And characterized in that:
[0013]
According to a fifth aspect of the present invention, there is provided the arithmetic apparatus for calculating the meshing stiffness of a helical tooth pair according to the second or fourth aspect, wherein the first approximate expression setting means converts the approximate expression to the helical position. It is a function that uses the coordinates on the equivalent action line of the tooth pair as a variable, and is characterized in that it is set to a function having second-order and fourth-order terms.
[0014]
According to a sixth aspect of the present invention, there is provided the helical tooth pair engagement stiffness calculating device according to the second, fourth or fifth aspect, wherein the engagement ratio comparing means sets the front engagement ratio to be smaller than the overlap engagement ratio. When the first approximate expression setting means determines that the value is also smaller, the first approximate expression setting means deforms the approximate function created based on the meshing stiffness curve of the predetermined tooth width-to-height ratio based on the coordinates of the bifurcation point. It is characterized in that an approximate expression is set.
[0015]
According to a seventh aspect of the present invention, there is provided the helical tooth meshing stiffness calculating apparatus according to the second, fourth, fifth, or sixth aspect, wherein the meshing ratio comparing means sets the front meshing ratio to the overlapping meshing. When it is determined that the torsion angle is larger than the ratio, the torsion angle comparing means has a torsion angle comparing means for comparing a magnitude relation between the torsion angle of the helical tooth pair and a preset torsion angle. The approximation formula is set to a different function according to the comparison result of the torsion angle comparison means.
[0016]
According to a ninth aspect of the present invention, in the helical tooth pair meshing stiffness calculating apparatus according to the seventh aspect, the torsion angle comparing means converts the torsion angle equivalently converted at a predetermined face width tooth height ratio. And comparing the magnitude relationship with the set torsion angle.
[0017]
According to a ninth aspect of the present invention, there is provided the helical tooth pair meshing rigidity calculating device according to the seventh or eighth aspect, wherein the torsion angle comparing means determines that the torsion angle is smaller than the set torsion angle. When the determination is made, the second approximation formula setting means sets the approximation formula as a linear function using the coordinates on the equivalent action line of the helical tooth pair as variables.
[0018]
According to a tenth aspect of the present invention, in the helical tooth pair meshing stiffness calculating apparatus according to the seventh or eighth aspect, the torsion angle comparing means determines that the torsion angle is larger than the set torsion angle. When the determination is made, the second approximate expression setting means is a function that uses the coordinates on the equivalent action line of the helical tooth pair as a variable, and defines the approximate expression based on the torsion angle. The function is set to a function having a predetermined-order exponential function and a predetermined-order function defined based on the torsion angle.
[0019]
According to the eleventh aspect of the present invention, in the helical tooth pair meshing stiffness calculating apparatus according to the tenth aspect of the present invention, the second approximate expression setting means performs equivalent conversion at a predetermined face width and tooth height ratio. It is characterized in that the order is defined using a twist angle.
[0020]
Further, according to a twelfth aspect of the present invention, there is provided a helical tooth meshing stiffness calculating apparatus according to the second aspect, wherein the meshing ratio comparison is performed. When the means determines that the front meshing rate is smaller than the overlapping meshing rate, the second approximation formula setting means sets the approximation formula to the coordinates on the equivalent action line of the helical tooth pair as a variable. A predetermined order exponential function defined on the basis of the torsion angle, the function having a predetermined order function defined on the basis of the torsion angle.
[0021]
According to a thirteenth aspect of the present invention, in the helical tooth pair meshing stiffness calculating device according to the twelfth aspect, the second approximate expression setting means performs equivalent conversion at a predetermined tooth width and tooth height ratio. It is characterized in that the order is defined using a twist angle.
[0022]
BEST MODE FOR CARRYING OUT THE INVENTION
Hereinafter, embodiments of the present invention will be described with reference to the drawings. 1 is a flowchart showing a routine for calculating the meshing stiffness of a helical tooth pair, FIG. 2 is a schematic configuration diagram of a meshing stiffness calculating device of a helical tooth pair, and FIG. FIG. 4 is a schematic diagram showing an example of a computer for realizing a meshing rigidity calculating device for a helical gear pair, FIG. 4 is an explanatory diagram showing an operation plane of a helical gear pair, and FIG. 5 is a helical gear in each case. FIG. 6 is a diagram showing a pair of meshing stiffness curves, FIG. 6 is an explanatory view showing a meshing branch point in a case where the frontal meshing ratio is larger than the overlapping meshing ratio (case 1), and FIG. 7 is a case where the frontal meshing ratio is equal to the overlapping meshing ratio. FIG. 8 is an explanatory diagram showing the meshing branch point in (Case 2), FIG. 8 is an explanatory diagram showing the meshing branch point in a case where the front meshing ratio overlaps and is smaller than the meshing ratio (Case 3), and FIG. FIG. 10 is a chart showing a meshing rigidity curve between points, and FIG. FIG. 11 is a chart showing the rigidity curves, FIG. 11 is a chart showing the respective meshing stiffness curves of FIG. 10 after coordinate conversion, and FIG. 12 is a chart showing the case where the branch point is near the mesh start point and the mesh end point, outside the branch point in Case 1. FIG. 13 is a chart showing an engagement rigidity curve, FIG. 13 is a chart showing an engagement rigidity curve outside the branch point in Case 1 when the branch point is near the engagement center point, and FIG. FIG. 15 is a chart showing the comparison result between the theoretical value and the approximate value of the meshing stiffness curve in Case 1, and FIG. 16 is a chart showing the comparison result between the theoretical value and the approximate value of the meshing stiffness curve in Case 2. FIG. 17 is a chart showing a comparison result between the theoretical value and the approximate value of the meshing rigidity curve in Case 3.
[0023]
Here, prior to describing the configuration of the meshing stiffness calculating apparatus for a helical tooth pair according to the present embodiment, the present applicants have analyzed the meshing stiffness value at various specifications of the helical tooth pair. Will be described.
[0024]
In FIG. 4, reference numeral 100 denotes a plane of action of a driving gear 101 and a driven gear 102 constituting a helical gear pair. As shown in FIG. 4, since the meshing contact line CC of the helical tooth pair is twisted with respect to the gear shaft, these meshing starts from point contact unlike the spur gear. That is, the meshing of the helical tooth pair starts from the point S, the oblique meshing contact line CC progresses in parallel on the action plane 100 while changing the length, and ends at the point E finally.
[0025]
First, the present applicant solves the following integral equation of the tooth bending deflection and the tooth surface contact deflection for the helical tooth pairs of various gear specifications, so that each meshing from the start of meshing to the end of meshing is performed. Deflection δ at contact line CC _x And the amount of deflection δ _x , The engagement rigidity value K (X) was determined.
[0026]
δ _x = ∫K _b (X, ξ) · P (ξ) dξ + ∫K _c (X = ξ) · P (ξ) dξ (1)
P _s = {P (ξ) d} (2)
K (X) = P _s / Δ _x … (3)
[0027]
Here, in equation (1), K is an influence function of the surface contact deflection. _c Is a free end load proposed by Suzuki et al. (See Suzuki and Umezawa, "Nearing by Gear Contact of One-Side Gears," Proceedings of the Japan Society of Mechanical Engineers, Vol. 52, No. 481 (1986), p. 2449). The theoretical formula between rollers considering the influence of distribution was used.
[0028]

[0029]
In addition, K is a bending bending influence function of the tooth. _b (See Kano and Saiki, "New bending deflection influence function of gear rack," Proceedings of the 2002 Annual Meeting of the Japan Society of Mechanical Engineers (V) No. 2314, p. 27). A high-precision formula that takes into account different bending characteristics in the tooth height direction was used.
[0030]

[0031]
The variables in the equations (1) to (6) are as follows.
[0032]
(X, y): coordinate value of the deflection observation point
(Ξ, η): Coordinate value of unit concentrated load point
P (ξ): Load distribution on the meshing contact line
E: Young's modulus [2.068 × 10 ¹¹ Nm ² ]
γ: Poisson's ratio [0.3]
ΔB: calculated division width on each meshing contact line
X: Coordinate value on the equivalent action line (see FIG. 4) of the helical tooth pair with the center of the contact tooth width being 0
U: Absolute value of deflection at the origin when load is concentrated at the origin
λ: Coordinate conversion coefficient of the bent elliptical distribution to the concentric distribution
r: radius of the deflection concentric distribution with the tooth tip as the origin
ν (r): deflection characteristic function of equivalent concentric distribution
G (η): Deflection characteristic function immediately below the concentrated load point in the tooth height direction
F (ξ): Deflection characteristic function just below the concentrated load point in the tooth width direction
[0033]
As a result of calculating and analyzing the theoretical value of each meshing stiffness from the start of meshing to the end of meshing of the helical tooth pair by various gear specifications using the above calculation formula, the present applicants found that the helical tooth pair There is a branch point (hereinafter referred to as a first branch point) where the behavior of the meshing rigidity curve changes between the meshing start point and the meshing center point, and the same between the meshing center point and the meshing end point. (Hereinafter referred to as a second branch point). Further, as a result of examining this branch point in more detail, the position was found to be the front meshing ratio ε. _α And the overlapping mesh ratio ε _β It has been found that, in addition to being determined by the magnitude relationship, the cases can be classified into three cases (case 1 to case 3) as shown in each example of FIG. 5, for example. In FIG. 5, the example shown in Case 1 is BH [ratio of tooth width to tooth height (teeth width / teeth length)] = 2, H _k (Tooth height ratio) = 2.35, α _n (Pressure angle) = 20.5 °, β ₀ (Twist angle on pitch cylinder) = Each mesh stiffness value K (X) at 4 ° is converted to mesh stiffness value K at the mesh center point. _p (K (0)) is a meshing stiffness curve normalized by (K (0)). In the example shown in Case 2, BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = 36 ° is the engagement rigidity value K (X) at the engagement center point. _p Is the meshing stiffness curve normalized by the following equation. The example shown in Case 3 is BH = 6, H _k = 2.35, α _n = 20.5 °, β ₀ = 36 ° is the engagement rigidity value K (X) at the engagement center point. _p It is an engagement rigidity curve standardized by.
[0034]
Case 1 has a front meshing ratio ε _α Overlap and the meshing ratio ε _β This is a larger case. In this case, as shown in FIG. 6, the length of the meshing contact line CC that moves in parallel from the meshing start point S to the meshing ending point E on the action plane 100 is such that the meshing contact line CC is toothed on the action plane 100. It is the longest when it crosses in the width direction. In this case, the first branch point exists at the moment when the length of the meshing contact line CC is the longest, and the second branch point exists at the moment when the length is not the longest.
[0035]
The case 2 has a front engagement ratio ε. _α Overlap and the meshing ratio ε _β Is the case. In this case, as shown in FIG. 7, the length of the meshing contact line CC that moves in parallel from the meshing start point S to the meshing ending point E on the action plane 100 is such that the meshing contact line CC is toothed on the action plane 100. It becomes the longest at the moment of crossing in the width direction and tooth height direction. In this case, at the moment when the length of the meshing contact line CC becomes the longest, the first and second branch points simultaneously exist.
[0036]
The case 3 has a front engagement ratio ε. _α Overlap and the meshing ratio ε _β Smaller case. In this case, as shown in FIG. 8, the length of the meshing contact line CC that moves in parallel from the meshing start point S to the meshing ending point E on the action plane 100 is such that the meshing contact line CC is toothed on the action plane 100. It is the longest when it crosses the length direction. In this case, the first branch point exists at the moment when the length of the meshing contact line CC is the longest, and the second branch point exists at the moment when the length is not the longest.
[0037]
Thus, each mesh stiffness curve is greatly affected by the difference in the branch point position. Further, the position of each branch point is one of the above three cases depending only on the gear specifications. Here, the coordinates on the action line of the mesh start point, the mesh end point, and the first and second branch points, that is, X _s , X _e , X _d1 , X _d2 Are calculated as follows.
[0038]
X _s = − (Ε _α + Ε _β ) / [(Ε _α / M _n H)… (7)
X _e = (Ε _α + Ε _β ) / [(Ε _α / M _n H)… (8)
X _d1 = − | (Ε _α −ε _β ) / [(Ε _α / M _n ) · H] |… (9)
X _d2 = | (Ε _α −ε _β ) / [(Ε _α / M _n ) · H] |… (10)
[0039]
Here, the variables in the equations (7) to (10) are as follows.
[0040]
m _n : Right angle module
H: Tooth length
Note that, as is clear from equations (9) and (10), the X coordinate of the first and second branch points is the meshing center point X _p , The representative of these is X _d2 Simply X _d Also called.
[0041]
Based on the above points, first, between the first and second branch points (X _d1 ≤X≤X _d2 The approximation function of each meshing stiffness value in ()) will be examined.
[0042]
FIG. 9 shows the meshing stiffness curves of the helical tooth pairs of various data classified as Case 1. In FIG. 9, BH = 2, H _k = 2.35, α _n = 20.5 °, β ₀ = Each engagement stiffness value K (X) at 20 ° is converted to the engagement stiffness value K at the engagement center point. _p BH = 4, H _k = 2.35, α _n = 20.5 °, β ₀ = Each engagement stiffness value K (X) at 20 ° is converted to the engagement stiffness value K at the engagement center point. _p And BH = 6, H _k = 2.35, α _n = 20.5 °, β ₀ = Each engagement stiffness value K (X) at 20 ° is converted to the engagement stiffness value K at the engagement center point. _p The meshing stiffness curves standardized in are respectively illustrated. As is clear from FIG. 9, in case 1, each meshing rigidity curve is on substantially the same parabola regardless of the face width-to-height ratio BH.
[0043]
FIG. 10 shows the meshing stiffness curves of the helical tooth pairs of various data classified as Case 3. In FIG. 10, BH = 4, H _k = 2.35, α _n = 20.5 °, β ₀ = 36 ° is the engagement rigidity value K (X) at the engagement center point. _p BH = 5, H _k = 2.35, α _n = 20.5 °, β ₀ = 36 ° is the engagement rigidity value K (X) at the engagement center point. _p And BH = 6, H _k = 2.35, α _n = 20.5 °, β ₀ = 36 ° is the engagement rigidity value K (X) at the engagement center point. _p The meshing stiffness curves standardized in are respectively illustrated. In this case, as shown in FIG. 10, since the meshing rigidity value K (X) of each helical tooth pair is affected by the tooth width B, the behavior differs according to the tooth width tooth height ratio BH. On the other hand, it can be seen that the meshing stiffness value curves of the respective helical tooth pairs substantially coincide on the same parabola when coordinate transformation is performed so as to match the positions of the branch points, as shown in FIG. FIG. 11 shows an example in which the first and second branch point positions of BH = 4 and 5 are matched with the first and second branch point positions of BH = 6 by coordinate conversion.
[0044]
As can be seen from the above, each mesh stiffness curve between branch points can be approximated by a bilaterally symmetric parabolic function having a quadratic term and a quartic term and having the meshing center point at the apex. In this case, the parabolic shape of case 1 is constant irrespective of the tooth width and tooth height ratio BH, but the parabolic shape of case 3 changes depending on the tooth width and tooth height ratio BH. Therefore, taking these factors into consideration, the present applicants set the distance between the branch points (−X _d1 ≤X≤X _d2 ) Was created as follows.
[0045]

[0046]

Here, the approximation functions defined by the equations (14) to (18) are based on the meshing stiffness curve with the tooth width and tooth height ratio BH = 6, and the branch point X _d By performing coordinate conversion based on the above, the meshing rigidity curve at each tooth width and tooth height ratio BH is approximated. That is, the helical tooth pair having the tooth width tooth height ratio BH = 6 is one of the frequently used helical tooth pairs, and the helical tooth pair is the tooth width tooth height. Since the ratio BH is large, the number of calculation of the engagement stiffness value can be set large. From these facts, in the equations (14) to (18), each tooth width tooth height ratio is determined based on the meshing rigidity curve of the helical teeth pair at the practical and highly approximate tooth width tooth height ratio BH = 6. Approximate the mesh stiffness curve at BH. In this case, the X coordinate (X _d ) Can be equivalently converted with the tooth width and tooth height ratio BH = 6 as in the following equation.
[0047]
X _{d (BH = 6)} = [Ε _β ・ (6 / BH) -ε _α ] / [(Ε _α / M _n H)… (19)
It is needless to say that an approximate function equivalent to the equations (14) to (18) may be created based on the meshing stiffness curve other than the tooth width and tooth height ratio BH = 6.
[0048]
Next, outside the first and second branch points (X _s ≤X≤X _d1 , X _d2 ≤X≤X _e The approximation function of each meshing stiffness value in ()) will be examined.
[0049]
In FIG. 12, among the meshing stiffness curves of the helical tooth pairs of the various data classified as Case 1, those in which the branch point is near the meshing start point and the meshing end point (that is, the torsion angle β) ₀ Is relatively small). In FIG. 12, BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = Each of the engagement stiffness values K (X) at 4 ° is the engagement stiffness value K at the engagement center point. _p BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = 8 ° is the engagement rigidity value K (X) at the engagement center point. _p BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = The engagement rigidity value K (X) at 12 ° is converted to the engagement rigidity value K at the engagement center point. _p BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = The engagement rigidity value K (X) at 16 ° is converted to the engagement rigidity value K at the engagement center point. _p BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = Each engagement stiffness value K (X) at 20 ° is converted to the engagement stiffness value K at the engagement center point. _p The meshing stiffness curves standardized in are respectively illustrated.
[0050]
FIG. 13 shows the meshing stiffness curves of the helical tooth pairs of the various data classified as Case 1 when the branch point is near the meshing center point (that is, the torsion angle β). ₀ Is relatively large). FIG. 13 shows BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = The engagement rigidity value K (X) at 24 ° is converted to the engagement rigidity value K at the engagement center point. _p BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = 28 ° is the engagement rigidity value K (X) at the engagement center point. _p BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = 32 ° is the engagement rigidity value K (X) at the engagement center point. _p The meshing stiffness curves standardized in are respectively illustrated.
[0051]
FIG. 14 shows the meshing stiffness curves of the helical tooth pairs of various data classified as Case 2. FIG. 14 shows BH = 3, H _k = 2.35, α _n = 20.5 °, β ₀ = 36 ° is the engagement rigidity value K (X) at the engagement center point. _p BH = 4, H _k = 2.35, α _n = 20.5 °, β ₀ = 30 °, the respective engagement stiffness values K (X) are converted to the engagement stiffness values K at the engagement center point. _p BH = 5, H _k = 2.35, α _n = 20.5 °, β ₀ = 26 ° is the engagement rigidity value K (X) at the engagement center point. _p And BH = 6, H _k = 2.35, α _n = 20.5 °, β ₀ = The engagement rigidity value K (X) at 22.5 ° is converted to the engagement rigidity value K at the engagement center point. _p The meshing stiffness curves standardized in are respectively illustrated. Here, the meshing stiffness curves of the helical tooth pairs of various specifications classified into Case 3 are substantially the same as those of Case 2 outside the branch point.
[0052]
As is clear from FIG. 12, the twist angle β ₀ Is relatively small, the meshing stiffness curve outside the branch point can be approximated by a straight line. 13 and 14, the twist angle β ₀ Is relatively large, it is difficult to approximate the mesh stiffness curve outside the branch point with a straight line. Based on these, the applicants conducted further studies on each meshing rigidity curve outside the branch point, and as a result, the torsion angle β ₀ With the tooth width and tooth height ratio BH = 6 (β _{0 (BH = 6)} ), Β _{0 (BH = 6)} <12 ° meshing stiffness curve can be linearly approximated, β _{0 (BH = 6)} It has been found that the meshing stiffness curve of ≧ 12 ° can be approximated by an exponential function having an A-order term and a function having a B-order term. Therefore, taking these factors into consideration, the present applicants set out of the branch point (X _s ≤X≤X _d1 , X _d2 ≤X≤X _e ) Was created as follows.
[0053]
Note that the equivalent conversion torsion angle β _{0 (BH = 6)} Is defined by the following equation.
[0054]
β _{0 (BH = 6)} = Tan ^-1 [(BH / 6) -tanβ ₀ ] ... (20)
[0055]
β _{0 (BH = 6)} <12 °
K (X) / K _p = A ₀ + A ₁ ・ | X |… (21)
a ₀ = (| X _d ｜ ・ Y _s -Y _d ・ ｜ X _s |) / (| X _d |-| X _s |)… (22)
a ₁ = (Y _d -Y _s ) / (| X _d |-| X _s |)… (23)
[0056]

[0057]
Here, in Equations (24) to (30), the orders A and B in each case are represented by the twist angle β. ₀ Is defined as follows based on
[0058]

In the equations (24) to (30), Y _s Indicates the ratio between the meshing rigidity value at the meshing start point and the meshing rigidity value at the meshing center point, and is obtained, for example, as follows.
[0059]

In the equations (24) to (30), Y _d Is the ratio of the mesh stiffness value at the branch point to the mesh stiffness value at the mesh center point [K (X _d ) / K _p Is obtained using an approximate expression set based on the expressions (11) to (13) or an approximate expression set based on the expressions (14) to (18).
[0060]
Next, a description will be given of an arithmetic unit for calculating the meshing stiffness of a helical tooth pair using each of the above-described approximation functions.
[0061]
In FIG. 2, reference numeral 1 denotes a meshing stiffness calculating device for a helical tooth pair. The meshing stiffness calculating device 1 includes an input unit 5 for inputting data of a helical tooth pair, A computing unit 6 for calculating the meshing stiffness value of the helical tooth pair based on the input data of the above, and a meshing stiffness calculation routine executed by the calculating unit 6 are stored. The storage unit 7 includes a storage unit 7 for appropriately storing the calculation result and the like in the calculation unit 6, and an output unit 8 for outputting the calculation result and the like in the calculation unit 6.
[0062]
Here, the meshing rigidity calculation device 1 is realized by, for example, a computer system 10 shown in FIG. The main part of the computer system 10, for example, is configured by connecting a keyboard 12, a display device 13, and a printer 14 to a computer main body 11 via a connection cable 15. In the computer system 10, for example, various drive devices, a keyboard 12, and the like provided in the computer main body 11 function as the input unit 5, and a CPU, ROM, RAM, and the like built in the computer main body 11 operate as an arithmetic unit. Functions as 6. In addition, a hard disk or the like built in the computer main body 11 functions as the storage unit 7, and the display device 13, the printer 14, and the like function as the output unit 8.
[0063]
In the present embodiment, the meshing stiffness calculation routine stored in the storage unit 7 sets and sets the approximation formula of the meshing stiffness according to the input data by appropriately using the above-described equations (7) to (38). This is for calculating the engagement stiffness value in each engagement state of the helical tooth pair based on the engagement stiffness approximation formula.
[0064]
The calculation unit 6 loads and executes the meshing rigidity calculation routine stored in the storage unit 7 to execute a branch point calculation unit, a meshing ratio comparison unit, and a first approximation expression setting unit (approximation expression setting unit). , A second approximation formula setting unit (approximation formula setting unit) and a twist angle comparison unit.
[0065]
Next, a description will be given of an engagement stiffness calculation routine executed by the calculation unit 6 in the engagement stiffness calculation device 1 having the above-described configuration.
[0066]
This routine is executed after the data of the helical tooth pair is input through the input unit 5. Here, in the present embodiment, the meshing rigidity calculating device 1 includes, for example, _n , Tooth height coefficient K _s , Top void coefficient C _k , Pressure angle α _n , Tooth width B and torsion angle β ₀ Is entered.
[0067]
When the routine is started, first, in step S101, the calculation unit 6 uses a well-known calculation method based on the input data to set the tooth height H, the tooth width tooth height ratio BH, and the front meshing ratio ε. _α , And the overlap meshing ratio ε _β Calculate specifications.
[0068]
In the following step S102, the calculation unit 6 uses the equations (7) and (8) to set the X coordinate (X) of the mesh start point and mesh end point of the helical tooth pair. _s , And X _e ) Is calculated.
[0069]
In the following step S103, the calculation unit 6 uses the equations (9) and (10) to set the X-coordinate (X coordinate) of the first and second branch points of the helical tooth pair. _d1 , And X _d2 ) Is calculated.
[0070]
Then, in step S104, the calculation unit 6 calculates the front meshing ratio ε obtained in step S101. _α And the overlapping mesh ratio ε _β Is compared with _α > Ε _β If it is determined that the helical tooth pair is based on the current input data, it is determined that the helical tooth pair is a helical tooth pair, and the process proceeds to step S105.
[0071]
When the process proceeds from step S104 to step S105, the arithmetic unit 6 uses the expressions (11) to (13) to calculate the distance between the first and second branch points (X _d1 ≤X≤X _d2 An approximate expression of the mesh stiffness curve in the section of ()) is set.
[0072]
In the subsequent step S106, the calculation unit 6 uses the approximation formula set in step S105 to calculate the ratio Y between the mesh stiffness value at the branch point and the mesh stiffness value at the mesh center point. _d [= K (X _d ) / K _p ] Is calculated.
[0073]
In the following step S107, the calculation unit 6 uses the equations (35) to (38) to calculate the ratio Y between the mesh stiffness value at the mesh start point and the mesh stiffness value at the mesh center point. _s Is calculated.
[0074]
In the following step S108, the calculation unit 6 calculates the equivalent conversion torsion angle β at the face width-to-height ratio BH = 6 using Expression (20). _{0 (BH = 6)} Is calculated.
[0075]
Then, in step S109, the arithmetic unit 6 sets the equivalent conversion torsion angle β _{0 (BH = 6)} Is compared with a preset torsion angle (for example, 12 °), and β _{0 (BH = 6)} If it is determined that the angle is <12 °, the helical tooth pair based on the current input data is outside the branch point (X _s ≤X≤X _d1 , X _d2 ≤X≤X _e ), It is determined that the mesh stiffness curve can be linearly approximated, and the process proceeds to step S110.
[0076]
When the process proceeds from step S109 to step S110, the calculation unit 6 sets an approximate expression of the mesh stiffness curve in a section outside the branch point using the expressions (21) to (23).
[0077]
Then, when the process proceeds from step S110 to step S117, the calculation unit 6 uses the approximation formula of the meshing stiffness curve in each section set in step S105 and step S110 to set the meshing start point X of the helical tooth pair. _s From meshing end point X _e After calculating the respective engagement stiffness values and outputting the calculation results through the output unit 8, the routine ends.
[0078]
On the other hand, in step S109, β _{0 (BH = 6)} If it is determined that ≧ 12 °, it is determined that it is difficult for the helical tooth pair based on the present input data to linearly approximate the mesh stiffness curve in a section outside the branch point. Then, the process proceeds to step S111.
[0079]
When the process proceeds from step S109 to step S111, the calculation unit 6 sets an approximation formula of the meshing stiffness curve in a section outside the branch point using formulas (24) to (32).
[0080]
Then, when the process proceeds from step S111 to step S117, the calculation unit 6 uses the approximation formula of the mesh stiffness curve in each section set in step S105 and step S111 to set the mesh start point X of the helical tooth pair. _s From meshing end point X _e After calculating the respective engagement stiffness values and outputting the calculation results through the output unit 8, the routine ends.
[0081]
Also, ε _α ≤ε _β If it is determined that the helical tooth pair is based on the present input data, it is determined that the helical tooth pair is a helical tooth pair classified into Case 2 or Case 3, and the process proceeds to Step S112.
[0082]
When the process proceeds from step S104 to step S112, the arithmetic unit 6 uses the expressions (14) to (19) to calculate the distance between the first and second branch points (X _d1 ≤X≤X _d2 An approximate expression of the mesh stiffness curve in the section of ()) is set.
[0083]
In the following step S113, the calculation unit 6 uses the approximation formula set in step S112 to calculate the ratio Y between the mesh stiffness value at the branch point and the mesh stiffness value at the mesh center point. _d [= K (X _d ) / K _p ] Is calculated.
[0084]
In the following step S114, the calculation unit 6 calculates the ratio Y between the mesh stiffness value at the mesh start point and the mesh stiffness value at the mesh center point using the equations (35) to (38). _s Is calculated.
[0085]
In the following step S115, the calculation unit 6 calculates the equivalent conversion torsion angle β at the face width-to-height ratio BH = 6 using Expression (20). _{0 (BH = 6)} Is calculated.
[0086]
Then, when the process proceeds from step S115 to step S116, the calculation unit 6 uses the equations (24) to (30), (33), and (34) to calculate the approximate expression of the mesh stiffness curve in the section outside the branch point. Set.
[0087]
Then, when the process proceeds from step S116 to step S117, the calculation unit 6 uses the approximation formula of the mesh stiffness curve in each section set in step S112 and step S116 to set the mesh start point X of the helical tooth pair. _s From meshing end point X _e After calculating the respective engagement stiffness values and outputting the calculation results through the output unit 8, the routine ends.
[0088]
Next, a specific calculation example of the engagement stiffness value using the above-described engagement stiffness operation device 1 will be described. Note that the calculation formulas and the calculation results that will be described below are slightly different from each other, because the calculation is performed on the meshing rigidity calculation device 1 by giving a predetermined calculation division number.
[0089]
(Operation example 1)
In this calculation example 1, as the data of the helical tooth pair, m _n = 1, K _s = 1, C _k = 0.35, α _n = 20.5 °, B = 4.7, β ₀ = 4 ° will be described.
[0090]
When the above specifications are input and the meshing rigidity calculation routine starts, first, in step S101, the calculation unit 6 determines the tooth height H, the face width and the tooth height ratio BH, and the front meshing ratio ε based on the input specifications. _α , And the overlap meshing ratio ε _β Calculate specifications.
[0091]
H = 2.35
BH = 2
ε _α = 1.919
ε _β = 0.104
[0092]
In the following step S102, the calculation unit 6 uses the equations (7) and (8) to set the X coordinate (X) of the mesh start point and mesh end point of the helical tooth pair. _s , And X _e ) Is calculated as follows.
[0093]

[0094]
In the following step S103, the calculation unit 6 uses the equations (9) and (10) to set the X-coordinate (X coordinate) of the first and second branch points of the helical tooth pair. _d1 , And X _d2 ) Is calculated as follows.
[0095]

[0096]
And ε _α = 1.919, ε _β = 0.104, the arithmetic unit 6 determines in step S104 that ε _α > Ε _β Is determined, the helical tooth pair based on the current input data is determined to be a helical tooth pair classified as Case 1, and the process proceeds to step S105.
[0097]
When the process proceeds from step S104 to step S105, the arithmetic unit 6 uses the expressions (11) to (13) to calculate the distance between the first and second branch points (X _d1 ≤X≤X _d2 The approximation formula of the mesh stiffness curve in the section of ()) is set as follows.
[0098]

[0099]
In the following step S106, the calculation unit 6 calculates the ratio Y between the mesh stiffness value at the branch point and the mesh stiffness value at the mesh center using the approximate expression (39). _d [= K (X _d ) / K _p ] Is calculated as follows.
[0100]

[0101]
In the following step S107, the calculation unit 6 uses the equations (35) to (38) to calculate the ratio Y between the mesh stiffness value at the mesh start point and the mesh stiffness value at the mesh center point. _s Is calculated as follows.
[0102]

[0103]
In the following step S108, the calculation unit 6 calculates the equivalent conversion torsion angle β at the face width-to-height ratio BH = 6 using Expression (20). _{0 (BH = 6)} Is calculated.
[0104]

[0105]
And β _{0 (BH = 6)} = 1.335 °, the arithmetic unit 6 determines in step S109 that β _{0 (BH = 6)} <12 °, and the helical tooth pair based on the current input data is outside the branch point (X _s ≤X≤X _d1 , X _d2 ≤X≤X _e ), It is determined that the mesh stiffness curve can be linearly approximated, and the process proceeds to step S110.
[0106]
When the process proceeds from step S109 to step S110, the calculation unit 6 sets an approximation formula of the meshing stiffness curve in the section outside the branch point as follows using formulas (21) to (23).
[0107]

[0108]
Then, when the process proceeds from step S110 to step S117, the calculation unit 6 uses the approximation formulas (39) and (40) of the meshing stiffness curve in each section set in step S105 and step S110 to set the helical tooth pair. Starting point X of _s From meshing end point X _e After calculating the respective engagement stiffness values and outputting the calculation results through the output unit 8, the routine ends.
[0109]
As a result, as shown in FIG. 15, a meshing stiffness value that very well matches the theoretical value of the meshing stiffness value calculated using Equations (1) to (6) is obtained.
[0110]
(Calculation example 2)
In this calculation example 2, the mesh stiffness calculation device 1 outputs m as the data of the helical tooth pair. _n = 1, K _s = 1, C _k = 0.35, α _n = 20.5 °, B = 7.05, β ₀ = 36 ° will be described.
[0111]
When the above specifications are input and the meshing rigidity calculation routine starts, first, in step S101, the calculation unit 6 determines the tooth height H, the face width and the tooth height ratio BH, and the front meshing ratio ε based on the input specifications. _α , And the overlap meshing ratio ε _β Calculate specifications.
[0112]
H = 2.35
BH = 3
ε _α = 1.347
ε _β = 1.319
[0113]
In the following step S102, the calculation unit 6 uses the equations (7) and (8) to set the X coordinate (X) of the mesh start point and mesh end point of the helical tooth pair. _s , And X _e ) Is calculated as follows.
[0114]

[0115]
In the following step S103, the calculation unit 6 uses the equations (9) and (10) to set the X-coordinate (X coordinate) of the first and second branch points of the helical tooth pair. _d1 , And X _d2 ) Is calculated as follows.
[0116]

Note that X _d1 , X _d2 Are actually given with a predetermined number of calculation divisions, so in this calculation example, they are all “0”.
[0117]
And ε _α = 1.347, ε _β = 1.319, the arithmetic unit 6 determines in step S104 that ε _α > Ε _β Is determined, the helical tooth pair based on the current input data is determined to be a helical tooth pair classified as Case 1, and the process proceeds to step S105.
[0118]
When the process proceeds from step S104 to step S105, the arithmetic unit 6 uses the expressions (11) to (13) to calculate the distance between the first and second branch points (X _d1 ≤X≤X _d2 The approximation formula of the mesh stiffness curve in the section of ()) is set as follows.
[0119]

[0120]
In the following step S106, the calculation unit 6 calculates the ratio Y between the mesh stiffness value at the branch point and the mesh stiffness value at the mesh center using the approximate expression (41). _d [= K (X _d ) / K _p ] Is calculated as follows.
[0121]

[0122]
In the following step S107, the calculation unit 6 uses the equations (35) to (38) to calculate the ratio Y between the mesh stiffness value at the mesh start point and the mesh stiffness value at the mesh center point. _s Is calculated as follows.
[0123]

[0124]
In the following step S108, the calculation unit 6 calculates the equivalent conversion torsion angle β at the face width-to-height ratio BH = 6 using Expression (20). _{0 (BH = 6)} Is calculated.
[0125]

[0126]
And β _{0 (BH = 6)} = 19.965 °, in step S109, the arithmetic unit 6 sets β _{0 (BH = 6)} It is determined that ≧ 12 °, and the helical tooth pair based on the current input data is outside the branch point (X _s ≤X≤X _d1 , X _d2 ≤X≤X _e It is determined that it is difficult to linearly approximate the meshing stiffness curve in the section of ()), and the process proceeds to step S111.
[0127]
When the process proceeds from step S109 to step S111, the calculation unit 6 sets an approximate expression of the meshing stiffness curve in a section outside the branch point as follows using Expressions (24) to (32).
[0128]

[0129]
Then, when proceeding from step S111 to step S117, the calculating unit 6 uses the approximation formulas (41) and (42) of the meshing stiffness curve in each section set in step S105 and step S111 to generate the helical tooth pair. Starting point X of _s From meshing end point X _e After calculating the respective engagement stiffness values and outputting the calculation results through the output unit 8, the routine ends.
[0130]
As a result, as shown in FIG. 16, a meshing stiffness value that very well matches the theoretical value of the meshing stiffness value calculated using Equations (1) to (6) is obtained.
[0131]
(Calculation example 3)
In the third calculation example, as the data of the helical teeth pair, m _n = 1, K _s = 1, C _k = 0.35, α _n = 20.5 °, B = 14.1, β ₀ = 36 ° will be described.
[0132]
When the above specifications are input and the meshing rigidity calculation routine starts, first, in step S101, the calculation unit 6 determines the tooth height H, the face width and the tooth height ratio BH, and the front meshing ratio ε based on the input specifications. _α , And the overlap meshing ratio ε _β Calculate specifications.
[0133]
H = 2.35
BH = 6
ε _α = 1.347
ε _β = 2.638
[0134]
In the following step S102, the calculation unit 6 uses the equations (7) and (8) to set the X coordinate (X) of the mesh start point and mesh end point of the helical tooth pair. _s , And X _e ) Is calculated as follows.
[0135]

[0136]
In the following step S103, the calculation unit 6 uses the equations (9) and (10) to set the X-coordinate (X coordinate) of the first and second branch points of the helical tooth pair. _d1 , And X _d2 ) Is calculated as follows.
[0137]

[0138]
And ε _α = 1.347, ε _β = 2.638, the arithmetic unit 6 determines in step S104 that ε _α ≤ε _β Is determined, the helical tooth pair based on the current input data is determined to be a helical tooth pair classified as Case 3, and the process proceeds to Step S112.
[0139]
When the process proceeds from step S104 to step S112, the arithmetic unit 6 uses the expressions (14) to (19) to calculate the distance between the first and second branch points (X _d1 ≤X≤X _d2 The approximation formula of the mesh stiffness curve in the section of ()) is set as follows.
[0140]

[0141]
In the following step S113, the calculation unit 6 calculates the ratio Y between the mesh stiffness value at the branch point and the mesh stiffness value at the mesh center point using the approximate expression (43). _d [= K (X _d ) / K _p ] Is calculated as follows.
[0142]

[0143]
In the following step S114, the calculation unit 6 calculates the ratio Y between the mesh stiffness value at the mesh start point and the mesh stiffness value at the mesh center point using the equations (35) to (38). _s Is calculated as follows.
[0144]

[0145]
In the following step S115, the calculation unit 6 calculates the equivalent conversion torsion angle β at the face width-to-height ratio BH = 6 using Expression (20). _{0 (BH = 6)} Is calculated.
[0146]

[0147]
Then, when the process proceeds from step S115 to step S116, the calculation unit 6 calculates the approximate expression of the mesh stiffness curve in the section outside the branch point using the equations (24) to (30), (33), and (34). Set as follows.
[0148]

[0149]
Then, when the process proceeds from step S116 to step S117, the calculation unit 6 uses the approximation formulas (43) and (44) of the meshing stiffness curve in each section set in step S112 and step S116 to set the helical tooth pair. Starting point X of _s From meshing end point X _e After calculating the respective engagement stiffness values and outputting the calculation results through the output unit 8, the routine ends.
[0150]
As a result, as shown in FIG. 17, a meshing stiffness value that very well matches the theoretical value of the meshing stiffness value calculated using Equations (1) to (6) is obtained.
[0151]
According to such an embodiment, the first and second branch points at which the behavior of the mesh stiffness curve changes are calculated, and the approximation of the mesh stiffness curve in the section from the first branch point to the second branch point is performed. By setting the formula and the approximation formula of the mesh stiffness curve in the section from the mesh start point to the first branch point and the section from the second branch point to the mesh end point to different functions, the actual mesh stiffness curve is extremely good. A matched approximation formula can be set, and by using the approximation formula set in this way, the meshing stiffness value of the helical tooth pair can be accurately grasped by simple calculation processing.
[0152]
Also, by comparing the magnitude relationship between the front meshing ratio and the overlapping meshing ratio of the helical tooth pair, and by setting an approximate formula of the meshing rigidity curve to a different function according to the comparison result, the actual meshing rigidity curve and the actual meshing rigidity curve are compared. It is possible to set an approximate expression that matches very well, and by using the thus set approximate expression, it is possible to accurately grasp the mesh stiffness value of the helical tooth pair with simple arithmetic processing.
[0153]
In these cases, by setting the approximate expression of the section from the first branch point to the second branch point to a function having quadratic and quartic terms, an approximation that is very consistent with the actual meshing rigidity curve Expressions can be set. At this time, if the front meshing ratio is smaller than the overlapping meshing ratio, an approximate function created based on the meshing stiffness curve of a predetermined tooth width and tooth height ratio (for example, BH = 6) is calculated based on the coordinates of the branch point. By deforming and setting an approximate expression, an approximate expression can be accurately set by a simple process.
[0154]
When the front meshing ratio is larger than the overlapping meshing ratio, the approximate expression of the meshing rigidity curve in the section from the meshing start point to the first branch point and the section from the second branching point to the meshing end point is expressed by: By setting different functions according to the comparison result of the magnitude relationship between the set value and the preset torsion angle, the approximate expression can be set with high accuracy. Specifically, when it is determined that the torsion angle is smaller than the set torsion angle, the approximate expression can be set with high accuracy by setting the approximate expression to a linear function. On the other hand, when it is determined that the torsion angle is larger than the set torsion angle, an exponential function of a predetermined order defined based on the torsion angle and a function of a predetermined order defined based on the torsion angle is provided. The approximate expression can be set with high accuracy by setting the approximate function. At this time, the torsion angle is compared with the set torsion angle by using the torsion angle equivalently converted at a predetermined tooth width / tooth height ratio (for example, BH = 6), so that the function to be used can be selected under a certain index. Can be performed uniformly. In addition, by defining the order when the torsion angle is determined to be larger than the set torsion angle by using the torsion angle equivalently converted at a predetermined tooth width / tooth height ratio (for example, BH = 6), the order is uniformly determined. A good order can be defined by simple calculations.
[0155]
If the front meshing ratio is smaller than the overlapping meshing ratio, the approximate expression of the meshing rigidity curve in the section from the meshing start point to the first branch point and the section from the second branch point to the meshing end point is expressed by: By setting the function to an exponential function of a predetermined order defined on the basis of the torsion angle and having a function of the predetermined order defined on the basis of the torsion angle, it is possible to accurately set the approximate expression. In this case, by defining the order using the torsion angle equivalently converted at a predetermined tooth width and tooth height ratio (for example, BH = 6), it is possible to define a good order by uniform calculation. .
[0156]
Then, based on the information obtained by performing various analyzes such as the natural frequency of the meshing and the vibration in the rotational direction using the meshing stiffness value obtained from the approximation formula set with high accuracy in this manner, the design of the helical gear pair and the specifications is performed. By doing so, a helical gear pair excellent in silence and strength can be obtained.
[0157]
【The invention's effect】
As described above, according to the present invention, the meshing rigidity value of a helical tooth pair can be grasped with high accuracy by simple arithmetic processing for a wide range of gear specifications.
[Brief description of the drawings]
FIG. 1 is a flowchart showing a routine for calculating the meshing rigidity of a helical tooth pair.
FIG. 2 is a schematic configuration diagram of a helical tooth pair meshing rigidity calculating device;
FIG. 3 is a schematic diagram showing an example of a computer for realizing a helical tooth pair engagement stiffness calculating device;
FIG. 4 is an explanatory diagram showing an operation plane of a helical gear pair.
FIG. 5 is a chart showing a meshing rigidity curve of a helical tooth pair in each case.
FIG. 6 is an explanatory diagram showing a meshing branch point in a case (case 1) where the front meshing ratio is larger than the overlapping meshing ratio.
FIG. 7 is an explanatory diagram showing a meshing branch point in a case (case 2) where the front meshing ratio is equal to the overlapping meshing ratio.
FIG. 8 is an explanatory diagram showing a meshing branch point in a case where the front meshing ratio overlaps and is smaller than the meshing ratio (case 3).
FIG. 9 is a chart showing an engagement stiffness curve between branch points in Case 1;
FIG. 10 is a chart showing an engagement stiffness curve between branch points in Case 3;
FIG. 11 is a table showing coordinate conversion of each meshing stiffness curve of FIG. 10;
FIG. 12 is a chart showing a mesh stiffness curve outside the branch point in Case 1 when the branch point is near the mesh start point and the mesh end point.
FIG. 13 is a chart showing a mesh stiffness curve outside the branch point in Case 1 when the branch point is near the mesh center point.
FIG. 14 is a chart showing an engagement stiffness curve outside a branch point in Case 2;
FIG. 15 is a table showing a comparison result between a theoretical value and an approximate value of an engagement stiffness curve in case 1;
FIG. 16 is a table showing a comparison result between a theoretical value and an approximate value of an engagement stiffness curve in case 2;
FIG. 17 is a table showing a comparison result between a theoretical value and an approximate value of an engagement stiffness curve in Case 3;
[Explanation of symbols]
1 ... meshing rigidity calculation device
6 ... calculation unit (branch point calculation means, meshing rate comparison means, first approximation expression setting means, second approximation expression setting means, torsion angle comparison means)

Claims (13)

An approximation formula of the mesh stiffness curve is set based on the specifications of the helical teeth pair that mesh with each other, and each mesh stiffness value between the mesh start point and the mesh end point is calculated based on the approximate formula. A tooth pair meshing stiffness calculating device,
Branch point calculating means for calculating first and second branch points at which the behavior of the meshing rigidity curve of the helical tooth pair fluctuates;
First approximation formula setting means for setting an approximation formula of the meshing stiffness curve in a section from the first branch point to the second branch point;
Second approximation formula setting means for setting an approximation formula of the mesh stiffness curve in a section from the mesh start point to the first branch point and in a section from the second branch point to the mesh end point. A helical tooth pair engagement stiffness calculating device characterized by the following. The above has an engagement ratio comparing means for comparing the magnitude relationship between the front engagement ratio of the helical teeth pair and the overlap engagement ratio,
The first and second approximate expression setting means sets an approximate expression of the engagement stiffness curve in each section to a different function according to a comparison result between the front engagement ratio and the overlap engagement ratio. The arithmetic device according to claim 1, wherein the meshing rigidity of the helical tooth pair is calculated. An approximation formula of the mesh stiffness curve is set based on the specifications of the helical teeth pair that mesh with each other, and each mesh stiffness value between the mesh start point and the mesh end point is calculated based on the approximate formula. A tooth pair meshing stiffness calculating device,
The above-mentioned meshing ratio comparing means for comparing the magnitude relationship between the front meshing ratio and the overlapping meshing ratio of the helical tooth pair,
Approximation formula setting means for setting an approximation formula of each of the meshing stiffness curves to a different function according to a comparison result between the front meshing ratio and the overlap meshing ratio. Engagement rigidity calculation device. The above-mentioned helical tooth pair has branch point calculating means for calculating first and second branch points at which the behavior of the meshing rigidity curve fluctuates;
The approximation formula setting means includes: first approximation formula setting means for setting an approximation formula of the meshing stiffness curve in a section from the first branch point to the second branch point; and 4. A second approximation formula setting means for setting an approximation formula of the mesh stiffness curve in a section up to a first branch point and in a section from the second branch point to the meshing end point. A device for calculating the mesh stiffness of a pair of helical teeth. The first approximation formula setting means sets the approximation formula to a function having a coordinate on the equivalent action line of the helical tooth pair as a variable and having a second and fourth order term. The arithmetic device for calculating the meshing rigidity of a helical tooth pair according to claim 2 or 4, wherein: When it is determined by the meshing ratio comparing means that the front meshing ratio is smaller than the overlapping meshing ratio,
The first approximation formula setting means sets the approximation formula by deforming an approximation function created based on a meshing stiffness curve of a predetermined tooth width-to-height ratio based on the coordinates of the branch point. The stiffness calculating device for a helical tooth pair according to claim 2, 4, or 5. When the engagement ratio comparing means determines that the front engagement ratio is greater than the overlap engagement ratio, a torsion angle comparison for comparing the magnitude relationship between the torsion angle of the helical tooth pair and a preset torsion angle is performed. Having means,
7. The lotus according to claim 2, wherein the second approximate expression setting means sets the approximate expression to a different function according to a result of the comparison by the torsion angle comparing means. A device for calculating the meshing stiffness of a pair of teeth. The helical tooth pair according to claim 7, wherein the torsion angle comparing means compares the magnitude relationship with the set torsion angle using a torsion angle equivalently converted at a predetermined tooth width / tooth ratio. For calculating the mesh stiffness. When the torsion angle is determined by the torsion angle comparing means to be smaller than the set torsion angle,
9. The method according to claim 7, wherein the second approximation expression setting means sets the approximation expression to a linear function having a coordinate on the equivalent action line of the helical tooth pair as a variable. A helical tooth pair meshing rigidity calculation device. When the torsion angle is determined by the torsion angle comparing means to be larger than the set torsion angle,
The second approximation formula setting means is a function that uses the approximation formula as a variable with coordinates on the equivalent action line of the helical tooth pair, and an exponential function of a predetermined order defined based on the torsion angle. 9. The apparatus according to claim 7, wherein a function of a predetermined order defined based on the torsion angle is set. 11. The arithmetic operation according to claim 10, wherein said second approximation formula setting means defines said order using a torsion angle equivalently converted at a predetermined tooth width and tooth height ratio. apparatus. When it is determined by the meshing ratio comparing means that the front meshing ratio is smaller than the overlapping meshing ratio,
The second approximation formula setting means is a function that uses the approximation formula as a variable with coordinates on the equivalent action line of the helical tooth pair, and an exponential function of a predetermined order defined based on the torsion angle. The helical tooth according to claim 2, wherein the function is set to a function of a predetermined order defined based on the torsion angle. Pair engagement rigidity calculation device. 13. The calculation of meshing rigidity of a helical tooth pair according to claim 12, wherein said second approximate expression setting means specifies the order using a torsion angle equivalently converted at a predetermined tooth width and tooth height ratio. apparatus.

Images