JP4182225B2 - Falling water film vibration suppression device and vibration suppression method - Google Patents
Falling water film vibration suppression device and vibration suppression method Download PDFInfo
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本発明は、堰やダム等から落下する水膜の振動を抑制する装置とその抑制方法に関し、特に、落下水膜の振動に起因する低周波音の発生を防止するものである。 The present invention relates to an apparatus for suppressing vibration of a water film falling from a weir, a dam, or the like and a method for suppressing the apparatus, and in particular, to prevent generation of low frequency sound due to vibration of a falling water film.
砂防ダムのような固定堰、あるいはゲートのような堰から水が膜状になって落下する場合、しばしば水膜振動が発生し、近隣の民家の障子や窓ガラスを振動させたり、ゲートそのものを振動させたりして問題となることがある。この水膜振動は、水膜背後空間の圧力変動が要因と考えられており、対策として、越流水膜をある幅で分割するスポイラー（水膜分離装置）を堰に設け、水膜背後と前方との圧力差を無くす方法が採られている。しかし、この方法を採用しても、水膜の厚さが厚い場合は、水が切れずに振動が発生する。 When water falls from a fixed weir such as a sabo dam or a weir such as a gate, water film vibration often occurs, causing the shoji and window glass of a nearby private house to vibrate, It may cause a problem by vibrating. This water film vibration is thought to be caused by pressure fluctuations in the space behind the water film. As a countermeasure, a spoiler (water film separation device) that divides the overflow water film by a certain width is installed in the weir, and behind and in front of the water film. The method of eliminating the pressure difference with However, even if this method is adopted, if the water film is thick, the water is not cut and vibration is generated.
本発明の発明者等は、先に、落下水膜の挙動について解析し、スポイラーに代わる落下水膜振動の抑制方法について提案している（下記非特許文献１）。
落下水膜の挙動解析には、図９の実験装置を使用した。この装置は、上部水槽１０と下部水槽１１とを備えており、上部水槽１０の底に開けた幅ａN、長さ３００ ｍｍのスリット１２から水が下部水槽１１へ落下し、落下高さｈの水膜１３が形成される。また、上部水槽１０と下部水槽１１との間には、スリット１２からｂだけ離れて垂直に立つ背面板１４と、水膜１３の両側に垂直に立つアクリル製の側板１５、１６とを具備しており、水膜１３、側板１５、１６、背面板１４及び下部水槽１１内の水によって閉空間が形成されている（但し、側板１５、１６及び背面板１４と上部水槽１０の底との間には僅かな隙間が存在する）。また、上部水槽１０は、ポンプで汲み上げた下部水槽１１の水を上部水槽１０に供給する給水管１７と、ハニカム状の整流板１８とを備えており、この整流板１８の作用で、上部水槽１０のスリット１２上の水は、水位ｄの静かな液面を形成する。
The inventors of the present invention have previously analyzed the behavior of a falling water film and proposed a method for suppressing falling water film vibration in place of a spoiler (Non-Patent Document 1 below).
The experimental apparatus of FIG. 9 was used for the behavior analysis of the falling water film. This apparatus includes an upper water tank 10 and a lower water tank 11, and water falls into the lower water tank 11 from a slit 12 having a width a N and a length of 300 mm opened at the bottom of the upper water tank 10. The water film 13 is formed. Further, between the upper water tank 10 and the lower water tank 11, there are provided a back plate 14 which stands vertically apart from the slit 12 by b, and acrylic side plates 15 and 16 which stand vertically on both sides of the water film 13. A closed space is formed by water in the water film 13, the side plates 15 and 16, the back plate 14 and the lower water tank 11 (however, between the side plates 15 and 16 and the back plate 14 and the bottom of the upper water tank 10. There is a slight gap). The upper water tank 10 includes a water supply pipe 17 for supplying water from the lower water tank 11 pumped up by a pump to the upper water tank 10, and a honeycomb-shaped rectifying plate 18. The water on the ten slits 12 forms a quiet liquid level d.
図１０は、水膜振動が十分発達した状態での振動の一周期にわたる水膜形状と、閉空間内の圧力変動とを示している。（ａ）〜（ｈ）の各水膜の右側が閉空間である。また、振動する水膜の山の一つに注目し、その山の各瞬間での位置を三角印で示している。
この図から、水膜振動が上から下へ向かう進行波的であること、水膜下部ほど振幅が増加していること、また、重力によって加速されるため、水膜下部ほど流速が速くなり波長が長くなっていることが分かる。閉空間内の圧力は、水膜振動と同じ振動数で周期的に変動しており、図１０（ａ）に示すように、水膜最下部が開空間側（左側）に変位したときに圧力は最高となり、図１０（ｅ）に示すように、閉空間側（右側）に変位したときに最低となる。また、この水膜最下部の変位が最大（最小）となる付近で閉空間圧力は急に増加あるいは減少している。
FIG. 10 shows the shape of the water film over one period of vibration and the pressure fluctuation in the closed space when the water film vibration is sufficiently developed. The right side of each water film of (a) to (h) is a closed space. Also, paying attention to one of the oscillating water film peaks, the position at each moment of the mountain is indicated by a triangle.
From this figure, the water film vibration is traveling wave from top to bottom, the amplitude is increasing at the bottom of the water film, and since it is accelerated by gravity, the flow velocity increases at the bottom of the water film and the wavelength It can be seen that is getting longer. The pressure in the closed space periodically fluctuates at the same frequency as the water film vibration. As shown in FIG. 10A, the pressure when the lowermost part of the water film is displaced to the open space side (left side). Becomes the highest and becomes the lowest when displaced toward the closed space (right side) as shown in FIG. Further, the closed space pressure suddenly increases or decreases near the maximum (minimum) displacement at the bottom of the water film.
また、振動していない静止状態の水膜が振動を始める場合に、いくつかの異なる振動数の水膜振動が発生する（それぞれの振動数で水膜振動する各状態を「モード」と呼ぶことにする）。図１１は、ａN＝３．１ｍｍ、ｈ＝６２０ｍｍ、ｄ＝２５０ｍｍの場合に発生した（ａ）９．８Ｈｚと（ｂ）１４．０Ｈｚの二つの振動モードの水膜振動を示している。他の実験条件では１〜４のモードが観察された。
実験条件を同じにしても複数のモードが現れる。振動発生直後の落下水膜には複数の振動モードが混在しているが、どのモードが選択されるかというメカニズムは分からないが、短時間（２〜５秒程度）でいずれか一つのモードに落ち着く。ただ、複数のモードの中には、その条件により、発生し易いモードと発生し難いモードとが存在している。
In addition, when a non-vibrating still water film starts to vibrate, water film vibrations of several different frequencies occur (each state in which a water film vibrates at each frequency is called a "mode") ). FIG. 11 shows water film vibrations in two vibration modes (a) 9.8 Hz and (b) 14.0 Hz, which occurred when a N = 3.1 mm, h = 620 mm, and d = 250 mm. In other experimental conditions, modes 1 to 4 were observed.
Even if the experimental conditions are the same, multiple modes appear. The falling water film immediately after the occurrence of vibration contains a plurality of vibration modes, but the mechanism of which mode is selected is not known, but it can be switched to any one mode in a short time (about 2 to 5 seconds). Calm down. However, among the plurality of modes, there are modes that are likely to occur and modes that are unlikely to occur depending on the conditions.
ここで、各モードにおいて、流体が上部水槽１０から落下して下部水槽１１の水面に達するまでの落下時間をＴ、その水膜１３の振動数をｆとして、Ｓ＝Ｔ×ｆを求めると、実験結果は、
Ｓ＝（整数値Ｋ）＋ (約１／４)
と表されることを示している。このＳは、落下時間の間に、ある位置の水膜が何回振動するかを示している。複数の水膜振動におけるＳが同じ場合は、それらの水膜に含まれる波の数は等しい。
以下、Ｓに含まれる整数値Ｋが１であるモードを１ｓｔモード、Ｋが２であるモードを２ｎｄモード、・・・と呼ぶことにする。図１２には、１ｓｔモード（ａ）、２ｎｄモード（ｂ）、３ｒｄモード（ｃ）、４ｔｈモード（ｄ）及び５ｔｈモード（ｅ）の各振動の形を示している。
Here, in each mode, when S = T × f is obtained, where T is the drop time until the fluid falls from the upper water tank 10 and reaches the water surface of the lower water tank 11, and f is the frequency of the water film 13. The experimental results are
S = (integer value K) + (about 1/4)
It is shown that. This S indicates how many times the water film at a certain position vibrates during the fall time. When S in a plurality of water film vibrations is the same, the number of waves contained in those water films is equal.
Hereinafter, a mode in which the integer value K included in S is 1 is referred to as a 1st mode, a mode in which K is 2 is referred to as a 2nd mode, and so on. FIG. 12 shows shapes of vibrations in the 1st mode (a), the 2nd mode (b), the 3rd mode (c), the 4th mode (d), and the 5th mode (e).
これらの水膜振動は、図１３（ａ）（実験装置（図９）の断面図）、図１３（ｂ）（図１３（ａ）のＡ−Ａ断面図）に示すように、水膜背後の閉空間を遮るバッフル板１９を、下部水槽１１の水面から、最低次モードの水膜振動の１／４波長に相当する位置に設けることで抑制できる（バッフル板１９の位置は、図９にも点線で示している）。
これは、次のような理由による。
図１４において、水膜の太線の領域は、落下水膜が閉空間の圧力に対して正の仕事をする領域を表し、水膜の細線の領域は、落下水膜が閉空間の圧力に対して負の仕事をする領域を表している。落下水膜中に含まれる１周期の波は、閉空間の圧力に対して、正の仕事をする領域と負の仕事をする領域との両方を含むから、正負の仕事がほぼ相殺される。従って、落下水膜に含まれるＫ個の波は、閉空間の圧力に大きな作用を及ぼさず、結果的に水面付近の１／４波長の水膜による仕事が、閉空間の圧力変動を生起し、水膜を振動させている。
As shown in FIG. 13A (cross-sectional view of the experimental apparatus (FIG. 9)) and FIG. 13B (cross-sectional view of AA in FIG. 13A), these water film vibrations The baffle plate 19 that blocks the closed space can be suppressed by providing it at a position corresponding to a quarter wavelength of the water film vibration in the lowest order mode from the water surface of the lower water tank 11 (the position of the baffle plate 19 is shown in FIG. 9). Is also indicated by a dotted line).
This is due to the following reason.
In FIG. 14, the thick line region of the water film represents a region where the falling water film works positively with respect to the pressure of the closed space, and the thin line region of the water film represents the pressure of the falling water film against the pressure of the closed space. Represents the area where negative work is done. Since one cycle wave contained in the falling water film includes both a positive work area and a negative work area with respect to the pressure in the closed space, the positive and negative work is almost canceled out. Therefore, the K waves contained in the falling water film do not have a large effect on the pressure in the closed space, and as a result, the work by the water film having a quarter wavelength near the water surface causes the pressure fluctuation in the closed space. The water film is vibrating.
これは、図１５の観測結果からも確かめることができる。図１５（ａ）は、静かに流れ落ちる水膜（Ｏ）の上部に開空間側より短時間だけ空気を送り（Ａ）、その外乱に基づいて継続的な水膜振動を発生させたときの水膜形状を示しており、図１５（ｂ）は、このときに測定した、水膜背後の閉空間における大気圧との圧力差ｐ（細線）と、水膜最下部の水面での変位ηh（太線）とを、横軸に時間を取って示している。ｐとηhとは高い相関を有しており、落下水膜背後の閉空間の圧力変動が、主に水膜最下部付近の変位によって齎されることを示している。
この水膜最下部付近の変位によって発生した圧力変動は、閉空間を介して水膜の上流側を揺する圧力の基となり、それによって発生した水膜の乱れが、水膜最下部付近の変位を助長し、水膜の乱れと閉空間の圧力変動とのフィードバックループが形成される。
This can be confirmed from the observation results of FIG. FIG. 15 (a) shows water when air is sent to the upper part of the water film (O) that gently flows down from the open space side (A) for a short time, and continuous water film vibration is generated based on the disturbance. FIG. 15 (b) shows the pressure difference p (thin line) between the atmospheric pressure in the closed space behind the water film and the displacement η h at the water surface at the bottom of the water film. (Bold line) shows the time on the horizontal axis. p and η h have a high correlation, indicating that the pressure fluctuation in the closed space behind the falling water film is drowned mainly by the displacement near the bottom of the water film.
The pressure fluctuation generated by the displacement near the bottom of the water film becomes the basis of the pressure that shakes the upstream side of the water film through the closed space, and the turbulence of the water film caused by this changes the displacement near the bottom of the water film. A feedback loop is formed between the turbulence of the water film and the pressure fluctuation in the closed space.
これに対して、水面から１／４波長付近の高さにバッフル板１９を取り付けた場合は、バッフル板１９の下方で発生する圧力変動の影響がバッフル板１９の上方に伝わり難くなる。その結果、フィードバックループが遮断され、水膜の乱れによる閉空間内の圧力変動が抑制され、水膜の振動が抑えられる。
また、下部水槽１１の水面から最低次モードの振動の１／４波長位置にバッフル板１９を取り付けた場合は、高次モードの振動の１／４波長位置が、すべてバッフル板１９の下になるため、高次モードの振動で発生する圧力変動の影響もバッフル板１９の上方には伝わらない。
Further, when the baffle plate 19 is attached to the quarter wavelength position of the vibration of the lowest order mode from the water surface of the lower water tank 11, all the quarter wavelength positions of the vibration of the higher order mode are below the baffle plate 19. Therefore, the influence of the pressure fluctuation generated by the vibration of the higher order mode is not transmitted above the baffle plate 19.
しかし、非特許文献１で提案した落下水膜の振動抑制方法では、バッフル板の下方で発生する圧力変動の影響がバッフル板の上方に伝わらないようにするため、バッフル板先端を落下水膜にできるだけ近づけて配置する必要があるが、バッフル板と水膜との距離が近いと、風圧等を受けた水膜がバッフル板に乗り上げる事態が起こり得る。このとき、バッフル板の上側において振動数がＳ（＝（整数値Ｋ）＋ (約１／４)）の水膜振動が発生し、非特許文献１の方法では、この振動を抑えることができない。
そのため、この方法を実際の堰等に適用した場合、落下水膜の振動を安定して抑制することが難しい。
However, in the method of suppressing vibration of the falling water film proposed in Non-Patent Document 1, the tip of the baffle plate is used as the falling water film so that the influence of pressure fluctuation generated below the baffle plate is not transmitted to the upper side of the baffle plate. Although it is necessary to arrange as close as possible, if the distance between the baffle plate and the water film is short, a situation may occur in which the water film subjected to wind pressure or the like rides on the baffle plate. At this time, a water film vibration having a frequency of S (= (integer value K) + (about 1/4)) is generated on the upper side of the baffle plate, and the method of Non-Patent Document 1 cannot suppress this vibration. .
Therefore, when this method is applied to an actual weir or the like, it is difficult to stably suppress the vibration of the falling water film.
本発明は、こうした問題点を解決するものであり、落下水膜の振動を安定して抑制することができる落下水膜の振動抑制方法と振動抑制装置とを提供することを目的としている。 The present invention has been made to solve these problems, and an object of the present invention is to provide a falling water film vibration suppressing method and a vibration suppressing apparatus capable of stably suppressing the vibration of the falling water film.
本発明は、水膜の背後に当該水膜の水膜振動によって圧力変動を受ける空間が形成される落下水膜の振動を抑制する振動抑制方法であって、前記落下水膜が水膜の形状で落下を開始する水膜発生位置から水膜の形状での落下を終了する落下終了位置までの落下距離が、前記落下水膜の幅方向において不均一となるように、前記落下水膜の落下位置の一部に水膜の形状での落下を終了させる１または複数個の障害物を配置し、当該障害物が存在しないときに前記落下水膜に発生する水膜振動において、落下終了位置の変位と逆位相の変位を発生する位置が、前記障害物による落下終了位置となるように設定することを特徴とする。
水膜背後の空間の圧力変動を抑えることにより、水膜振動は抑制される。
The present invention relates to a the suppressing vibration suppressing method vibrations fall water film space subjected to pressure variations by the water film vibration of the water film is formed behind the water film, the falling water film is the water film shape The falling water film drops so that the falling distance from the water film generation position where the falling starts to the end position where the falling in the shape of the water film ends is uneven in the width direction of the falling water film. One or a plurality of obstacles for ending the fall in the form of a water film are arranged at a part of the position , and in the water film vibration generated in the falling water film when the obstacle does not exist , The position where the displacement having the opposite phase to the displacement is set to be the position where the obstacle is dropped is set .
By suppressing the pressure fluctuation in the space behind the water film, the water film vibration is suppressed.
また、本発明は、水膜の背後に当該水膜の水膜振動によって圧力変動を受ける空間が形成される落下水膜の振動を抑制する振動抑制方法であって、前記落下水膜が水膜の形状で落下を開始する水膜発生位置から水膜の形状での落下を終了する落下終了位置までの落下距離が、前記落下水膜の幅方向において不均一となるように、前記落下水膜の落下位置に水膜の形状での落下を終了させる、前記幅方向に傾斜した傾斜面を配置し、当該傾斜面が存在しないときに前記落下水膜に発生する水膜振動において、前記空間に及ぼす圧力変動が打消し合う変位を発生する位置が、前記傾斜面による落下終了位置に含まれるように設定することを特徴とする。
この傾斜面により、水膜の落下距離は連続的に変化し、無限段の段差を設けたことに相当する。
Further, the present invention provides a the suppressing vibration suppressing method vibrations fall water film space subjected to pressure variations by the water film vibration of the water film is formed behind the water film, the falling water film is the water film The falling water film so that the falling distance from the water film generating position where the falling starts in the shape of the water to the falling end position where the falling in the shape of the water film ends is not uniform in the width direction of the falling water film In the water film vibration generated in the falling water film when there is no inclined surface, the inclined surface inclined in the width direction is disposed at the falling position of the water film, and in the space, It is characterized in that the position where the displacement in which the applied pressure fluctuations cancel each other is included in the fall end position by the inclined surface is set .
By this inclined surface, the falling distance of the water film changes continuously, which corresponds to providing an infinite step.
また、本発明は、水膜の背後に当該水膜の水膜振動によって圧力変動を受ける空間が形成される落下水膜の振動を抑制する振動抑制装置であって、前記落下水膜が水膜の形状で落下を開始する水膜発生位置から水膜の形状での落下を終了する落下終了位置までの落下距離が、前記落下水膜の幅方向において不均一となるように、前記落下水膜の落下位置の一部に水膜の形状での落下を終了させる１または複数個の障害物を配置し、当該障害物が存在しないときに前記落下水膜に発生する水膜振動において、落下終了位置の変位と逆位相の変位を発生する位置が、前記障害物による落下終了位置となるように設定したことを特徴とする。
障害物の作用で水膜背後の空間の圧力変動が抑えられ、水膜振動が抑制される。
The present invention is also a vibration suppressing device that suppresses vibration of a falling water film in which a space subject to pressure fluctuation is formed behind the water film due to vibration of the water film , wherein the falling water film is a water film. The falling water film so that the falling distance from the water film generating position where the falling starts in the shape of the water to the falling end position where the falling in the shape of the water film ends is not uniform in the width direction of the falling water film One or a plurality of obstacles for ending the fall in the form of a water film are arranged at a part of the fall position of the water, and the drop ends in the water film vibration generated in the fall water film when the obstacle does not exist The position at which the displacement having the opposite phase to the displacement of the position is set to be the falling end position by the obstacle.
The pressure fluctuation of the space behind the water film is suppressed by the action of the obstacle , and the water film vibration is suppressed.
また、本発明は、水膜の背後に当該水膜の水膜振動によって圧力変動を受ける空間が形成される落下水膜の振動を抑制する振動抑制装置であって、前記落下水膜が水膜の形状で落下を開始する水膜発生位置から水膜の形状での落下を終了する落下終了位置までの落下距離が、前記落下水膜の幅方向において不均一となるように、前記落下水膜の落下位置に水膜の形状での落下を終了させる、前記幅方向に傾斜した傾斜面を配置し、当該傾斜面が存在しないときに前記落下水膜に発生する水膜振動において、前記空間に及ぼす圧力変動が打消し合う変位を発生する位置が、前記傾斜面による落下終了位置に含まれるように設定したことを特徴とする。
この傾斜面が、水膜の落下距離を連続的に変化させる。
The present invention is also a vibration suppressing device that suppresses vibration of a falling water film in which a space subject to pressure fluctuation is formed behind the water film due to vibration of the water film , wherein the falling water film is a water film. The falling water film so that the falling distance from the water film generating position where the falling starts in the shape of the water to the falling end position where the falling in the shape of the water film ends is not uniform in the width direction of the falling water film In the water film vibration generated in the falling water film when there is no inclined surface, the inclined surface inclined in the width direction is disposed at the falling position of the water film, and in the space, It is characterized in that the position where the displacement in which the applied pressure fluctuation cancels is generated is set to be included in the drop end position by the inclined surface.
This inclined surface continuously changes the falling distance of the water film.
本発明の落下水膜の振動抑制方法及び振動抑制装置は、水膜の水量が多い場合でも、また、落下水膜に風などの外乱が加わった場合でも、安定して水膜振動を抑えることができる。 The falling water film vibration suppressing method and vibration suppressing apparatus of the present invention can stably suppress water film vibration even when the amount of water in the water film is large or when a disturbance such as wind is applied to the falling water film. Can do.
（第１の実施形態）
本発明の第１の実施形態では、水膜の落下距離を段階的に違えるように構成した落下水膜の振動抑制装置について説明する。
この装置は、図１に示すように、下部水槽１１に、落下水膜１３の一部分だけが乗り上げる箱状の障害物２０を備えている。その他の構成は、図９と変わりがない。この装置では、水膜１３、側板１５、１６、背面板１４、障害物２０及び下部水槽１１内の水によって閉空間が形成される（但し、側板１５、１６及び背面板１４と上部水槽１０の底との間には僅かな隙間が存在する）。
(First embodiment)
In the first embodiment of the present invention, a description will be given of a falling water film vibration suppression device configured to change the falling distance of the water film stepwise.
As shown in FIG. 1, this apparatus includes a box-shaped obstacle 20 on which only a part of the falling water film 13 rides on the lower water tank 11. Other configurations are the same as in FIG. In this apparatus, a closed space is formed by the water in the water film 13, the side plates 15 and 16, the back plate 14, the obstacle 20 and the water in the lower water tank 11 (however, the side plates 15 and 16, the back plate 14 and the upper water tank 10 There is a slight gap between the bottom).
図２は、水膜１３の正面方向から障害物２０を見た図であり、図２（ａ）は１つの障害物２０１を配置して、水膜の落下距離を２段階に設定した場合、図２（ｂ）は２つの障害物２０２、２０３を配置して、水膜の落下距離を３段階に設定した場合を示している。
図３は、図１の装置の落下高さｈ＝６２０ｍｍ、流出速度（上部水槽１０のスリット１２から放出される水膜の初速度）ｕN＝１．８２ｍ／ｓに設定し、この装置の下部水槽１１に、図２（ａ）の１つの障害物２０１を配置したときの障害物２０１の高さＣ１（ｍｍ）と、発生する水膜振動との関係を表に纏めている。
FIG. 2 is a view of the obstacle 20 viewed from the front direction of the water film 13, and FIG. 2A shows a case where one obstacle 201 is arranged and the falling distance of the water film is set in two stages. FIG. 2B shows a case where two obstacles 202 and 203 are arranged and the falling distance of the water film is set in three stages.
FIG. 3 shows that the height of the apparatus of FIG. 1 is set to h = 620 mm, the outflow speed (the initial speed of the water film discharged from the slit 12 of the upper water tank 10) u N = 1.82 m / s. The relationship between the height C1 (mm) of the obstacle 201 when one obstacle 201 in FIG. 2A is arranged in the lower water tank 11 and the generated water film vibration is summarized in a table.
図３の左端にはＣ１（ｍｍ）の値、次欄には「系の安定性」を記載し、その右側に、発生した水膜振動のＳ（＝振動数ｆ×落下時間Ｔ）の値を１ｓｔモード、２ｎｄモード、３ｒｄモード、４ｔｈモード及び５ｔｈモードの各モードに分けて記載している。
「系の安定性」は、常に振動が発生し「不安定」な状態を「Ｕ」、水膜を扇いだりして外乱を加えたときにのみ振動が発生する「準安定」な状態を「ＱＳ」、外乱を加えても振動が発生しない「安定」な状態を「Ｓ」で表している。
また、Ｓ＝ｆＴの計算に用いたｆ（周波数）は、水膜を撮影した高速ビデオの映像を解析して求めており、また、Ｔ（落下時間）は、初速度ｕNの流体が重力ｇの影響を受けて距離ｈだけ落下するときの時間を表す数式１に、便宜的にｈ＝６２０ｍｍを代入して算出している。
Ｔ＝｛√（２ｇｈ＋ｕN 2）−ｕN｝／ｇ （数式１）
＝０．２１５秒
そのため、Ｓにおける少数点以下の値は、大きな意味を持たず、Ｓの整数部分、即ち、どのモードの水膜振動が発生したかが意味を持つ。
The value of C1 (mm) is described at the left end of FIG. 3, “system stability” is described in the next column, and the value of S (= frequency f × drop time T) of the generated water film vibration is displayed on the right side. Are divided into 1st mode, 2nd mode, 3rd mode, 4th mode and 5th mode.
“System stability” means “U” when the vibration is always generated and “unstable”, and “metastable” state where the vibration is generated only when a disturbance is applied by fanning the water film. “QS” and “S” represents a “stable” state in which no vibration is generated even when a disturbance is applied.
In addition, f (frequency) used in the calculation of S = fT is obtained by analyzing a high-speed video image obtained by photographing a water film, and T (falling time) is obtained when the fluid at the initial velocity u N is gravity. It is calculated by substituting h = 620 mm for the sake of convenience in Formula 1 that represents the time when it falls by the distance h under the influence of g.
T = {√ (2gh + u N 2 ) −u N } / g (Formula 1)
= 0.215 seconds Therefore, the value below the decimal point in S does not have a large meaning, and it has a meaning that the integer part of S, that is, in which mode water film vibration has occurred.
図３において、Ｃ１＝０ｍｍは障害物２０１を設置していない状態であり、この場合、２ｎｄモードまたは３ｒｄモードの振動が現れる。Ｃ１＝０〜５０ｍｍの範囲では常に振動が発生し「不安定」である。また、Ｃ１＝７０、９０ｍｍでは「準安定」である。このように、障害物２０１の高さＣ１によって発生モードと安定性とが変化する。 In FIG. 3, C1 = 0 mm is a state in which no obstacle 201 is installed. In this case, vibration in the 2nd mode or the 3rd mode appears. In the range of C1 = 0 to 50 mm, vibration always occurs and is “unstable”. Further, it is “metastable” when C1 = 70 and 90 mm. Thus, the generation mode and the stability change depending on the height C1 of the obstacle 201.
図４は、各モードの波形を、水面に相当する図の下端から上方向に正を取る座標で表し、各波形の上に図３で記載した各モードの振動発生位置（灰色部分）を重ね合わせて表示している。図中の破線は、各モードの水膜の変位が、水面での変位と逆位相になる高さを結んだ線である。
この図４から、障害物２０１の高さＣ１が破線近傍にあれば、水膜振動は発生せず、逆に、障害物２０１の高さＣ１が破線から離れていると、振動が発生し易い不安定な状態にあることが分かる。
FIG. 4 shows the waveform of each mode as a positive coordinate upward from the lower end of the figure corresponding to the water surface, and the vibration generation position (gray part) of each mode described in FIG. 3 is superimposed on each waveform. Displayed together. The broken line in the figure is a line connecting the heights at which the displacement of the water film in each mode is opposite in phase to the displacement on the water surface.
From FIG. 4, if the height C1 of the obstacle 201 is in the vicinity of the broken line, water film vibration does not occur. Conversely, if the height C1 of the obstacle 201 is far from the broken line, vibration is likely to occur. It turns out that it is in an unstable state.
従って、あるモードにおける水面での水膜の変位と逆位相に当たる高さ付近に段差を設ければ、そのモードの振動発生が抑制できることを示している。
これは、水面に正（負）方向に変位した水膜が衝突して正 (負)の圧力が発生する場合に、段差部分では、負（正）方向に変位した水膜が障害物２０１に衝突して負 (正)の圧力を発生し、これらが相殺されて水膜を揺する圧力変化が生じないためである。
Therefore, if a step is provided in the vicinity of the height corresponding to the phase opposite to the displacement of the water film on the water surface in a certain mode, the generation of vibrations in that mode can be suppressed.
This is because when a water film displaced in the positive (negative) direction collides with the water surface and a positive (negative) pressure is generated, the water film displaced in the negative (positive) direction on the obstacle 201 at the stepped portion. This is because a negative (positive) pressure is generated by collision, and these pressures are canceled out to cause no pressure change that shakes the water film.
このことから、様々な高さの段差を設け、各モードの振動発生を抑制することにより、水膜に外乱を加えても振動が発生しない「安定」な状態を作り出せることが分かる。
図５（ａ）は、この装置の下部水槽１１に、図２（ｂ）のように高さＣ２＝１５０ｍｍの障害物２０２と、高さＣ１の障害物２０３とを配置し、障害物２０３の高さＣ１を変えたときに発生する水膜振動を表に纏めている。この場合は、Ｃ１＝５０、７０ｍｍのときに「安定」な状態になる。
From this, it can be seen that by providing steps with various heights and suppressing the occurrence of vibration in each mode, it is possible to create a “stable” state in which no vibration is generated even when a disturbance is applied to the water film.
In FIG. 5A, an obstacle 202 having a height C2 = 150 mm and an obstacle 203 having a height C1 are arranged in the lower tank 11 of the apparatus as shown in FIG. The water film vibration generated when the height C1 is changed is summarized in the table. In this case, a “stable” state is obtained when C1 = 50 and 70 mm.
この装置では、障害物２０２の高さＣ２が１ｓｔモード、２ｎｄモード及び３ｒｄモードの振動発生を抑えており、障害物２０３の高さＣ１が残りの４ｔｈモード及び５ｔｈモードの振動発生を抑える高さのとき「安定」状態になる。
また、図５（ｂ）は、同様に、この装置の下部水槽１１に、高さＣ２＝２００ｍｍの障害物２０２と、高さＣ１の障害物２０３とを配置し、障害物２０３の高さＣ１を変えたときに発生する水膜振動を表示している。この場合は、Ｃ１＝５０〜９０ｍｍのときに「安定」な状態になる。
In this apparatus, the height C2 of the obstacle 202 suppresses the occurrence of vibrations in the 1st mode, 2nd mode, and 3rd mode, and the height C1 of the obstacle 203 suppresses the occurrence of vibrations in the remaining 4th mode and 5th mode. It becomes “stable” state.
Similarly, FIG. 5B shows that an obstacle 202 having a height C2 = 200 mm and an obstacle 203 having a height C1 are arranged in the lower water tank 11 of the apparatus, and the height C1 of the obstacle 203 is shown. The water film vibration that occurs when changing the is displayed. In this case, a “stable” state is obtained when C1 = 50 to 90 mm.
この装置では、障害物２０２の高さＣ２が１ｓｔモード、２ｎｄモード及び５ｔｈモードの振動発生を抑えており、障害物２０３の高さＣ１が残りの３ｒｄモード及び４ｔｈモードの振動発生を抑える高さのとき「安定」状態になる。
また、図５（ｃ）は、同様に、この装置の下部水槽１１に、高さＣ２＝２５０ｍｍの障害物２０２と、高さＣ１の障害物２０３とを配置し、障害物２０３の高さＣ１を変えたときに発生する水膜振動を表示している。この場合は、Ｃ１＝７０ｍｍのときに「安定」な状態になる。
In this device, the height C2 of the obstacle 202 suppresses the occurrence of vibrations in the 1st mode, 2nd mode, and 5th mode, and the height C1 of the obstacle 203 suppresses the occurrence of vibrations in the remaining 3rd mode and 4th mode. It becomes “stable” state.
Similarly, in FIG. 5C, an obstacle 202 having a height C2 = 250 mm and an obstacle 203 having a height C1 are arranged in the lower water tank 11 of the apparatus, and the height C1 of the obstacle 203 is shown. The water film vibration that occurs when changing the is displayed. In this case, a “stable” state is obtained when C1 = 70 mm.
この装置では、障害物２０２の高さＣ２が１ｓｔモード及び４ｔｈモードの振動発生を抑えており、障害物２０３の高さＣ１が残りの２ｎｄモード、３ｒｄモード及び５ｔｈモードの振動発生を抑える高さのとき「安定」状態になる。
このように、この落下水膜の振動抑制装置では、水膜最下部が水面に落下衝突したときに発生する圧力変動に対して、位相差を持つ圧力変動を発生するように水膜の下端に段差を設けており、こうすることで、水膜背後の閉空間内部の圧力変動を相殺し、水膜の振動の発生を抑制することができる。
なお、水膜の下端における段差は、３以上に設定しても良い。
In this apparatus, the height C2 of the obstacle 202 suppresses the occurrence of vibrations in the 1st mode and the 4th mode, and the height C1 of the obstacle 203 suppresses the occurrence of vibrations in the remaining 2nd mode, 3rd mode, and 5th mode. It becomes “stable” state.
As described above, in this falling water film vibration suppression device, the pressure fluctuation at the lower end of the water film is generated so as to generate a pressure fluctuation having a phase difference with respect to the pressure fluctuation generated when the lowermost part of the water film collides with the water surface. A step is provided, and by doing so, pressure fluctuations in the closed space behind the water film can be offset, and vibration of the water film can be suppressed.
The step at the lower end of the water film may be set to 3 or more.
（第２の実施形態）
本発明の第２の実施形態では、水膜の落下距離を連続的に変えて落下水膜の振動を抑制する装置について説明する。
この装置は、図６（斜視図）及び図７（ａ）（正面図）、図７（ｂ）（側面図）に示すように、下部水槽１１の水膜１３の落下位置に斜めの底板２２を備えており、また、上部水槽１０の水出口の片側にアクリル製の三角板２１を備えている。その他の構成は、図９と変わりがない。
(Second Embodiment)
In the second embodiment of the present invention, an apparatus for continuously controlling the falling distance of the water film to suppress the vibration of the falling water film will be described.
As shown in FIG. 6 (perspective view), FIG. 7 (a) (front view), and FIG. 7 (b) (side view), this apparatus has an inclined bottom plate 22 at the position where the water film 13 of the lower tank 11 drops. In addition, an acrylic triangular plate 21 is provided on one side of the water outlet of the upper water tank 10. Other configurations are the same as in FIG.
この装置では、上部水槽１０のスリット１２から流れ出た水が、三角板２１に沿って流れ落ち、三角板２１の斜辺を通過して始めて振動可能な水膜１３となる。この水膜１３の落下距離は、三角板２１の斜辺位置から底板２２までの長さとなる。また、水膜背後の閉空間は、水膜１３、側板１５、１６、背面板１４、底板２２及び下部水槽１１内の水によって形成される（但し、側板１５、１６及び背面板１４と上部水槽１０の底との間には僅かな隙間が存在する）。 In this apparatus, the water flowing out from the slit 12 of the upper water tank 10 flows down along the triangular plate 21 and becomes a vibrated water film 13 only after passing through the oblique side of the triangular plate 21. The falling distance of the water film 13 is the length from the hypotenuse position of the triangular plate 21 to the bottom plate 22. The closed space behind the water film is formed by water in the water film 13, the side plates 15 and 16, the back plate 14, the bottom plate 22 and the lower water tank 11 (however, the side plates 15 and 16, the back plate 14 and the upper water tank There is a slight gap between the bottom of 10).
図８は、三角板２１及び底板２２の傾斜を種々変えたときに発生する水膜振動を表に纏めている。表中のＳ（＝ｆＴ）の値は、水膜を撮影した高速ビデオの映像から求めたｆ（周波数）と、数式１のｈに、便宜的に図７（ａ）の（６２０−Ｃｄ）（但し、Ｃｄは立てかけた底板２２の高さ）の値を代入して求めたＴ（落下時間）とを乗算して算出している。Ｓにおける少数点以下の値は、大きな意味を持たず、Ｓの整数部分、即ち、どのモードの水膜振動が発生したかが意味を持つ。
図８（ａ）は、三角板２１の垂直な辺の長さＣｕをＣｕ＝０ｍｍ（即ち、三角板２１が存在しない状態）に設定し、底板２２を立てかける高さＣｄを変えたときに発生する水膜振動を表に纏めている。この場合、Ｃｄ＝１８０ｍｍ〜３００ｍｍのときに「安定」状態になる。
FIG. 8 summarizes the water film vibrations generated when the inclinations of the triangular plate 21 and the bottom plate 22 are variously changed. The value of S (= fT) in the table is f (frequency) obtained from the video of the high-speed video obtained by photographing the water film, and h in Equation 1 for the sake of convenience (620-Cd) in FIG. (However, Cd is calculated by multiplying by T (fall time) obtained by substituting the value of the leaning bottom plate 22). A value below the decimal point in S does not have a large meaning, but has an integer part of S, that is, which mode of water film vibration has occurred.
FIG. 8A shows water generated when the length Cu of the vertical side of the triangular plate 21 is set to Cu = 0 mm (that is, the triangular plate 21 is not present) and the height Cd against which the bottom plate 22 is leaned is changed. The membrane vibration is summarized in the table. In this case, a “stable” state is obtained when Cd = 180 mm to 300 mm.
また、図８（ｂ）は、三角板２１をＣｕ＝８０ｍｍに設定し、底板２２の高さＣｄを変えたときに発生する水膜振動を表示している。この場合も、Ｃｄ＝１８０ｍｍ〜３００ｍｍのときに「安定」状態になる。
また、図８（ｃ）は、三角板２１をＣｕ＝１２０ｍｍに設定し、底板２２の高さＣｄを変えたときに発生する水膜振動を表示している。この場合には、Ｃｄ＝２００ｍｍ〜３００ｍｍのときに「安定」状態になる。
また、図８（ｄ）は、三角板２１をＣｕ＝１６０ｍｍに設定し、底板２２の高さＣｄを変えたときに発生する水膜振動を表示している。この場合、Ｃｄ＝０〜６０ｍｍ及びＣｄ＝２４０ｍｍ〜３００ｍｍのときに「安定」状態になる。
また、図８（ｅ）は、三角板２１をＣｕ＝２００ｍｍに設定し、底板２２の高さＣｄを変えたときに発生する水膜振動を表示している。この場合は、Ｃｄ＝０〜３００ｍｍの全てにおいて「安定」状態になる。
FIG. 8B shows water film vibration that occurs when the triangular plate 21 is set to Cu = 80 mm and the height Cd of the bottom plate 22 is changed. Also in this case, a “stable” state is obtained when Cd = 180 mm to 300 mm.
FIG. 8C shows water film vibration that occurs when the triangular plate 21 is set to Cu = 120 mm and the height Cd of the bottom plate 22 is changed. In this case, a “stable” state is obtained when Cd = 200 mm to 300 mm.
FIG. 8D shows water film vibration that occurs when the triangular plate 21 is set to Cu = 160 mm and the height Cd of the bottom plate 22 is changed. In this case, a “stable” state is obtained when Cd = 0 to 60 mm and Cd = 240 mm to 300 mm.
FIG. 8E shows the water film vibration that occurs when the triangular plate 21 is set to Cu = 200 mm and the height Cd of the bottom plate 22 is changed. In this case, all of Cd = 0 to 300 mm are in a “stable” state.
図８から明らかなように、三角板２１の傾斜が増加すると水膜振動は発生しにくくなり、実験範囲の中で最も傾斜のきついＣｕ＝２００ｍｍ（図８（ｅ））の条件では、どのＣｄにおいても水膜振動が発生しない。
一方、三角板２１の斜辺と底板とが平行に近いと、水膜振動が成長し易く、不安定性が強い。
また、図８（ａ）から明らかなように、Ｃｄ＝１８０ｍｍ以上の底板２２のみを取り付けたときに振動は完全に抑制される。
また、図８（ｄ）から明らかなように、Ｃｕ＝１６０ｍｍの三角板２１のみを取り付けたときも完全に振動が抑制される。
As can be seen from FIG. 8, when the inclination of the triangular plate 21 increases, water film vibrations are less likely to occur, and at any Cd under the condition of the most inclined Cu = 200 mm (FIG. 8 (e)) in the experimental range. No water film vibration occurs.
On the other hand, if the oblique side of the triangular plate 21 and the bottom plate are nearly parallel, the water film vibration is likely to grow and the instability is strong.
As is clear from FIG. 8A, vibration is completely suppressed when only the bottom plate 22 having Cd = 180 mm or more is attached.
Further, as apparent from FIG. 8D, vibration is completely suppressed even when only the triangular plate 21 with Cu = 160 mm is attached.
このように、水膜の発生箇所に三角板２１を配置し、あるいは、水面に斜めの底板２２を設けることにより、高い振動抑制の効果が得られる。これは、三角板２１や底板２２の設置が無限段の段差を設けることに相当し、その間に含まれる全ての振動モードが抑制されるためである。
即ち、上部水槽１０のスリット１２に三角板２１を取り付けた場合は、水膜振動の波面が三角板２１の斜辺と平行に斜めになり、水面に衝突したときの水膜の各点の変位に位相差が生じる。このとき各点で発生する圧力にも同じく位相差が生じて打ち消しあうために、水膜背後の閉空間における圧力変動が発生せず、水膜振動は抑制される。
As described above, by arranging the triangular plate 21 at the location where the water film is generated or by providing the inclined bottom plate 22 on the water surface, a high vibration suppressing effect can be obtained. This is because the installation of the triangular plate 21 and the bottom plate 22 corresponds to providing an infinite step, and all the vibration modes included therebetween are suppressed.
That is, when the triangular plate 21 is attached to the slit 12 of the upper water tank 10, the wave front of the water film vibration is inclined in parallel with the oblique side of the triangular plate 21, and the phase difference is caused by the displacement of each point of the water film when it collides with the water surface. Occurs. At this time, the pressure generated at each point also has the same phase difference and cancels out. Therefore, the pressure fluctuation in the closed space behind the water film does not occur, and the water film vibration is suppressed.
また、下部水槽１１に斜めの底板２２を設けた場合は、底板２２に衝突した水膜の各点の変位に位相差が生じる。このとき各点で発生する圧力にも同じく位相差が生じて打ち消しあうために、水膜背後の閉空間での圧力変動が発生せず、水膜振動は抑制される。
一方、三角板２１の斜辺と底板２２とが平行に近い場合は、落下水膜の各点での落下距離が等しくなり、波面の衝突時の位相が揃ってしまうために水膜振動が発生する。
Further, when the inclined bottom plate 22 is provided in the lower water tank 11, a phase difference occurs in the displacement of each point of the water film colliding with the bottom plate 22. At this time, the pressure generated at each point also has a phase difference and cancels out. Therefore, the pressure fluctuation in the closed space behind the water film does not occur, and the water film vibration is suppressed.
On the other hand, when the oblique side of the triangular plate 21 and the bottom plate 22 are nearly parallel, the falling distance at each point of the falling water film is equal, and the phase at the time of the collision of the wave fronts is uniform, so water film vibration occurs.
このように、落下水膜の振動は、水膜の落下距離を不均一に設定し、落下終了位置における水膜の水平方向の変位に位相差を発生させて、水膜背後の閉空間の圧力変動を相殺することにより、抑制できる。
この方法に依れば、落下水膜の厚さに関わらず、水膜振動を安定的に抑えることが可能である。
なお、底板２２の傾斜面は曲面であっても良く、三角板２１の傾斜辺は曲線であっても良い。
Thus, the vibration of the falling water film sets the water film falling distance non-uniformly, generates a phase difference in the horizontal displacement of the water film at the end position of the falling, and the pressure in the closed space behind the water film It can be suppressed by offsetting the fluctuations.
According to this method, it is possible to stably suppress water film vibration regardless of the thickness of the falling water film.
The inclined surface of the bottom plate 22 may be a curved surface, and the inclined side of the triangular plate 21 may be a curved line.
本発明の落下水膜の振動抑制方法及び振動抑制装置は、堰やダムから落下する水膜の振動を効果的に抑制することが可能であり、この振動に起因する低周波音公害を防止するために、各所の堰やダム等で利用することができる。 The vibration suppression method and the vibration suppression device of the falling water film of the present invention can effectively suppress the vibration of the water film falling from the weir or dam, and prevent the low frequency sound pollution caused by this vibration. Therefore, it can be used at dams and dams in various places.
１０ 上部水槽
１１ 下部水槽
１２ スリット
１３ 水膜
１４ 背面板
１５ 側板
１６ 側板
１７ 給水管
１８ 整流板
１９ バッフル板
２０ 障害物
２１ 三角板
２２ 底板
２０１ 障害物
２０２ 障害物
２０３ 障害物
DESCRIPTION OF SYMBOLS 10 Upper water tank 11 Lower water tank 12 Slit 13 Water film 14 Back board 15 Side board 16 Side board 17 Water supply pipe 18 Current plate 19 Baffle board 20 Obstacle 21 Triangle board 22 Bottom board 201 Obstacle 202 Obstacle 203 Obstacle
Claims (4)
前記落下水膜が水膜の形状で落下を開始する水膜発生位置から水膜の形状での落下を終了する落下終了位置までの落下距離が、前記落下水膜の幅方向において不均一となるように、前記落下水膜の落下位置の一部に水膜の形状での落下を終了させる１または複数個の障害物を配置し、当該障害物が存在しないときに前記落下水膜に発生する水膜振動において、落下終了位置の変位と逆位相の変位を発生する位置が、前記障害物による落下終了位置となるように設定することを特徴とする落下水膜の振動抑制方法。 A vibration suppression method for suppressing vibration of a falling water film in which a space that receives pressure fluctuation is formed behind the water film due to water film vibration of the water film ,
Fall distance to fall end position where the falling water film finishes falling in the form of water film from the water film generated at which to begin dropping in water film shape, it becomes uneven in the width direction of the falling water film As described above, one or a plurality of obstacles for ending the fall in the shape of the water film are arranged at a part of the falling position of the falling water film, and the falling water film is generated when the obstacle does not exist. A method for suppressing vibration of a falling water film , wherein a position where a displacement having a phase opposite to a displacement of a falling end position occurs in the water film vibration is a falling end position due to the obstacle .
前記落下水膜が水膜の形状で落下を開始する水膜発生位置から水膜の形状での落下を終了する落下終了位置までの落下距離が、前記落下水膜の幅方向において不均一となるように、前記落下水膜の落下位置に水膜の形状での落下を終了させる、前記幅方向に傾斜した傾斜面を配置し、当該傾斜面が存在しないときに前記落下水膜に発生する水膜振動において、前記空間に及ぼす圧力変動が打消し合う変位を発生する位置が、前記傾斜面による落下終了位置に含まれるように設定することを特徴とする落下水膜の振動抑制方法。 A vibration suppression method for suppressing vibration of a falling water film in which a space that receives pressure fluctuation is formed behind the water film due to water film vibration of the water film ,
The falling distance from the water film generation position where the falling water film starts falling in the shape of the water film to the end position where the falling in the shape of the water film ends is non-uniform in the width direction of the falling water film As described above, an inclined surface inclined in the width direction is disposed at the falling position of the falling water film to end the falling in the shape of the water film, and water generated in the falling water film when the inclined surface does not exist In the membrane vibration, the falling water film vibration suppressing method is characterized in that a position where a displacement at which pressure fluctuations exerted on the space cancel each other is included in a falling end position by the inclined surface .
前記落下水膜が水膜の形状で落下を開始する水膜発生位置から水膜の形状での落下を終了する落下終了位置までの落下距離が、前記落下水膜の幅方向において不均一となるように、前記落下水膜の落下位置の一部に水膜の形状での落下を終了させる１または複数個の障害物を配置し、当該障害物が存在しないときに前記落下水膜に発生する水膜振動において、落下終了位置の変位と逆位相の変位を発生する位置が、前記障害物による落下終了位置となるように設定したことを特徴とする落下水膜の振動抑制装置。The falling distance from the water film generation position where the falling water film starts falling in the shape of the water film to the end position where the falling in the shape of the water film ends is non-uniform in the width direction of the falling water film As described above, one or a plurality of obstacles for ending the fall in the shape of the water film are arranged at a part of the falling position of the falling water film, and the falling water film is generated when the obstacle does not exist. In the water film vibration, a falling water film vibration suppressing device is set such that a position where a displacement having a phase opposite to that of the falling end position is generated is a falling end position due to the obstacle.
前記落下水膜が水膜の形状で落下を開始する水膜発生位置から水膜の形状での落下を終了する落下終了位置までの落下距離が、前記落下水膜の幅方向において不均一となるように、前記落下水膜の落下位置に水膜の形状での落下を終了させる、前記幅方向に傾斜した傾斜面を配置し、当該傾斜面が存在しないときに前記落下水膜に発生する水膜振動において、前記空間に及ぼす圧力変動が打消し合う変位を発生する位置が、前記傾斜面による落下終了位置に含まれるように設定したことを特徴とする落下水膜の振動抑制装置。The falling distance from the water film generation position where the falling water film starts falling in the shape of the water film to the end position where the falling in the shape of the water film ends is non-uniform in the width direction of the falling water film As described above, an inclined surface inclined in the width direction is disposed at the falling position of the falling water film to end the falling in the shape of the water film, and water generated in the falling water film when the inclined surface does not exist In the membrane vibration, the falling water film vibration suppressing device is set such that a position where a displacement in which pressure fluctuations exerted on the space cancel each other is included in the falling end position by the inclined surface.
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