CDH1 MP3 63 36 200A1X B1CGEMWW重載型液壓缸設計【液壓油缸設計】

CDH1 MP3 63 36 200A1X B1CGEMWW重載型液壓缸設計【液壓油缸設計】,液壓油缸設計,CDH1,MP3,63,36,200A1X,B1CGEMWW重載型液壓缸設計【液壓油缸設計】,a1x,b1cgemww,重載,液壓缸,設計,液壓油 寧XX大學 畢業(yè)設計(論文) CDH1 MP3/63/36/200A1X/B1CGEMWW 重載型液壓缸設計 所在學院 專 業(yè) 班 級 姓 名 學 號 指導老師 年 月 日 摘 要 本液壓油缸以傳遞動力為主，保證足夠的動力是其基本要求。另外，還要考慮油缸的穩(wěn)定性、可靠性、可維護性、安全性及效率。其中穩(wěn)定是指系統(tǒng)工作時的運動平穩(wěn)性及系統(tǒng)性能的穩(wěn)定性(如環(huán)境溫度對油液的影響等因素)?？煽啃允侵赶到y(tǒng)不因意外的原因而無法工作(如油管破裂、無電等情況)。可維護性是指系統(tǒng)盡可能簡單，元件盡可能選標準件，結(jié)構上盡可能使維護方便．安全性是指不因液壓缸的故障導致后車廂蓋的其它事故．效率是指液壓缸的各種能量損失盡可能的小。上述要求中，除滿足系統(tǒng)的動力要求外，最重要的是保證系統(tǒng)的安全性和可靠性。 關鍵詞：液壓缸，油缸 27 Abstract The hydraulic system to transfer power, ensure adequate power is its basic requirement. In addition, consider the system stability, reliability, maintainability, safety and efficiency. The stabilizing means when the system works steady motion and system performance stability (such as environmental temperature on the influence of oil etc). Reliability refers to the system is not due to accident reason to work ( such as tubing rupture without electricity, etc. ). Maintainability is referred to the system as simple as possible, element is chosen as far as possible standard parts, structure as much as possible so that the maintenance is convenient. Security is not due to the fault of the hydraulic system causes the antenna frame collapse or other accidents (such as the drop out of control, antenna due to gravity acceleration whereabouts ) . Efficiency refers to the hydraulic system of the various energy loss as small as possible. The above requirements, in addition to meet the power requirements, the most important thing is to ensure the safety and reliability of the system. Keywords: hydraulic system, 目 錄 摘 要 II Abstract III 目 錄 IV 1 油缸主參數(shù)的確定 5 2油缸主要部位的計算校核 9 2.1液壓缸穩(wěn)定性計算 9 2.2 缸筒的加工要求 10 2.3法蘭設計 11 2.4(缸筒端部）法蘭連接螺栓的強度計算 11 3 活塞的設計 13 4 導向套的設計與計算 14 5 端蓋和缸底的設計與計算 16 6 缸體長度的確定 17 7 緩沖裝置的設計 17 8 排氣裝置 18 9 密封件的選用 19 10 防塵圈 21 11 液壓缸的安裝連接結(jié)構 22 總 結(jié) 25 致 謝 26 參考文獻 27 基本技術數(shù)據(jù)，是根據(jù)用途及結(jié)構類型來確定的，它反映了工作能力及特點，也基本上上確定了輪廓尺寸及本體總質(zhì)量等。 1 油缸主參數(shù)的確定 本課題設計的油缸CDH1 MP3/63/36/200A1X/B1CGEMWW重載型液壓缸設計重載型液壓缸設計，查表 查得油缸的液壓缸的內(nèi)徑為63mm,活塞桿直徑為36mm,有效行程為200 mm 表4.1 液壓缸內(nèi)徑系列 mm 8 10 12 16 20 25 32 40 50 63 80 100 125 160 200 250 320 400 500 1. 液壓缸缸體厚度計算 缸體是液壓缸中最重要的零件，當液壓缸的工作壓力較高和缸體內(nèi)經(jīng)較大時，必須進行強度校核。缸體的常用材料為20、25、35、45號鋼的無縫鋼管。在這幾種材料中45號鋼的性能最為優(yōu)良，所以這里選用45號鋼作為缸體的材料。 式中，——實驗壓力，MPa。當液壓缸額定壓力Pn5.1MPa時，Py=1.5Pn，當Pn 16MPa時，Py=1.25Pn。 []——缸筒材料許用應力，N/mm。[]=，為材料的抗拉強度。 注：1.額定壓力Pn 額定壓力又稱公稱壓力即系統(tǒng)壓力，Pn=25MPa 2.最高允許壓力Pmax Pmax1.5Pn=1.2525=31.25MPa 液壓缸缸筒材料采用45鋼，則抗拉強度：σb=600MPa 安全系數(shù)n按《液壓傳動與控制手冊》P243表2—10，取n=5。 則許用應力[]==120MPa = =8.2mm ,滿足。取液壓缸厚度10mm。 取液壓缸缸體外徑為83mm。 4.液壓缸長度的確定 液壓缸工作行程長度可以根據(jù)執(zhí)行機構實際工作的最大行程確定，并參照表4-4選取標準值。液壓缸活塞行程參數(shù)優(yōu)先次序按表4-4中的a、b、c選用。 表4-4（a）液壓缸行程系列（GB 2349-80）[6] 25 50 80 100 125 160 200 250 320 400 500 630 800 1000 1250 1600 2000 2500 3200 4000 表4-4（b） 液壓缸行程系列（GB 2349-80）[6] 40 63 90 110 140 180 220 280 360 450 550 700 900 1100 1400 1800 2200 2800 3600 表4-4（c） 液壓缸形成系列（GB 2349-80）[6] 240 260 300 340 380 420 480 530 600 650 750 850 950 1050 1200 1300 1500 1700 1900 2100 2400 2600 3000 3400 3800 液壓缸長度L根據(jù)工作部件的行程長度確定。 L==200mm 查油缸參數(shù)得到的 5. 活塞桿直徑的設計 查《液壓傳動與控制手冊》根據(jù)桿徑比d/D，一般的選取原則是：當活塞桿受拉時，一般選取d/D=0.3-0.5，當活塞桿受壓時，一般選取d/D=0.5-0.7。本設計我選擇d/D=0.55，即d=0.7D=0.55×63=34.65mm。 表4.2 活塞桿直徑系列 4 5 6 8 10 12 14 16 18 20 22 25 28 32 36 40 45 50 56 63 70 80 90 100 110 125 140 160 180 200 220 250 280 320 360 400 故取d=36mm。 2.活塞桿強度計算： 式中 ————許用應力；（Q235鋼的抗拉強度為375-500MPa，取400MPa，為位安全系數(shù)取5，即活塞桿的強度適中） 3．活塞桿的結(jié)構設計 活塞桿的外端頭部與負載的拖動電機機構相連接，為了避免活塞桿在工作生產(chǎn)中偏心負載力，適應液壓缸的安裝要求，提高其作用效率，應根據(jù)負載的具體情況，選擇適當?shù)幕钊麠U端部結(jié)構。 4.活塞桿的密封與防塵 活塞桿的密封形式有Y形密封圈、U形夾織物密封圈、O形密封圈、V形密封圈等[6]。采用薄鋼片組合防塵圈時，防塵圈與活塞桿的配合可按H9/f9選取。薄鋼片厚度為0.5mm。為方便設計和維護，本方案選擇O型密封圈。 2油缸主要部位的計算校核 2.1液壓缸穩(wěn)定性計算 液壓缸穩(wěn)定性計算 活塞桿受軸向壓縮負載時，它所承受的力不能超過使它保持穩(wěn)定工作所允許的臨界負載，以免發(fā)生縱向彎曲，破壞液壓缸的正常工作。的值與活塞桿材料性質(zhì)、截面形狀、直徑和長度以及液壓缸的安裝方式等因素有關。若活塞桿的長徑比且桿件承受壓負載時，則必須進行液壓缸穩(wěn)定性校核?；钊麠U穩(wěn)定性的校核依下式進行 式中，為安全系數(shù)，一般取=2~4。 a.當活塞桿的細長比時 b.當活塞桿的細長比時 式中，為安裝長度，其值與安裝方式有關，見表1；為活塞桿橫截面最小回轉(zhuǎn)半徑，；為柔性系數(shù)，其值見表3-2； 為由液壓缸支撐方式?jīng)Q定的末端系數(shù)，其值見表1；為活塞桿材料的彈性模量，對鋼?。粸榛钊麠U橫截面慣性矩；為活塞桿橫截面積；為由材料強度決定的實驗值，為系數(shù)，具體數(shù)值見表3-3。 表3-2液壓缸支承方式和末端系數(shù)的值 支承方式 支承說明 末端系數(shù) 一端自由一端固定 1/4 兩端鉸接 1 一端鉸接一端固定 2 兩端固定 4 表3-3 、、的值 材料 鑄鐵 5.6 1/1600 80 鍛鐵 2.5 1/9000 110 鋼 4.9 1/5000 85 c.當時,缸已經(jīng)足夠穩(wěn)定，不需要進行校核。 此設計安裝方式中間固定的方式，此缸已經(jīng)足夠穩(wěn)定，不需要進行穩(wěn)定性校核。 2.2 缸筒的加工要求 缸筒內(nèi)徑采用H7級配合，表面粗糙度為0.16，需要進行研磨； 熱處理：調(diào)制，HB240； 缸筒內(nèi)徑的圓度、錐度、圓柱度不大于內(nèi)徑公差之半； 剛通直線度不大于0.03mm； 油口的孔口及排氣口必須有倒角，不能有飛邊、毛刺； 在缸內(nèi)表面鍍鉻，外表面刷防腐油漆。 2.3法蘭設計 液壓缸的端蓋形式有很多，較為常見的是法蘭式端蓋。本次設計選擇法蘭式端蓋 （缸筒端部）法蘭厚度根據(jù)下式進行計算： 式（3-8） 式中， -法蘭厚度（m）； —密封環(huán)內(nèi)經(jīng)d=40mm（m）； 密封環(huán)外徑（m）；=50mm 系統(tǒng)工作壓力（pa）；=25MPa 附加密封力（Pa）；值取其材料屈服點353MPa； 螺釘孔分布圓直徑（m）；=55mm 密封環(huán)平均直徑（m）；=45mm 法蘭材料的許用應力（Pa）；[]=/n=353/5=70.6MPa —法蘭受力總合力（m） 所以=13.2mm 為了安全取=14mm 2.4(缸筒端部）法蘭連接螺栓的強度計算 連接圖如下： 圖3-1缸體端部法蘭用螺栓連接 1-前端蓋；2-缸筒 螺栓強度根據(jù)下式計算： 螺紋處的拉應力: (MPa) 式（3-9） 螺紋處的剪應力 (MPa) 式（3-10） 合成應力 (MPa) 式（3-11） 式中, —液壓缸的最大負載，=A，單桿時，雙桿是 —螺紋預緊系數(shù)，不變載荷=1.25~1.5，變載荷=2.5~4； —液壓缸內(nèi)徑； —缸體螺紋外徑； —螺紋內(nèi)經(jīng)； —螺紋內(nèi)摩擦因數(shù)，一般取=0.12;變載荷取=2.5~4； —材料許用應力，,為材料的屈服極限，n為安全系數(shù)，一般取n=1.2~1.5； Z—螺栓個數(shù)。 最大推力為： 使用4個螺栓緊固缸蓋，即：=4 螺紋外徑和底徑的選擇： =10mm =8mm 系數(shù)選擇：選取=1.3=0.12 根據(jù)式（3-9）得到螺紋處的拉應力為： = 根據(jù)式（3-10）得到螺紋處的剪應力為： 根據(jù)式（3-11）得到合成應力為： ==367.6MPa 由以上運算結(jié)果知，應選擇螺栓等級為12.9級； 查表的得：抗拉強度極限=1220MP;屈服極限強度=1100MP； 不妨取安全系數(shù)n=2 可以得到許用應力值：[]=/n=1100/2=550MP 證明選用螺栓等級合適。 3 活塞的設計 活塞的寬度一般取=（0.6-1.0） 即=（0.6-1.0）×63=（37.5-63）mm 取=40mm 由于活塞在液壓力的作用下沿缸筒往復滑動，因此，它與缸筒的配合應適當，既不能過緊，也不能間隙過大。配合過緊，不僅使最低啟動壓力增大，降低機械效率，而且容易損壞缸筒和活塞的配合表面；間隙過大，會引起液壓缸內(nèi)部泄露，降低容積效率，使液壓缸達不到要求的設計性能。 活塞與缸體的密封形式分為：間隙密封（用于低壓系統(tǒng)中的液壓缸活塞的密封）、活塞環(huán)密封（適用于溫度變化范圍大、要求摩擦力小、壽命長的活塞密封）、密封圈密封三大類。其中密封圈密封又包括O形密封圈（密封性能好，摩擦因數(shù)小，安裝空間?。?、Y形密封圈（用在20Mpa壓力下、往復運動速度較高的液壓缸密封）、形密封圈（耐高壓，耐磨性好，低溫性能好，逐漸取代Y形密封圈）、V形密封圈（可用于50Mpa壓力下，耐久性好，但摩擦阻力大）。綜合以上因素，考慮選用O型密封圈。 4 導向套的設計與計算 1.最小導向長度H的確定 當活塞桿全部伸出時，從活塞支承面中點到到導向套滑動面中點的距離稱為最小導向長度[1]。如果導向長度過短，將使液壓缸因間隙引起的初始撓度增大，影響液壓缸工作性能和穩(wěn)定性。因此，在設計時必須保證液壓缸有一定的最小導向長度。根據(jù)經(jīng)驗,當液壓缸最大行程為L,缸筒直徑為D時,最小導向長度為: （4-5） 一般導向套滑動面的長度A，在缸徑小于80mm時取A=(0.6~1.0)D，當缸徑大于80mm時取A=(0.6~1.0)d.?；钊麑挾菳取B=(0.6~1.0)D。若導向長度H不夠時,可在活塞桿上增加一個導向套K(見圖4-1)來增加H值。隔套K的寬度。 圖4-1 液壓缸最小導向長度[1] 2.導向套的結(jié)構 導向套有普通導向套、易拆導向套、球面導向套和靜壓導向套等，可按工作情況適當選擇。 1）普通導向套 這種導向套安裝在支承座或端蓋上，油槽內(nèi)的壓力油起潤滑作用和張開密封圈唇邊而起密封作用[6]。 2）易拆導向套 這種導向套用螺釘或螺紋固定在端蓋上。當導向套和密封圈磨損而需要更換時，不必拆卸端蓋和活塞桿就能進行，維修十分方便。它適用于工作條件惡劣，需經(jīng)常更換導向套和密封圈而又不允許拆卸液壓缸的情況下。 3）球面導向套 這種導向套的外球面與端蓋接觸，當活塞桿受一偏心負載而引起方向傾斜時，導向套可以自動調(diào)位，使導向套軸線始終與運動方向一致，不產(chǎn)生“憋勁“現(xiàn)象。這樣，不僅保證了活塞桿的順利工作，而且導向套的內(nèi)孔磨損也比較均勻。 4）靜壓導向套 活塞桿往復運動頻率高、速度快、振動大的液壓缸，可以采用靜壓導向套。由于活塞桿與導向套之間有壓力油膜，它們之間不存在直接接觸，而是在壓力油中浮動，所以摩擦因數(shù)小、無磨損、剛性好、能吸收振動、同軸度高，但制造復雜，要有專用的靜壓系統(tǒng)。 5 端蓋和缸底的設計與計算 在單活塞液壓缸中，有活塞桿通過的端蓋叫端蓋，無活塞桿通過的缸蓋叫缸頭或缸底。端蓋、缸底與缸筒構成密封的壓力容腔，它不僅要有足夠的強度以承受液壓力，而且必須具有一定的連接強度。端蓋上有活塞桿導向孔（或裝導向套的孔）及防塵圈、密封圈槽，還有連接螺釘孔，受力情況比較復雜，設計的不好容易損壞。 1.端蓋的設計計算 端蓋厚h為： 式中 D1——螺釘孔分布直徑，cm； P——液壓力，； ——密封環(huán)形端面平均直徑，cm； ——材料的許用應力，。 2.缸底的設計 缸底分平底缸，橢圓缸底，半球形缸底。 2.端蓋的結(jié)構 端蓋在結(jié)構上除要解決與缸體的連接與密封外，還必須考慮活塞桿的導向，密封和防塵等問題[6]。缸體端部的連接形式有以下幾種： A．焊接 特點是結(jié)構簡單，尺寸小，質(zhì)量小，使用廣泛。缸體焊接后可能變形，且內(nèi)缸不易加工。主要用于柱塞式液壓缸。 B．螺紋連接（外螺紋、內(nèi)螺紋） 特點是徑向尺寸小，質(zhì)量較小，使用廣泛。缸體外徑需加工，且應與內(nèi)徑同軸；裝卸徐專用工具；安裝時應防止密封圈扭曲。 C．法蘭連接 特點是結(jié)構較簡單，易加工、易裝卸，使用廣泛。徑向尺寸較大，質(zhì)量比螺紋連接的大。非焊接式法蘭的端部應燉粗。 D．拉桿連接 特點是結(jié)構通用性好。缸體加工容易，裝卸方便，使用較廣。外形尺寸大，質(zhì)量大。用于載荷較大的雙作用缸。 E．半球連接，它又分為外半環(huán)和內(nèi)半環(huán)兩種。外半環(huán)連接的特點是質(zhì)量比拉桿連接小，缸體外徑需加工。半環(huán)槽消弱了缸體，為此缸體壁厚應加厚。內(nèi)半環(huán)連接的特點是結(jié)構緊湊，質(zhì)量小。安裝時端部進入缸體較深，密封圈有可能被進油口邊緣擦傷。 F．鋼絲連接 特點是結(jié)構簡單，尺寸小，質(zhì)量小。 6 缸體長度的確定 液壓缸缸體內(nèi)部長度應等于活塞的行程與活塞的寬度之和。缸體外形長度還需要考慮到兩端端蓋的厚度[1]。一般液壓缸缸體長度不應大于缸體內(nèi)經(jīng)的20~30倍。 7 緩沖裝置的設計 液壓缸的活塞桿（或柱塞桿）具有一定的質(zhì)量，在液壓力的驅(qū)動下運動時具有很大的動量。在它們的行程終端，當桿頭進入液壓缸的端蓋和缸底部分時，會引起機械碰撞，產(chǎn)生很大的沖擊和噪聲。采用緩沖裝置，就是為了避免這種機械撞擊，但沖擊壓力仍然存在，大約是額定工作壓力的兩倍，這就必然會嚴重影響液壓缸和整個液壓缸的強度及正常工作。緩沖裝置可以防止和減少液壓缸活塞及活塞桿等運動部件在運動時對缸底或端蓋的沖擊，在它們的行程終端能實現(xiàn)速度的遞減，直至為零。 當液壓缸中活塞活塞運動速度在6m/min以下時，一般不設緩沖裝置，而運動速度在12m/min以上時，不需設置緩沖裝置。在該組合機床液壓缸中，動力滑臺的最大速度為4m/min,因此沒有必要設計緩沖裝置。 8 排氣裝置 如果排氣裝置設置不當或者沒有設置排氣裝置，壓力油進入液壓缸后，缸內(nèi)仍會存在空氣[6]。由于空氣具有壓縮性和滯后擴張性,會造成液壓缸和整個液壓缸在工作中的顫振和爬行,影響液壓缸的正常工作。比如液壓導軌磨床在加工過程中，這不僅會影響被加工表面的光潔程度和精度，而且會損壞砂輪和磨頭等機構。為了避免這種現(xiàn)象的發(fā)生，除了防止空氣進入液壓缸外，還必須在液壓缸上設置排氣裝置。配氣裝置的位置要合理，由于空氣比壓力油輕，總是向上浮動，因此水平安裝的液壓缸，其位置應設在缸體兩腔端部的上方；垂直安裝的液壓缸，應設在端蓋的上方。 一般有整體排氣塞和組合排氣塞兩種。整體排氣塞如圖4-2（a）所示。 表4-5 排氣閥（塞）尺寸[6] d 閥座 閥桿 孔 c D M16 6 11 6 19.2 9 3 2 31 17 10 8.5 3 48 4~6 23 M20x2 8 14 7 25.4 11 4 3 39 22 13 11 4 59 4~8 28 圖4-2 (a) 整體排氣孔 圖4-2（b） 組合排氣孔 圖4-2（c） 整體排氣閥零件結(jié)構尺寸 由于螺紋與缸筒或端面連接，靠頭部錐面起密封作用。排氣時，擰松螺紋，缸內(nèi)空氣從錐面空隙中擠出來并經(jīng)過斜孔排除缸外。這種排氣裝置簡單、方便，但螺紋與錐面密封處同軸度要求較高，否則擰緊排氣塞后不能密封，造成外泄漏。組合排氣塞如圖4-2（b）所示，一般由絡螺塞和錐閥組成。螺塞擰松后，錐閥在壓力的推動下脫離密封面排出空氣。排氣裝置的零件圖及尺寸圖見4-2（c）以及表4-2（d）。 圖4-2（d） 組合排氣閥零件結(jié)構尺寸 9 密封件的選用 1.對密封件的要求 液壓缸工作中要求達到零泄漏、摩擦小和耐磨損的要求。在設計時，正確地選擇密封件、導向套（支承環(huán)）和防塵圈的結(jié)構形式和材料是很重要的。從現(xiàn)在密封技術來分析，液壓缸的活塞和活塞桿及密封、導向套和防塵等應作為一個綜合的密封系統(tǒng)來考慮，具有可靠的密封系統(tǒng)，才能式液壓缸具有良好的工作狀態(tài)和理想的使用壽命。 在液壓元件中，對液壓缸的密封要求是比較高的，特別是一些特殊材料液壓缸，如擺動液壓缸等。液壓缸中不僅有靜密封，更多的部位是動密封，而且工作壓力高，這就要求密封件的密封性能要好，耐磨損，對溫度適應范圍大，要求彈性好，永久變形小，有適當?shù)臋C械強度，摩擦阻力小，容易制造和裝卸，能隨壓力的升高而提高密封能力和利于自動補償磨損。 密封件一般以斷面形狀分類。有O形、U形、V形、J形、L形和Y形等。除O形外，其他都屬于唇形密封件。 2.O形密封圈的選用 液壓缸的靜密封部位主要是活塞內(nèi)孔與活塞桿、支承座外圓與缸筒內(nèi)孔、缸蓋與缸體端面等處[6]。這些部位雖然是靜密封，但因工作由液壓力大，稍有意外，就會引起過量的內(nèi)漏和外漏。 靜密封部位使用的密封件基本上都是O形密封圈。O形密封圈雖小，確實一種精密的橡膠制品，在復雜使用條件下，具有較好的尺寸穩(wěn)定性和保持自身的性能。在設計選用時，根據(jù)使用條件選擇適宜的材料和尺寸，并采取合理的安裝維護措施，才能達到較滿意的密封效果。 安裝O形圈的溝槽有多種形式，如矩形、三角形、V形、燕尾形、半圓形、斜底形等，可根據(jù)不同使用條件選擇，不能一概而論。使用最多的溝槽是矩形，其加工簡便，但容易引起密封圈咬邊、扭轉(zhuǎn)等現(xiàn)象。 2.動密封部位密封圈的選用 液壓缸動密封部位主要有活塞與缸筒內(nèi)孔的密封、活塞桿與支承座（導向套）的密封等。 形密封圈是我國液壓缸行業(yè)使用極其廣泛的往復運動密封圈。它是一種軸、孔互不通用的密封圈。一般，使用壓力低于16MPa時，可不用擋圈而單獨使用。當超過16MPa并用于活塞動密封裝置時，應使用擋圈，以防止間隙“擠出”。 10 防塵圈 防塵圈設置與活塞桿或柱塞密封外側(cè)，用于防止外界塵埃、沙粒等異物侵入液壓缸，從而可以防止液壓油被污染導致元件磨損。 1.防塵圈 A型防塵圈 是一種單唇無骨架橡膠密封圈，適于在A型密封結(jié)構形式內(nèi)安 裝，起防塵作用。 B型防塵密封圈 是一種單唇帶骨架橡膠密封圈，適于在B型密封結(jié)構形式 內(nèi)安裝，起防塵作用。 C型防塵圈 是一種雙唇密封橡膠圈，適于在C型結(jié)構形式內(nèi)安裝，起防塵 和輔助密封的作用。 2.防塵罩 防塵罩采用橡膠或尼龍、帆布等材料制作。在高溫工作時，可用氯丁橡膠，可在130℃以下工作。如果溫度再高時，可用耐火石棉材料。當選用防塵伸縮套時，要注意在高頻率動作時的耐久性，同時注意在高速運動時伸縮套透氣孔是否能及時導入足夠的空氣。但是，安裝伸縮套給液壓缸的裝配調(diào)整會帶來一些困難。 11 液壓缸的安裝連接結(jié)構 液壓缸的安裝連接結(jié)構包括液壓缸的安裝結(jié)構、液壓缸近處有口的連接等。1.液壓缸的安裝形式 液壓缸的安裝形式很多，但大致可以分為以下兩類。 1）軸線固定類 這類安裝形式的液壓缸在工作時，軸線位置固定不變。機床上的液壓缸絕大多數(shù)是采用這種安裝形式。 A 通用拉桿式。在兩端缸蓋上鉆出通孔，用雙頭螺釘將缸和安裝座連接拉緊。一般短行程、壓力低的液壓缸。 B 法蘭式。用液壓缸上的法蘭將其固定在機器上。 C 支座式。將液壓缸頭尾兩端的凸緣與支座固定在一起。支座可置于液壓缸左右的徑向、切向，也可置于軸向底部的前后端。 2）周線擺動類 液壓缸在往復運動時，由于機構的相互作用使其軸線產(chǎn)生擺動，達到調(diào)整位置和方向的要求。安裝這類液壓缸，安裝形式也只能采用使其能擺動的鉸接方式。工程機械、農(nóng)用機械、翻斗汽車和船舶甲板機械等所用的液壓缸多用這類安裝形式。 A 耳軸式。將固定在液壓缸上的鉸軸安裝在機械的軸座內(nèi)，使液壓缸軸線能在某個平面內(nèi)自由擺動。 B 耳環(huán)式。將液壓缸的耳環(huán)與機械上的耳環(huán)用銷軸連接在一起，使液壓缸能在某個平面內(nèi)自由擺動。耳環(huán)在液壓缸的尾部，可以是單耳環(huán)，也可以是雙耳環(huán)，還可以做成帶關節(jié)軸承的單耳環(huán)或雙耳環(huán)。 C 球頭式。將液壓缸尾部的球頭與機械上的球座連接在一起，使液壓缸能在一定的空間錐角范圍內(nèi)任意擺動。 2.液壓缸油口設計 油口孔是壓力油進入液壓缸的直接通道，雖然只是一個孔，但不能輕視其作用[6]。如果孔小了，不僅造成進油時流量供不應求，影響液壓缸的活塞運動速度，而且會造成回油時受阻，形成背壓，影響活塞的退回速度，減少液壓缸的負載能力。對液壓缸往復速度要求較嚴的設計，一定要計算孔徑的大小。 液壓缸的進出油口，可以布置在缸筒和前后端蓋上。對于活塞桿固定的液壓缸，進出油口可以設在活塞桿端部。如果液壓缸無專用排氣裝置，進出油口應設在液壓缸的最高處，以便空氣能首先從液壓缸排出。液壓缸進出油口的鏈接形式有螺紋、方形法蘭和矩形法蘭等。 B. O形密封圈的選用 液壓缸的靜密封部位主要有活塞內(nèi)孔與活塞桿、支撐座外圓與缸筒內(nèi)孔、端蓋與缸體端面等處。靜密封部位使用的密封件基本上都是O形密封圈。 C.動密封部位密封圈的選用 由于O型密封圈用于往復運動存在起動阻力大的缺點，所以用于往復運動的密封件一般不用O形圈，而使用唇形密封圈或金屬密封圈。 液壓缸動密封部位主要有活塞與缸筒內(nèi)孔的密封、活塞桿與支撐座（或?qū)蛱祝┑拿芊獾取? 活塞環(huán)是具有彈性的金屬密封圈，摩擦阻力小，耐高溫，使用壽命長，但密封性能差，內(nèi)泄漏量大，而且工藝復雜，造價高。對內(nèi)泄漏量要求不嚴而要求耐高溫的液壓缸，使用這種密封圈較合適。 V形圈的密封效果一般，密封壓力通過壓圈可以調(diào)節(jié)，但摩擦阻力大，溫升嚴重。因其是成組使用，模具多，也不經(jīng)濟。對于運動速度不高、出力大的大直徑液壓缸，用這種密封圈較好。 U形圈雖是唇形密封圈，但安裝時需用支撐環(huán)壓住，否則就容易卷唇，而且只能在工作壓力低于10MPa時使用，對壓力高的液壓缸不適用。 比較而言，能保證密封效果，摩擦阻力小，安裝方便，制造簡單經(jīng)濟的密封圈就屬Yx型密封圈了。它屬于不等高雙唇自封壓緊式密封圈 ，分軸用和孔用兩種。 綜上，所以本設計選用Yx型圈，聚氨酯和聚四氟乙烯密封材料組合使用，可以顯著提高密封性能： a.降低摩擦阻力，無爬行現(xiàn)象； b.具有良好的動態(tài)和靜態(tài)密封性，耐磨損，使用壽命長； c.安裝溝槽簡單，拆裝簡便。 這種組合的特別之處就是允許活塞外園和缸筒內(nèi)壁有較大間隙，因為組合式密封的密封圈能防止擠入間隙內(nèi)，降低了活塞與缸筒的加工要求，密封方式圖如下： 圖3-2 密封方式圖 總 結(jié) 在這不到一周的設計中，能學到的東西真的很有限，但是不能說一點收獲都沒有，我想我知道了一般機床液壓缸的設計框架而且我也掌握了設計一個液壓缸的步驟，我想本次課程設計是我們對所學知識運用的一次嘗試，是我們在液壓知識學習方面的一次有意義的實踐。 在本次設計中，我獨立完成了自己的設計任務，通過這次設計，弄懂了一些以前書本中難以理解的內(nèi)容，加深了對以前所學知識的鞏固。在設計中，通過老師的指導，使自己在設計思想、設計方法和設計技能等方面都得到了一次良好的訓練。 致 謝 這次畢業(yè)設計可以圓滿的完成，離不開導師XXX的悉心指導。從課題的提出和論證到論文完成，X導師淵博的學識、先進的學術思想、對待研究的嚴謹態(tài)度和無私的奉獻精神都是學生的楷模，使我受益匪淺，在論文完成之際，謹向尊敬的X導師致以崇高的敬意和由衷的感謝。 在進行畢業(yè)設計的過程中 ，我的感激之情無以言表，僅以此文獻給他們，感謝我的朋友們，大家這四年來無論深處何地，距離多遠，我始終感受的到與你們大家在一起；感謝我的老師，四年來對我的關心幫助讓我在學校的生活和學習中都能有親人般的感覺；感謝我的同學們，大家雖然來自不同的地方，但是大家始終相親相愛，團結(jié)一致。我很慶幸能有了你們大家陪我一路走過艱難的歷程?；厥状髮W四年，往事歷歷在目，心緒難以平復，如此多的關心和幫助讓我感到莫大的幸運，感覺充滿力量，無論是身邊的同學老師還是遠方的親人朋友們，他們的支持是我可以努力和堅持的最大動力，有了他們才真正讓我感受到這個世界是無與倫比的美麗，這些都將支持我走向新的崗位，為社會為他人貢獻我的綿薄之力。 參考文獻 [1] 李碩衛(wèi),張國賢.現(xiàn)代液壓技術的發(fā)展現(xiàn)狀[J]. 機械工程師，2009(2):54～57 [2] 郭麗穎.液壓自動換向回路及其應用[J]. 煤礦機械，2005(3):90～91 [3] 董傳軍,楊延水,劉艷霞.一種液壓增壓缸的應用[J]. 制造技術與基礎，2009(7):108～109 [4] 王建軍.一種液壓增壓缸的介紹[J]. 液壓與氣動，1992(4):36 [5] 周曉君,袁輝.單井增注液壓增壓系統(tǒng)設計[J]. 液壓與氣動，2003(7):12～14 [6] 隋文臣.自控式雙作用增壓器的研究[J]. 煤礦機械，2004(9):106～108 [7] 許福玲,陳堯明.液壓與氣壓傳動（第3版）[M].北京：機械工業(yè)出版社，2007.143～144 [8] 李振軍，劉建英.液壓傳動與控制.北京：機械工業(yè)出版社，2009.101 [9]Y.He,P.5.K.Chua,G.H.Lim.Fault Diagnosisof Loaded Water hydraulie Aetuators by Online Testing with LABVIEW[J]. Journal of Testing and Evaluation,2003,31(5):378～387 [10]Bill Savela.Digital Control Aids hydraulie-Press Produetivity[J]. Metal Forming,2005,39(2):20～22 [11]K.Heister，P.J.Kleingeld，U.5.Keijzer.A new laboratory set-up for measurements of electrical，hydraulic and osmotic fluxes in clays.Engineering Geology，2005，77（3/4）：295～303 KSME International Journal, Vol. 11, No. 4, pp. 397-407, 1997 A Study on the Stability Analysis of a PWM Controlled Hydraulic Equipment 397 Jun-Young Huh* and G. Wennmacher* (Received August 16 1996) PWM control, which is inherently nonlinear digital control, has been used for hydraulic equipment control because of the robustness and the availability of the low priced high speed on-off valve which is required for the system. Since this valve can be directly controlled without any D/A converter, it is easily implemented to hydraulic equipments with microcomputers. The objectives of this study is to analyze the limit cycle which ordinarily appears in the position control system using high speed on-off valve, and to give a criterion for the stability of this sy,;tem. The nonlinear characteristics of PWM and cylinder friction are described by harmonic linearization and the effects of parameter variations to the system stability are investigated theoretically. Experimental results demonstrate the feasibility of the proposed method. Key Words: Electro-Hydraulic Servo System, High Speed On-Off Solenoid Valve, PWM Method, Stability Analysis, Limit Cycle Nomenclature Ns : Transfer function of valve-cylinder sys- tem As : Piston area P1, P2 : Forward and return pressure of cylinder ca : Discharge coefficient Ps : Supply pressure D : Duty (:acting time/T) Pr : Tank pressure Fc : Coulombs friction force q : Input signal of PWM FL : External load c : Amplitude of PWM input signal Fe : Friction force of piston Q, Q2 : Forward and return flows G : Viscous damping coefficient qmax : Magnitude of PWM input signal of duty kl : Flow gain of valve : 1 k2 : Flow-pressure coefficient T : Period of PWM carrier wave k: : Proportional gain Tq : Period of PWM input signal wave km : Maximum displacement of valve poppet Ts : Period of sampling time K : System gain (:k: 9 k,) : Amplitude of PWM output pulse Kq : Sizing factor of valve v : Velocity of piston Ls : Static friction 7 : Amplitude of piston velocity wave M : Piston mass Vt : Total volume of pipe, valve and cylinder MRa : Critical speed starting viscous damping chamber friction w : Area gradient of valve port NpwM : Describing function of PWM x : Displacement of piston xv : Displacement of valve poppet (i:a, b, 9 Department of Mechanical Engineering, Korea c, d) University of Technology and Education, Chonan P.O.B. 55, Chungnam, 330-600, Korea Be Effective bulk modulus of fluid and pipe 9 * Liebherr-Aerospace Lindenberg GmbH, Pfaender : Synchronizing degree Str. 50-52, D-88161 Lindenberg, Germany r : Valve delay time 398 Jun-Young Huh and G. Wennrnacher r(q) : Width of PWM output co : Angular frequency of PWM input signal ton.ea : Switch on delay ton,d,so : Switch on displacement time tof.dead : Switch off delay toff,dsp : Switch off displacement time 1. Introduction The control method using Pulse Width Modu- lation(PWM) is one of the nonlinear control schemes. Hydraulic equipments which are operat- ed in PWM control usually use high speed on-off valves as control valves. The usage of this valve gives several advantages; this valve has very sim- ple structure so that it has robustness to oil contamination and its price is low. And it can be easily implemented to the system by using micro- computer since the control of this valve can be carried out digitally without D/A converter (Wennmacher, 1992a, b; Wennmacher, 1994; Muto, et. al., 1988). Since the hydraulic cylinder system controlled by PWM has highly nonlinear characteristics such as the nonlinearity of PWM, the phase delay of valve poppet and the friction in the cylinder, the nonlinear characteristics should be considered in the analysis of the system stabil- ity. Tanaka(1988) considered this system as a linear time invariant discrete system and derived transfer function using z-transform, then, showed the stable limiting gain against the damping ratio on every sampling time. Noritsugu(1983) sim- plified the nonlinearity of PWM as a saturation function and considered it with the phase lag of valve poppet and the cylinder friction in the velocity control of pneumatic cylinder. But the nonlinearity of PWM signal generation is not considered fully. Prochnio (1986) utilized describ- ing function based on the harmonic linearization in depicting the nonlinearity of PWM and consid- ered the friction in cylinder, did not include the phase lag of the valve poppet. The objective of this study is to present a method to predict the stability of hydraulic servo system. The electro-hydraulic position control system is theoretically modelled, which have high speed on-off valves controlled by PWM. In this process, the highly nonlinear characteristics of the system such as the dynamics of PWM, the phase lag of valve poppet and the cylinder friction are modelled mathematically. Especially the non- linear characteristics of PWM and cylinder fric- tion are expressed with describing function by the harmonic linearization. The validity of the presented method is investigated for the variation of the system parameters theoretically and experi- mentally. 2. Hydraulic Servo System Description and Analysis 2.1 Dynamic modelling The schematic diagram of the hydraulic servo system employed in this study is shown in Fig. 1. This system consists of 4 high speed on-off valves and a hydraulic actuator(double rod double acting cylinder) as the main parts. The piston displacement (x), which is output of the system is feedbacked to the comparator, which is im- plemented by software inside the computer, to be compared with the reference input value. For the PWM operation with the high speed on-off valve, since the switching characteristic of valve poppet makes a significant influence to the performance of this system, the lag of the valve poppet move- ment should be considered in dynamic modelling. The hydraulic system is modelled in Fig. 2. When the ON signal is applied to the valves a and c, since the valves b and d receive the OFF signal simultaneously, the forward oil flow(Q1) is oc- IIII Fig. 1 ? Electro-hydraulic servo system with high speed on-off valves. A Study on the Stability Analysis of a PWM Controlled Hydraulic Equipment 399 X,= X,)%,R P1 P2 . l I 7- / I 7 TM 5 r I ELI Xvd Xvc I Xvb Xvo I I PT I PT PS Fig. 2 Modelling diagram of hydraulic system. ing, because it is generally not critical or of particular interest in system design, it cannot be altered appreciably by design, and it does not affect system stability. To derive the transfer func- tion of the system, the load flow Eq. in (5) which is a nonlinear equation is linealized by the Taylors expansion in the vicinity of an operating point(xv0, PLo) of interest, It can be written as (Merrit, 1967) QL = kx - k2PL (8) curred through the valves a and c, and this makes the positive piston displacement. Under the assumption that the supply pressure(Ps) is con- stant, the leakage flows between a valve poppet and its seat and that of cylinder are negligible, and the valves are operated simultaneously, the flow rates (Q, Qz) through the orifices of the high speed on-off valves are given by Q = caw v 9 xe (1) Q2=cawvf. xaPUA=-P (2) where the load pressure(PL) and the load flow (QL) are defined as pL = p- p2 (3) QL :- Q1 + Qz (4) 2 The load flow equation can be deriven as, QL = Kqx,/-Ps- PL (5) where Kq - CdW v The continuity equation of the valve-cylinder system is given by QL=AhI lit dPL (6) 4fie dt The force balance equation of the piston is d2x M-t- = AhPL - FR (7) where the viscous damping friction of the piston is included in the term of total friction force of cylinder (FR), and the external load (FL) is omit- ted. The external force is neglected in the modell- 2.2 Describing function of PWM When a high speed on-off valve is operated in PWM mode, the input signal which is inflicted on the valve from the PWM module is a type of pulse row which has a constant pulse magnitude but a different pulse width and sign according to the magnitude of the input signal of PWM module. And the pulse row has the same interval between the pulse starting points. To investigate the stabil- ity of this system analytically, the characteristics of the PWM signal generation should be depicted analytically. Under the assumption that the on- off operations of the valves are symmetric and the linear part of this hydraulic system has the enough low pass filter function (Follinger, 1991), we derive the describing function of PWM signal generation which has the highly nonlinear charac- teristic. The PWM module catches the input sig- nal(q) at every time interval and knows the end of the pulse width previously. The input signal (q (t) of the PWM module can be represented as the following in the harmonic vibration balance state. q (kT) = c7 9 sin (w 9 kT + q5) (9) where the synchronizing degree (if) represents the phase lag between the first output pulse of PWM module and the input signal of PWM module in the harmonic vibration balance state. For exam- ple, the case of if=90 is shown in Fig. 3. The input signal q(t) of PWM module which is depicted as Eq. (9) is a periodic signal which have a period (Tq). If the ratio between the period (Tq) and the period of PWM carrier wave(T) comes to 2n (n=l, 2, 3, .), Fourier series method can be utilized in calculation of the 400 Jun-Young Huh and G. Wennmacher describing function of the PWM signal genera- tion. The output pulse equation of PWM module u (t) = 0 sgn(q (kT) ) is given by kT Nt kT +rq(kT) kT+rq(kT)t (k+l) T (lO) where rq(kT)= .D DI D Do u (t) is the function of the output signal of PWM module, q(kT) is the Kth value of the input signal of PWM module, and D is a duty, which is determined as I q(kT)/qmaxl. Do is the duty value corresponds to the delay time in the valve operation. A vibration which continues steadily with a constant amplitude and frequency is called a limit cycle. Especially, when the ratio of (Tq/ T) is 2 as shown in Fig. 3, we call it 1 pulse IT I (k -G. Fig. 3 ? i l)T (k+2)T Output pulse trains of 1 pulse limit cycle. fi -0 (k+n)r (k+n+1)T (k+2n-1)T kT (k+l)T (k+n-1)T t Fig. 4 Output pulse trains of n pulse limit cycle. limit cycle. The Fourier coefficient al and bl of basic vibration is given by 2s sin( r (c7) (11) al- 7/ 2s / r 1 b, =-LCOST. r () - (12) then the describing function of PWM signal gen- eration is given as the following for the case of 1 pulse limit cycle. r 2s . ;r , NwM(#, l, TJ-Tgksn,r ) ) Meanwhile, the n pulse limit cycle has succes- sively n positive pulses and n negative pulses for one period time(Tq) of limit cycle as shown in Fig. 4. The describing function of this case is derived as the following, 2s . n-t . NpwM (c7, n, b)= _ e-k_oe- lrq = - e-Jk+ nT r(q(kT) (14) As shown in Eq. (14), NPwM(ff, n, b) is a function of the amplitude () of the input signal of PWM module, the pulse number(n) of limit cycle and synchronizing degree(b), but it is in- dependent to angular frequency (w). 2.3 Describing function of friction charac- teristic The friction characteristic of hydraulic cylinder which is modelled as shown in Fig. 5 consists of static friction, coulombs friction and viscous damping friction which is dependent upon piston velocity. Supposing the hysterisis in friction char- acteristic is negligible, the friction characteristic shown in Fig. 5 can be formulated as the follow- ing: (Back6, 1992) I ( Fe()=Gk. + F+L. l- sgn(,7) (15) A Study on the Stability Analysis of a PWM Controlled Hydraulic Equipment 401 In order to derive the describing function, the above equation could be approximated as (Proch- nio, 1986) FR():Gg+Fc4 L 5 , sgn(v) (16) Where c is a constant required to approximate Eq. (15) to Eq. (16). In harmonic balance, the piston velocity v(t) is given by v(t) = 7-sin(wt) (17) the describin function of friction characteristic is NR() = a(g) (18) qvhere a(7) is the Fourier coefficient of basic vibration as follows: 1 ,-2r _ +F a(g)=-J 1Gv sin(x) L Ls + 1 +( g sin(X)c .)2- sgn(sin X) sin(x) y Fig. 5 Model of cylinder friction. Eq. (18) the describing function of friction char- acteristic becomes as follows: NR () =Gk+ 4Fc 4 2Lsc 2 . In xv x2/ 2 + c2 t7 + (20) 2.4 Analysis of the control loop In order to investigate the stability of the system which have highly nonlinear characteristics such as PWM signal generation and friction in cylin- der, the describing functions of these nonlinear characteristics are derived. Hence, the well known linear analysis method can be utilized. The above derived Eqs. (6), (7), (8), (14) and (20) are combined to construct the block diagram of the overall system as shown in Fig. 6. In this figure the nonlinear characteristics which is depicted by the describing functions are shown with double square frame. The transfer function Ns of the valve-cylinder system is derived as Eq. (21), the input of which is the valve spool displacement X and the output is cylinder displacement x, Ns(W, g)=- c (21) s ( s2 + c2s + c3 where, C1- k c2= FRM c _ 4J - VM The unstable phenomenon of a vibration sys- tem which has an element with a highly nonlinear characteristic tlsually starts with the limit cycle, Nsr ), F Fig. 6 Block diagram of PWM hydraulic cylinder system. 402 Jun-Young Huh and G. Wennmacher that is, the output of the system oscillate continu- ously with a constant amplitude and frequency. Whether the limit cycle will occur or not in a certain condition can be determined through the investigation of the solution of the closed loop characteristic equation. That is, in the harmonic vibration balance state, the limit cycle will occur if the solution of the closed loop characteristic equation exists (Prochnio, 1986; Follinger, 1991). The objective of this study is to present a method to predict the stability of the system so that it can be used to guarantee the stable operation in a wide range of operating conditions. The charac- teristic equation of the position control system shown in Fig. 6 is given as 1 + Ke - 9 NewM Ns :0 (22) where K is obtained by the product of the propor- tional gain(k1) and the maximum displacement of valve poppet(kin). The characteristic Eq. (22) is so complicated that it is very difficult to solve it analytically. Therefore, a graphical method is utilized to get the solution of the characteristic equation. wave(T) comes to 2n (n=l, 2, 3, .-.), hence, the relation between angular frequency(co) and PWM carrier wave period(T) is derived as the following: z n=l, 2, 3, . (24) co= nZ In the above equation, when the pulse number (n) of limit cycle is fixed, angular frequency(co) is uniquely determined. Therefore the unfixed parameters of Eq. (22) are the gain(K), the amplitude () of the input signal of PWM mod- ule, synchronizing degree(e) and pulse number (n) of limit cycle. At first to investigate whether the 1 pulse limit cycle occur or not, the pulse number of limit cycle is set as n=l. Then, we changed K value variously in the range of interest for the fixed r and value, otherwise we chan- ged value variously for the fixed r and K value, and investigated the solutions which satisfy the Eq. (22). And for the cases of n=2, 3, . it is investigated in the same way. The values of each parameters which is used in computer simu- lation are shown in Table 1. 3. Computer Simulation and Experiment In order to solve the characteristic Eq. (22) by graphical method which is utilized in this study, the first step is to investigate the Eq. (22) and look for the unfixed parameters. In the second term of the left hand side of the Eq. (22) NewM (c7, n, r is the function of the amplitude(p) of the input signal of PWM module, the pulse num- ber(n) of limit cycle and synchronizing degree (r but it is independent to angular frequency (co). Ns(co, g) is the function of angular fre- quency(w) and piston velocity amplitude(f). The relation between the amplitude(p) of the input signal of PWM module and piston velocity amplitude is given by t = k/ g (23) co When this system is operated in harmonic vibra- tion balance state, the ratio between the input signal period (Tq) and the period of PWM carrier Table 1 System parameters used in computer simulation. Parameters Value Dimension A 7.65 cm 2 c 1.5 cm/s Fc 0 kgf L 21 kgf G, 5.46 kgf s/cm kl 4666.7 cm2/s k2 0. 3536 cmS/kgf s k, 0.015 cm M 5-80 kg T 5 ms Ts 1 ms Vt 306 cm 3 Va 50 cm 3 r 1 ms fie 12000 kgf/cm 2 A Study on the Stability Analysis of a PWM Controlled Hydraulic Equipment 403 I I I I I I I- ! J I r-1 I I I ! I I I I I I I I I I I Ace.2 I I I I I I I I L. -3 r I I I I I I .3 Fig. 7 Hydraulic circuit of experimental equipment. The experimental apparatus is constructed as shown in Fig. 7, which has 4 high speed on-off valves, a double rod hydraulic cylinder, and a microcomputer as the major components. Piston displacement is measured by the digital optical Table 2 Specification of ex 9erimental apparatus Equipment Specification Hydraulic System Electric Equipment Electric Motor 5 PS Hydraulic Pump Qmx:8.9 l/min Actuator Ak = 7.65 cm 2 St.= 10 cm High speed On-off valve Relief valve Filter Accmulator 1 Accumulator 2 Displacement Transducer Microcomputer Controller Qmax:4.2 1/min Xvmax:0.015 cm ton,dead :0.46 ms /on,dlp :0.18 ms toll,dead = 0.30 ms tou,dlsp :0.16 ms Pmax :230 kgf/cm 2 10 m 4 liter 0.075 1, 70 bar 12 bit (0.315ram) CPU386 sensor with + l#m accuracy. Pressure transducer is represented by P/T in Fig. 7, of which accuracy is - bar. It is an analog sensor so that the analog to digital converter is required to transfer measured values in the computer. Tile controller in control loop gets the sampled data on every millisecond, calculates the control law and gives it to 4 channel PWM modulator. PWM carrier frequency is 200 Hz and the period is 5ms. For the various inertia loads, we looked for the propor- tional gain that makes the system on the threshold from stability to instability with limit cycle. The major specifications of the components which is used in this experiment are shown in Table 2. 4. Performance of the Control System In the cases of inertia load 6kg and 50kg, the experimental results for step input are showed with the solid line and the dotted line respectively in Fig. 8 (ks= 1 5). The responses shows instabil- ity with almost regular oscillations which have almost constant amplitudes. For the inertia load 6kg, we can find that the response shows 1 pulse limit cycle because it vibrated with 100Hz fre- quency which have 10 peaks per 0.1 second, and 404 Jun-Young Huh and G. Wennmacher ? i. ,. . I . . . i . i i ; i i . 0.5 ; i i ! 0 0 0.05 0.1 0.15 0.2 .me (sac) Fig. 8 Experimental results of n pulse limit cycle (n= 1, 2). this is the half of the PWM carrier frequency 200Hz. For the inertia load 50kg, the response shows almost 2 pulse limit cycle to have about 5 peaks per 0.1 second. Here the reason that the response does not show the exact 5 peaks per 0.1 second is guessed due to the followirtg vibration of high speed on-off valve after its opening and closing operation. Figure 9 is one of the result figures which is utilized to investigate analytically the existence of 1 pulse limit cycle which is the solution of Eq. (22). One solid line of Fig. 9 (a) is obtained for the case with the inertia load of 11 kg, synchronizing degree (b) of 88 and propor- tional gain(ks) of 0.7, by plotting the value of Ke -s 9 NpWM Ns in s-plane for the variation of the velocity amplitude(7) from 0.01 to 25, and the other solid line curves are obtained by repeat- ing this process for the increased proportional gain(ks) from 0.7 to 1.5 by step of 0.2.Similarly when the proportional gain (kl) varies from 0.1 to 4.5 for the increased amplitude(O) from 0.2 to 3.2 by the step of 0.6, the values ofKe - 9 NpwM 9 Ns are plotted with dotted lines. Herein, when the curve crosses the point (-1, 0) of s-plan, there exist the solution of Eq. (22) which is given by the values of the proportional gain (kj) and veloc- ity amplitude (7). One solid line of Fig. 9 (b) is obt