2附錄 A液壓支架的最優(yōu)化設(shè)計摘 要本文介紹了從兩組不同參數(shù)的采礦工程所使用的液壓支架中選優(yōu)的流程。這種流程建立在一定的數(shù)學(xué)模型之上。第一步，尋找四連桿機構(gòu)的最理想的結(jié)構(gòu)參數(shù)以便確保支架的理想的運動軌跡有最小的橫向位移。第二步，計算出四連桿有最理想的參數(shù)時的最大誤差，以便得出最理想的、最滿意的液壓支架。關(guān)鍵詞：四連桿機構(gòu)； 優(yōu)化設(shè)計； 精確設(shè)計； 模糊設(shè)計； 誤差 1.前言設(shè)計者的目的時尋找機械系統(tǒng)的最優(yōu)設(shè)計。導(dǎo)致的結(jié)果是一個系統(tǒng)所選擇的參數(shù)是最優(yōu)的。一個數(shù)學(xué)函數(shù)伴隨著一個合適的系統(tǒng)的數(shù)學(xué)模型的出現(xiàn)而出現(xiàn)。當(dāng)然這數(shù)學(xué)函數(shù)建立在這種類型的系統(tǒng)上。有了這種數(shù)學(xué)函數(shù)模型，加上一臺好的計算機的支持，一定能找出系統(tǒng)最優(yōu)的參數(shù)。Harl 描述的液壓支架是斯洛文尼亞的 Velenje 礦場的采煤設(shè)備的一個組成部分，它用來支護采煤工作面的巷道。它由兩組四連桿機構(gòu)組成，如圖 2 所示.四連桿機構(gòu)AEDB 控制絞結(jié)點 C 的運動軌跡，四連桿機構(gòu) FEDG 通過液壓泵來驅(qū)動液壓支架。圖 2 中，支架的運動，確切的說，支架上絞結(jié)點 C 點豎向的雙紐線的運動軌跡要求橫向位移最小。如果不是這種情況，液壓支架將不能很好的工作，因為支架工作在運動的地層上。圖 2 液壓支架實驗室測試了一液壓支架的原型。支架表現(xiàn)出大的雙紐線位移，這種雙紐線位移的方式回見少支架的承受能力。因此，重新設(shè)計很有必要。如果允許的話，這會減少3支架的承受能力。因此，重新設(shè)計很有必要。如果允許的話，這種設(shè)計還可以在最少的成本上下文章。它能決定去怎樣尋找最主要的圖 2 兩四連桿機構(gòu)四連桿機構(gòu)數(shù)學(xué)模型 AEDB 的最有問題的參數(shù) 。否則的話這將有必要在最421,a小的機構(gòu) AEDB 改變這種設(shè)計方案。上面所羅列出的所有問題的解決方案將告訴我們關(guān)于最理想的液壓支架的答案。真正的答案將是不同的，因為系統(tǒng)有各種不同的參數(shù)的誤差，那就是為什么在數(shù)學(xué)模型的幫助下，參數(shù) 允許的最大的誤差將被計算出來。421,a2.液壓支架的確定性模型首先，有必要進一步研究適當(dāng)?shù)囊簤褐Ъ艿臋C械模型。它有可能建立在下面所列假設(shè)之上：（1）連接體是剛性的，（2）單個獨立的連接體的運動是相對緩慢的.液壓支架是只有一個方向自由度的機械裝置。它的運動學(xué)規(guī)律可以通過同步的兩個四連桿機構(gòu) FEDG 和 AEDB 的運動來模擬。最主要的四連桿機構(gòu)對液壓支架的運動規(guī)律有決定性的影響。機構(gòu) 2 只是被用來通過液壓泵來驅(qū)動液壓支架。絞結(jié)點 C 的運動軌跡 L 可以很好地來描述液壓支架的運動規(guī)律。因此，設(shè)計任務(wù)就是通過使點 C的軌跡盡可能地接近軌跡 K 來找到機構(gòu) 1 的最理想的連接長度值。四連桿機構(gòu) 1 的綜合可以通過 Rao 和 Dukkipati 給出運動的運動學(xué)方程式的幫助來完成。在水壓機等4液壓元件中，空穴或氣穴意味著，在低壓區(qū)液壓液體會出有空腔或氣泡形成以及崩潰在高壓地區(qū)，這將導(dǎo)致噪聲，振動，這將會降低效率。空蝕對泵的使用是極為不利的，這是因為倒塌形成的沖擊波可能像炸彈一樣足以損壞元件。當(dāng)其壓力過低或溫度過高時，液壓油會蒸發(fā)。在實踐中，許多方法大多用于處理這些問題，比如：（1）提高油箱中的液位高度，（2 油箱加壓，（3）提高泵的進口壓力，（4）降低泵內(nèi)流體的溫度，（5）特意設(shè)計的柱塞泵本身，對其結(jié)構(gòu)進行優(yōu)化設(shè)計。在液壓機設(shè)計中的氣蝕現(xiàn)象，許多研究成果已取得一定的成果。在柱塞泵中，氣蝕主要可以分為兩種類型：一是與困油現(xiàn)象有關(guān)（這種現(xiàn)象可通過適當(dāng)?shù)脑O(shè)計配流盤來阻止困油現(xiàn)象的發(fā)生）和所觀察到的層上收縮或擴大后的流動通道(由于旋轉(zhuǎn)設(shè)計所造成的)。在這項研究中處理氣蝕和測量氣缸壓力之間的關(guān)系。Edge and Darling 報道了關(guān)于軸向柱塞泵內(nèi)的氣缸壓力的實驗研究。其中包括流體勢效應(yīng)和氣蝕在氣缸內(nèi)高速度和高負荷條件的預(yù)測。另一項研究概述了液壓流體影響進氣條件和汽蝕潛力的觀點。它表明,物理屬性(如蒸汽壓力、粘度、密度和體積彈性模量)對適當(dāng)?shù)卦u估影響潤滑和氣蝕是至關(guān)重要的。一個相似的氣蝕模型在熱力學(xué)性質(zhì)液體和蒸汽基礎(chǔ)上的用來理解了基本的物理現(xiàn)象的質(zhì)量流量減少和波動產(chǎn)生影響的液壓工具和噴射系統(tǒng)。Dular et al 開發(fā)了一套專業(yè)系統(tǒng)用它來監(jiān)測和控制的液壓機械和調(diào)查氣蝕的可能性通過使用運用計算流體動力學(xué)(CFD)工具。通過一個簡單的單翼配置在一個空化隧道，氣蝕侵蝕作用已經(jīng)被測量和驗證。它假定了嚴重侵蝕經(jīng)常是由于一個主要的空穴飛轉(zhuǎn)的漩渦重復(fù)的崩潰所產(chǎn)生的。然后,在汽蝕強度通過一套簡單流參數(shù)可能擴大: 上游速度 ,空腔長度和壓力。一個新的空蝕裝置,稱為漩渦汽蝕生成器,介紹了各種侵蝕情況。更多的先前的研究已經(jīng)被集中在閥板的設(shè)計,在軸向柱塞泵中活塞和泵壓動力學(xué)與空穴現(xiàn)象相關(guān)聯(lián)?？刂企w積的方法和瞬時流(泄漏)正在深刻地研究中。Berta et al 采用有限體積的概念發(fā)展了一個數(shù)學(xué)模型，壓力平衡槽的形式已經(jīng)被效仿和氣態(tài)的汽蝕被認為是在一個簡化的方式。一種改進的模型已經(jīng)被提出且實驗驗證了其結(jié)果。該模型可以分析氣缸壓力和流量的漣漪影響壓力平衡槽的設(shè)計。四種不同的數(shù)值模型的重點是液壓液體的特點,考慮到空穴以不同的方法協(xié)助減少流量振蕩。柱塞泵發(fā)展的經(jīng)驗表明,優(yōu)化的空穴現(xiàn)象應(yīng)當(dāng)包括下列問題: 發(fā)生氣蝕和空氣釋放、泵聲學(xué)引起的噪聲誘導(dǎo)、最大振幅的壓力波動,轉(zhuǎn)動力矩進展等。然而,這項研究的目的是修改配流盤的設(shè)計來防止氣蝕造成侵蝕蒸汽或空氣泡沫崩潰的墻壁上的軸向泵組件。與文學(xué)研究相反，這項5研究主要集中在配流盤的幾何形狀和氣蝕分析之間的關(guān)系的發(fā)展。此優(yōu)化方法應(yīng)用于分析的壓力脈沖與活塞孔內(nèi)飽和蒸汽壓。圖 3 描述了一般的情況。圖 3 機械運動一般情況點 C 的軌跡 L 的方程式將在同一框架下被打印出來。點 C 的相對應(yīng)的坐標(biāo) x 和 y隨著四連桿機構(gòu)的獨有的參數(shù) … 一起被打印出來。,21a6點 B 和 D 的坐標(biāo)分別是xB=x - cos (1)5a?yB=y - sin (2)xD=x - cos( ) (3)6??yD=y - sin( ) (4)a?參數(shù) … 也彼此相關(guān),216xB2 +yB2= (5)(xD-α 1)2+ yD2= (6)4a把(1) － (4)代入（5）－（6）即可獲得支架的最終方程式(x- cos )2+ (y- sin )2- =0 (7)a?5?a[x- cos( )- ]2+[ y- sin( )]2- =0 (8)6??16??4a6此方程式描述了計算參數(shù) 的理想值的最基本的數(shù)學(xué)模型。421,a2.1 數(shù)學(xué)模型Haug 和 Arora 提議，系統(tǒng)的數(shù)學(xué)模型可以用下面形式的公式表示min f(u,v), (9)約束于gi(u,v) 0, i=1,2,…,l, (10)?和響應(yīng)函數(shù)hi(u,v)=0, j=1,2,…,m. (11)向量 u=[u 1,u2,…,un]T 響應(yīng)設(shè)計時的變量, v=[v 1,v2,…,vm]T是可變響應(yīng)向量，(9)式中的 f 是目標(biāo)函數(shù)。為了使設(shè)計的主導(dǎo)四連桿機構(gòu) AEDB 達到最佳，設(shè)計時的變量可被定義為u=[ ]T, (12)1a24可變響應(yīng)向量可被定義為v=[x y]T. (13)相應(yīng)復(fù)數(shù) α 3，α 5，α 6的尺寸是確定的。目標(biāo)函數(shù)被定義為理想軌跡 K 和實際軌跡 L 之間的一些“有差異的尺寸”f(u,v) =max[g0(y)-f0(y)]2, (14)式中 x= g0(y) 是曲線 K 的函數(shù)，x= f 0(y)是曲線 L 的函數(shù)。我們將為系統(tǒng)挑選一定局限性。這種系統(tǒng)必須滿足眾所周知的最一般的情況。(15)??2143???aa(16)042不等式表達了四連桿機構(gòu)這樣的特性：復(fù)數(shù) 只可能只振蕩的。42,a這種情況：(17)u?給出了設(shè)計變量的上下約束條件。用基于梯度的最優(yōu)化式方法不能直接的解決(9)–(11)的問題。min un+1 (18)7從屬于gi(u,v) 0, i=1,2,…,l, (19)?f(u,v)- un+1 0 (20)并響應(yīng)函數(shù)hj(u,v)=0, j=1,2,…,m, (21)式中： u=[u1 … un un+1]Tv=[v1 … vn vn+1]T因此，主導(dǎo)四連桿機構(gòu) AEDB 的一個非線性設(shè)計問題可以被描述為：minα 7, (22)從屬于約束(23)??02143???aa(24)??432, 11?22(25)44a(26)??),(0)(7200 yyfg???并響應(yīng)函數(shù)：(27)0)sin()cos(2525 ??aax??(28)0)]i([][246216 ???? aya??有了上面的公式，使得點 C 的橫向位移和軌跡 K 之間的有最微小的差別變得可能。結(jié)果是參數(shù) 有最理想的值。421,3.液壓支架的隨機模型數(shù)學(xué)模型可以用來計算比如參數(shù) 確保軌跡 L 和 K 之間的距離保持最小。421,a然而端點C的計算軌跡L可能有些偏離，因為在運動中存在一些干擾因數(shù)?？催@些偏離到底合時與否關(guān)鍵在于這個偏差是否在參數(shù) 容許的公差范圍內(nèi)。421,8響應(yīng)函數(shù)（27）－（28）允許我們考慮響應(yīng)變量 v 的矢量，這個矢量依賴設(shè)計變量 v 的矢量。這就意味著 v＝h (v)，函數(shù) h 是數(shù)學(xué)模型（22）－（28）的基礎(chǔ)，因為它描述出了響應(yīng)變量 v 的矢量和設(shè)計變量 v 的矢量以及和數(shù)學(xué)模型中 v 的關(guān)系。同樣，函數(shù) h 用來考慮參數(shù) 的誤差值 的最大允許421,a421,a?值。 在隨機模型中，設(shè)計變量的矢量 u=[u1,…,un]T可以被看作 U=[U1,…,Un]T的隨機矢量，也就是意味著響應(yīng)變量的矢量 v=[v1,…,vn]T也是一個隨機矢量 V=[V1,V2,…,Vn]T v=h(u) (29)假設(shè)設(shè)計變量 U1,…,Un 從概率論的觀點以及正常的分類函數(shù)U k～ ),(kN??(k=1,2,…,n)中獨立出來。主要參數(shù) 和 (k=1,2,…,n)可以與如測量這類科學(xué)概k??念和公差聯(lián)系起來，比如 = , 。所以只要選擇合適的存在概率k3??， k=1,2,…,n (30)k??????式（30）就計算出結(jié)果。隨機矢量 V 的概率分布函數(shù)被探求依賴隨機矢量 U 概率分布函數(shù)及它實際不可計算性。因此，隨意矢量 V 被描述借助于數(shù)學(xué)特性，而這個特性被確定是利用Taylor的有關(guān)點 u=[u1,…,un]T 的函數(shù)h逼近描述，或者借助被Oblak和Harl在設(shè)計提出的Monte Carlo 的方法。3.1 數(shù)學(xué)模型用來計算液壓支架最優(yōu)化的容許誤差的數(shù)學(xué)模型將會以非線性問題的獨立的變量 w=[ ] (31)1a?24T和目標(biāo)函數(shù)(32)421)(awf???的型式描述出來。約束條件(33)0??EY?9, 11a??22a??(34)44在式（33）中，E 是是坐標(biāo) C 點的 x 值的最大允許偏差 ，其中Y?jAjjiY aag?????241),(61??A={1,2,4} (35)非線性工程問題的計算公差定義式如下：(36))11min(42a??它服從以下條件：(37)0??EY?, (38)11a?22a??(39)444.有數(shù)字的實列液壓支架的工作阻力為1600kN。以及四連桿機構(gòu)AEDB及FEDG 必須符合以下要求：－它們必須確保鉸接點C 的橫向位移控制在最小的范圍內(nèi)，－它們必須提供充分的運動穩(wěn)定性圖2中的液壓支架的有關(guān)參數(shù)列在表1 中。支撐四桿機構(gòu) FEDG 可以由矢量(mm) (39)??TTdb]130,25)13(,40[,321 ??來確定。四連桿 AEDB 可以通過下面矢量關(guān)系來確定。(mm) (40)??TTa]130,826,74[,4321?在方程(39)中，參數(shù)d是液壓支架的移動步距，為925mm .四連桿AEDA的桿系的有關(guān)參數(shù)列于表2中。表 1 液壓支架的參數(shù) 表 2 四連桿 AEDA 的參數(shù)104.1 四連桿 AEDA 的優(yōu)化四連桿的數(shù)學(xué)模型 AEDA 的相關(guān)數(shù)據(jù)在方程(22)-(28)中都有表述。(圖 3)鉸接點 C雙紐線的橫向最大偏距為 65mm。那就是為什么式(26)為(41)0)65(7??ax桿AA與桿AE之間的角度范圍在76.8 o和94.8 o之間，將數(shù) … 依次導(dǎo)入公式,21x19(41)中所得結(jié)果列于表3中。這些點所對應(yīng)的角 … 都在角度范圍[76.8 o,94.8o]內(nèi)而且它們每個角度,21?19之差為1 o設(shè)計變量的最小和最大范圍是(mm) (42)Tu]0,1283,64[?(mm) (43)970非線性設(shè)計問題以方程(22)與(28)的形式表述出來。這個問題通過Kegl et al(1991)提出的基于近似值逼近的優(yōu)化方法來解決。通過用直接的區(qū)分方法來計算出設(shè)計派生數(shù)據(jù)。設(shè)計變量的初始值為(mm) (44)TTa]30,16,74[],,[704201?優(yōu)化設(shè)計的參數(shù)經(jīng)過25次反復(fù)計算后是表3 絞結(jié)點C對應(yīng)的x與y 的值角度)(2o?x 初值(mm)y 初值(mm)x 終值(mm)y 終值(mm)1176.8 66.78 1784.87 69.47 1787.5077.8 65.91 1817.67 68.74 1820.4078.8 64.95 1850.09 67.93 1852.9279.8 63.92 1882.15 67.04 1885.0780.8 62.84 1913.85 66.12 1916.8781.8 61.75 1945.20 65.20 1948.3282.8 60.67 1976.22 64.29 1979.4483.8 59.65 2006.91 63.46 2010.4384.8 58.72 2037.28 62.72 2040.7085.8 57.92 2067.35 62.13 2070.8786.8 57.30 2097.11 61.73 2100.7487.8 56.91 2126.59 61.57 2130.3288.8 56.81 2155.80 61.72 2159.6389.8 57.06 2184.74 62.24 2188.6790.8 57.73 2213.42 63.21 2217.4691.8 58.91 2241.87 64.71 2246.0192.8 60.71 2270.08 66.85 2274.3393.8 63.21 2298.09 69.73 2302.4494.8 66.56 2325.89 70.50 2330.36(mm) (45)Tu]65.3,8109,74.36,2.[*?在表3中C點x值與y 值分別對應(yīng)開始設(shè)計變量和優(yōu)化設(shè)計變量。圖 4 用圖表示了端點 C開始的雙紐線軌跡 L(虛線)和垂直的理想軌跡K(實線)。12圖 4 絞結(jié)點 C 的軌跡4.2 四連桿機構(gòu) AEDA 的最優(yōu)誤差在非線性問題(36)-(38),選擇的獨立變量 的最小值和最大值為421,a?(mm) (46)Tw]01.,.0[?(mm) (47)3,獨立變量的初始值為(mm) (48)Tw]1.0,.[0?軌跡偏離選擇了兩種情況 E=0.01 和 E=0.05。在第一種情況，設(shè)計變量 的421,a理想公差經(jīng)過 9 次反復(fù)的計算，已初結(jié)果。第二種情況也在 7 次的反復(fù)計算后得到了理想值。這些結(jié)果列在表 4 和表 5 中。圖 5 和圖 6 的標(biāo)準偏差已經(jīng)由 Monte Carlo 方法計算出來并表示在圖中(圖中雙點劃線示)同時比較泰勒近似法的曲線(實線)。13圖 5 E=0.01 時的標(biāo)準誤差圖 6 E=0.05 時的標(biāo)準誤差5.結(jié)論通過選用系統(tǒng)的合適的數(shù)學(xué)模型以及采用數(shù)學(xué)函數(shù)，讓液壓支架的設(shè)計得到改良，而且產(chǎn)品的性能更加可靠。然而，由于理想誤差的結(jié)果的出現(xiàn)，將有理由再考慮一個新的問題。這個問題在四連桿的問題上表現(xiàn)的尤為突出，因為一個公差變化稍微都能導(dǎo)致產(chǎn)品成本的升高。14附錄 BStruct Multidisc Optim 20, 76–82 Springer-Verlag 2000Optimal design of hydraulic supportM. Oblak, B. Harl and B. ButinarAbstract ：This paper describes a procedure for optimal determi- nation of two groups of parameters of a hydraulic support employed in the min- ing industry. The procedure is based on mathematical programming methods. In the?rst step, the optimal values of some parameters of the leading four-bar mec- hanism are found in order to ensure the desired motion of the support with min- imal transversal displacements. In the second step, maximal tolerances of the optimal values of the leading four-bar mechanism are calculated, so the response of hydraulic support will be satisfying.Key words：four-bar mechanism, optimal design, mathematical program- mming, approximation method, tolerance1 IntroductionThe designer aims to ?nd the best design for the mechanical system considered. Part of this e?ort is the optimal choice of some selected parameters of a system. Methods of mathematical programming can be used, if a suitable mathematical model of the system is made. Of course, it depends on the type of the system. With this formulation , good computer support is assured to look for optimal parameters of the system.The hydraulic support (Fig. 1) described by Harl(1998) is a part of the mining industry equipment in the mine Velenje-Slovenia, used for protection of working environment in the gallery. It consists of two four-bar mechanisms FEDG and AEDB as shown in Fig. 2. The mechanism AEDB de?nes the path of coupler point C and the mechanism FEDG is used to drive the support by a hydraulic actuator.15Fig. 1 Hydraulic supportIt is required that the motion of the support, more precisely, the motion of point C in Fig. 2, is vertical with minimal transversal displacements. If this is not the case,the hydraulic support will not work properly because it is stranded on removal of the earth machine.A prototype of the hydraulic support was tested in a 333laboratory (Grm 1992). The support exhibited large transversal displacements, which would reduce its employability. Therefore, a redesign was necessary. The project should be improved with minimal cost if possible. It was decided to ?nd the best values for the most problematic parameters , a2, a4 of the leading four-bar mechanism AEDB with methods of mathematical program- mming. Otherwise it would be necessary to change the project, at least mechan- ism AEDB.The solution of above problem will give us the response of hydraulic support for the ideal system. Real response will be di?erent because of toleran- ces of various parameters of the system, which is why the maximal allowed tole- rances of parameters a1, a2, a4 will be calculated, with help of methods of math- ematical programming.16Fig. 2 Two four-bar mechanisms2 The deterministic model of the hydraulic supportAt ?rst it is necessary to develop an appropriate mechanical model of the hydraulic support. It could be based on the following assumptions: – the links are rigid bodies,– the motion of individual links is relatively slow.The hydraulic support is a mechanism with one degree of freedom. Its kinematics can be modelled with syn-chronous motion of two four-bar mech- anisms FEDG and AEDB (Oblak et al. 1998). The leading four-bar mechanism AEDB has a decisive in?uence on the motion of the hydraulic support. Mechan- ism 2 is used to drive the support by a hydraulic actuator. The motion of the sup- port is well described by the trajectory L of the coupler point C. Therefore, the task is to ?nd the optimal values of link lengths of mechanism 1 by requiring that the trajectory of the point C is as near as possible to the desired trajectory K.In hydrostatic machines, cavitations mean that cavities or bubbles form in the hydraulic liquid at the low pressure and collapse at the high pressure region, which causes noise, vibration, and less efficiency. Cavitations are undesirable in the pump since the shock waves formed by collapsed may be strong enough to damage components. The hydraulic fluid will vaporize when its pressure becomes too low or when the temperature is too high. In practice, a number of approaches 17are mostly used to deal with the problems: (1) raise the liquid level in the tank, (2) pressurize the tank, (3) booster the inlet pressure of the pump,(4) lower the pumping fluid temperature, and (5) design deliberately the pump itself.Many research efforts have been made on cavitation phenomena in hydraulic machine designs. The cavitation is classified into two types in piston pumps: trapping phenomenon related one (which can be prevented by the proper design of the valve plate) and the one observed on the layers after the contraction or enlargement of flow passages (caused by rotating group designs) in Ref. (1). The relationship between the cavitation and the measured cylinder pressure is addressed in this study. Edge and Darling (2) reported an experimental study of the cylinder pressure within an axial piston pump. The inclusion of fluid momentum effects and cavitations within the cylinder bore are predicted at both high speed and high load conditions. Another study in Ref. (3) provides an overview of hydraulic fluid impacting on the inlet condition and cavitation potential. It indicates that physical properties (such as vapor pressure, viscosity, density, and bulk modulus) are vital to properly evaluate the effects on lubrication and cavitation. A homogeneous cavitation model based on the thermodynamic properties of the liquid and steam is used to understand the basic physical phenomena of mass flow reduction and wave motion influences in the hydraulic tools and injection systems (4). Dularetal. (5, 6) developed an expert system for monitoring and control of cavitations in hydraulic machines and investigated the possibility of cavitation erosion by using the computational fluid dynamics (CFD) tools. The erosion effects of cavitations have been measured and validated by a simple single hydrofoil configuration in a cavitation tunnel. It is assumed that the severe erosion is often due to the repeated collapse of the traveling vortex generated by a leading edge cavity in Ref. (7). Then, the cavitation erosion intensity may be scaled by a simple set of flow parameters: the upstream velocity, the Strouhal number, the cavity length, and the pressure. A new cavitation erosion device, called vortex cavitation generator, is introduced to comparatively study various erosion situations (8).More previous research has been concentrated on the valve plate designs, piston, and pump pressure dynamics that can be associated with cavitations in axial piston pumps. The control volume approach and instantaneous flows (leakage) are profoundly studied in Ref. [9]. 18Berta et al. [10] used the finite volume concept to develop a mathematical model in which the effects of port plate relief grooves have been modeled and the gaseous cavitation is considered in a simplified manner. An improved model is proposed in Ref. [11] and validated by experimental results. The model may analyze the cylinder pressure and flow ripples influenced by port plate and relief groove design. Manring compared principal advantages of various valve plate slots (i.e., the slots with constant, linearly varying, and quadratic varying areas) in axial piston pumps [12]. Four different numerical models are focused on the characteristics of hydraulic fluid, and cavitations are taken into account in different ways to assist the reduction in flow oscillations [13].The experiences of piston pump developments show that the optimization of the cavitations/aerations shall include the following issues: occurring cavitation and air release, pump acoustics caused by the induced noises, maximal amplitudes of pressure fluctuations, rotational torque progression, etc. However, the aim of this study is to modify the valve plate design to prevent cavitation erosions caused by collapsing steam or air bubbles on the walls of axial pump components. In contrast to literature studies, the research focuses on the development of analytical relationship between the valve plate geometrics and cavitations. The optimization method is applied to analyze the pressure undershoots compared with the saturated vapor pressure within the piston bore.The appropriate design of instantaneous flow areas between the valve plate and barrel kidney can be decided consequently.The synthesis of the four-bar mechanism 1 has been performed with help of kinematics equations of motion given by Rao and Dukkipati (1989). The general situation is depicted in Fig. 3.19Fig. 3 Trajectory L of the point CEquations of trajectory L of the point C will be written in the coordinate frame considered. Coordinates x and y of the point C will be written with the typical parameters of a four-bar mechanism a1, a2, . . . , a6. The coordinates of points B and D are ous parameters of the system, which is why the maximal allowed tolerances of parameters a1, a2, a4 will be calculated, with help of methods of mathematical programming.The parameters a1, a2, . . . , a6 are related to each other byBy substituting (1)–(4) into (5)–(6) the response equations of the support are obtained as20This equation represents the base of the mathematical model for calculating the optimal values of parameters a1,a2, a4.2.1 Mathematical modelThe mathematical model of the system will be formulated in the form proposed by Haug and Arora (1979):subject to constraintsand response equationsThe vector u = [u1 . . . un]T is called the vector of design variables, v = [v1 . . . vm]T is the vector of response variables and f in (9) is the objective function. To perform the optimal design of the leading four-bar mechanism AEDB, the vector of design variables is de?ned as and the vector of response variables asThe dimensions a3,a5,a6 of the corresponding links are kept ?xed. The Objective function is de?ned as some “measure of di?erence” between the trajectory Land the desired trajectory K aswhere x = g0(y) is the equation of the curve K and x =f0(y) is the equation of the curve L.Suitable limitations for our system will be chosen. The system must satisfy the well- 21known Grassho? conditions.Inequalities (15) and (16) express the property of a four-bar mechanism, where the links a2, a4 may only oscillate.The conditionprescribes the lower and upper bounds of the design variables.The problem (9)–(11) is not directly solvable with the usual gradient-based optimization methods. This could be circumvented by introducing an arti?cial design variable un+1 as proposed by Hsieh and Arora (1984). The new formulation exhibiting a more convenient for m may be written assubject toand response equationswhere 11[,]Tnu???and 1[]TmVv??.A nonlinear programming problem of the leading fourbar mechanism AEDB can therefore be de?ned asmin a7 , (22)subject to constraints(a3 + a4) ? (a1 + a2) ≤ 0 , (23)22(a2 + a3) ? (a1 + a4) ≤ 0 , (24)and response equations(x ? a5 cos Θ)2 + (y ? a5 sin Θ)2 ? a22 = 0 , (27)[x ? a6 cos(Θ + γ) ? a1]2+ [y ? a6 sin(Θ + γ)]2 ? = 0 . (28)This formulation enables the minimization of the di?erence between the transversal displacement of the point C and the prescribed trajectory K. The result is the optimal values of the parameters a1, a2, a4.3 The stochastic model of the hydraulic supportThe mathematical model (22)–(28) may be used to calculate such values of the parameters a1, a2, a4, that the “di?erence between trajectories L and K” is mi- nimal. However, the real trajectory L of the point C could deviate from the calc- ulated values because of di?erent in?uences. The suitable mathematical model deviation could be treated dependently on tolerances of parameters a1, a2, a4.The r